Theorem poimirlem28 34063

Description: Lemma for poimir 34068, a variant of Sperner's lemma. (Contributed by Brendan Leahy, 21-Aug-2020.)

Hypotheses

Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem28.1	⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
poimirlem28.2	⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
poimirlem28.3	⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛)
poimirlem28.4	⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
poimirlem28.5	⊢ (𝜑 → 𝐾 ∈ ℕ)

Assertion

Ref	Expression
poimirlem28	⊢ (𝜑 → ∃𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)

Distinct variable groups:   𝑓,𝑖,𝑗,𝑛,𝑝,𝑠   𝜑,𝑗,𝑛   𝑗,𝑁,𝑛   𝜑,𝑖,𝑝,𝑠   𝐵,𝑓,𝑖,𝑗,𝑛,𝑠   𝑓,𝐾,𝑖,𝑗,𝑛,𝑝,𝑠   𝑓,𝑁,𝑖,𝑝,𝑠   𝐶,𝑖,𝑛,𝑝

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)   𝐶(𝑓,𝑗,𝑠)

Proof of Theorem poimirlem28

Dummy variables 𝑘 𝑚 𝑞 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimir.0	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	nnnn0d 11702	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
3	1	nnred 11391	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
4	3	leidd 10941	. . . . 5 ⊢ (𝜑 → 𝑁 ≤ 𝑁)
5	2, 2, 4	3jca 1119	. . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 𝑁))
6		oveq2 6930	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (1...𝑘) = (1...0))
7		fz10 12679	. . . . . . . . . . . . . . . 16 ⊢ (1...0) = ∅
8	6, 7	syl6eq 2830	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (1...𝑘) = ∅)
9	8	oveq2d 6938	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 ∅))
10		fzofi 13092	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐾) ∈ Fin
11		map0e 8179	. . . . . . . . . . . . . . . 16 ⊢ ((0..^𝐾) ∈ Fin → ((0..^𝐾) ↑𝑚 ∅) = 1o)
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((0..^𝐾) ↑𝑚 ∅) = 1o
13		df1o2 7856	. . . . . . . . . . . . . . 15 ⊢ 1o = {∅}
14	12, 13	eqtri 2802	. . . . . . . . . . . . . 14 ⊢ ((0..^𝐾) ↑𝑚 ∅) = {∅}
15	9, 14	syl6eq 2830	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = {∅})
16		eqidd 2779	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → 𝑓 = 𝑓)
17	16, 8, 8	f1oeq123d 6386	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:∅–1-1-onto→∅))
18		eqid 2778	. . . . . . . . . . . . . . . . 17 ⊢ ∅ = ∅
19		f1o00 6425	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:∅–1-1-onto→∅ ↔ (𝑓 = ∅ ∧ ∅ = ∅))
20	18, 19	mpbiran2 700	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:∅–1-1-onto→∅ ↔ 𝑓 = ∅)
21	17, 20	syl6bb 279	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓 = ∅))
22	21	abbidv 2906	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓 = ∅})
23		df-sn 4399	. . . . . . . . . . . . . 14 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅}
24	22, 23	syl6eqr 2832	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {∅})
25	15, 24	xpeq12d 5386	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = ({∅} × {∅}))
26		0ex 5026	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
27	26, 26	xpsn 6672	. . . . . . . . . . . 12 ⊢ ({∅} × {∅}) = {⟨∅, ∅⟩}
28	25, 27	syl6req 2831	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → {⟨∅, ∅⟩} = (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}))
29		elsni 4415	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → 𝑠 = ⟨∅, ∅⟩)
30	26, 26	op1std 7455	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ⟨∅, ∅⟩ → (1st ‘𝑠) = ∅)
31	29, 30	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (1st ‘𝑠) = ∅)
32	31	oveq1d 6937	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ((1st ‘𝑠) ∘𝑓 + ∅) = (∅ ∘𝑓 + ∅))
33		f0 6336	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅:∅⟶∅
34		ffn 6291	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
35	33, 34	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ∅ Fn ∅)
36	26	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ∅ ∈ V)
37		inidm 4043	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∩ ∅) = ∅
38		0fv 6486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅‘𝑛) = ∅
39	38	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ {⟨∅, ∅⟩} ∧ 𝑛 ∈ ∅) → (∅‘𝑛) = ∅)
40	35, 35, 36, 36, 37, 39, 39	offval 7181	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘𝑓 + ∅) = (𝑛 ∈ ∅ ↦ (∅ + ∅)))
41		mpt0 6267	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ∅ ↦ (∅ + ∅)) = ∅
42	40, 41	syl6eq 2830	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘𝑓 + ∅) = ∅)
43	32, 42	eqtrd 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ((1st ‘𝑠) ∘𝑓 + ∅) = ∅)
44	43	uneq1d 3989	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) = (∅ ∪ ((1...𝑁) × {0})))
45		uncom 3980	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∪ ((1...𝑁) × {0})) = (((1...𝑁) × {0}) ∪ ∅)
46		un0 4193	. . . . . . . . . . . . . . . 16 ⊢ (((1...𝑁) × {0}) ∪ ∅) = ((1...𝑁) × {0})
47	45, 46	eqtri 2802	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ ((1...𝑁) × {0})) = ((1...𝑁) × {0})
48	44, 47	syl6req 2831	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ((1...𝑁) × {0}) = (((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})))
49	48	csbeq1d 3758	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
50	49	eqeq2d 2788	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
51		oveq2 6930	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (0...𝑘) = (0...0))
52		0z 11739	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
53		fzsn 12700	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ℤ → (0...0) = {0})
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (0...0) = {0}
55	51, 54	syl6eq 2830	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (0...𝑘) = {0})
56		oveq2 6930	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...0))
57	56	imaeq2d 5720	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...0)))
58	57	xpeq1d 5384	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))
59	58	uneq2d 3990	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 0 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0})))
60	59	oveq2d 6938	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))))
61		oveq1 6929	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
62		0p1e1 11504	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 + 1) = 1
63	61, 62	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → (𝑘 + 1) = 1)
64	63	oveq1d 6937	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 0 → ((𝑘 + 1)...𝑁) = (1...𝑁))
65	64	xpeq1d 5384	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (((𝑘 + 1)...𝑁) × {0}) = ((1...𝑁) × {0}))
66	60, 65	uneq12d 3991	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})))
67	66	csbeq1d 3758	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
68	67	eqeq2d 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
69	55, 68	rexeqbidv 3327	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ {0}𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
70		c0ex 10370	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
71		oveq2 6930	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 = 0 → (1...𝑗) = (1...0))
72	71, 7	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 0 → (1...𝑗) = ∅)
73	72	imaeq2d 5720	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd ‘𝑠) “ ∅))
74		ima0 5735	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2nd ‘𝑠) “ ∅) = ∅
75	73, 74	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ (1...𝑗)) = ∅)
76	75	xpeq1d 5384	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = (∅ × {1}))
77		0xp 5447	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ × {1}) = ∅
78	76, 77	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = ∅)
79		oveq1 6929	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
80	79, 62	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 = 0 → (𝑗 + 1) = 1)
81	80	oveq1d 6937	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 = 0 → ((𝑗 + 1)...0) = (1...0))
82	81, 7	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 0 → ((𝑗 + 1)...0) = ∅)
83	82	imaeq2d 5720	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ ((𝑗 + 1)...0)) = ((2nd ‘𝑠) “ ∅))
84	83, 74	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ ((𝑗 + 1)...0)) = ∅)
85	84	xpeq1d 5384	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}) = (∅ × {0}))
86		0xp 5447	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ × {0}) = ∅
87	85, 86	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}) = ∅)
88	78, 87	uneq12d 3991	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 0 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0})) = (∅ ∪ ∅))
89		un0 4193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∪ ∅) = ∅
90	88, 89	syl6eq 2830	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 0 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0})) = ∅)
91	90	oveq2d 6938	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 0 → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) = ((1st ‘𝑠) ∘𝑓 + ∅))
92	91	uneq1d 3989	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 0 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})))
93	92	csbeq1d 3758	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 0 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
94	93	eqeq2d 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 0 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
95	70, 94	rexsn 4451	. . . . . . . . . . . . . . 15 ⊢ (∃𝑗 ∈ {0}𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
96	69, 95	syl6bb 279	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
97	55, 96	raleqbidv 3326	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ {0}𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
98		eqeq1 2782	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
99	70, 98	ralsn 4450	. . . . . . . . . . . . 13 ⊢ (∀𝑖 ∈ {0}𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
100	97, 99	syl6rbb 280	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (0 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
101	50, 100	sylan9bbr 506	. . . . . . . . . . 11 ⊢ ((𝑘 = 0 ∧ 𝑠 ∈ {⟨∅, ∅⟩}) → (0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
102	28, 101	rabeqbidva 3393	. . . . . . . . . 10 ⊢ (𝑘 = 0 → {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})
103	102	eqcomd 2784	. . . . . . . . 9 ⊢ (𝑘 = 0 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})
104	103	fveq2d 6450	. . . . . . . 8 ⊢ (𝑘 = 0 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))
105	104	breq2d 4898	. . . . . . 7 ⊢ (𝑘 = 0 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})))
106	105	notbid 310	. . . . . 6 ⊢ (𝑘 = 0 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})))
107	106	imbi2d 332	. . . . 5 ⊢ (𝑘 = 0 → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))))
108		oveq2 6930	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (1...𝑘) = (1...𝑚))
109	108	oveq2d 6938	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...𝑚)))
110		eqidd 2779	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → 𝑓 = 𝑓)
111	110, 108, 108	f1oeq123d 6386	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)))
112	111	abbidv 2906	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)})
113	109, 112	xpeq12d 5386	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}))
114		oveq2 6930	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (0...𝑘) = (0...𝑚))
115		oveq2 6930	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑚 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑚))
116	115	imaeq2d 5720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑚 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)))
117	116	xpeq1d 5384	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))
118	117	uneq2d 3990	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0})))
119	118	oveq2d 6938	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))))
120		oveq1 6929	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
121	120	oveq1d 6937	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → ((𝑘 + 1)...𝑁) = ((𝑚 + 1)...𝑁))
122	121	xpeq1d 5384	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑚 + 1)...𝑁) × {0}))
123	119, 122	uneq12d 3991	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})))
124	123	csbeq1d 3758	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
125	124	eqeq2d 2788	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
126	114, 125	rexeqbidv 3327	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
127	114, 126	raleqbidv 3326	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
128	113, 127	rabeqbidv 3392	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})
129	128	fveq2d 6450	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))
130	129	breq2d 4898	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
131	130	notbid 310	. . . . . 6 ⊢ (𝑘 = 𝑚 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
132	131	imbi2d 332	. . . . 5 ⊢ (𝑘 = 𝑚 → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
133		oveq2 6930	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (1...𝑘) = (1...(𝑚 + 1)))
134	133	oveq2d 6938	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))))
135		eqidd 2779	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → 𝑓 = 𝑓)
136	135, 133, 133	f1oeq123d 6386	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))))
137	136	abbidv 2906	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))})
138	134, 137	xpeq12d 5386	. . . . . . . . . 10 ⊢ (𝑘 = (𝑚 + 1) → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}))
139		oveq2 6930	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → (0...𝑘) = (0...(𝑚 + 1)))
140		oveq2 6930	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑚 + 1) → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...(𝑚 + 1)))
141	140	imaeq2d 5720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑚 + 1) → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))))
142	141	xpeq1d 5384	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑚 + 1) → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))
143	142	uneq2d 3990	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑚 + 1) → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0})))
144	143	oveq2d 6938	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑚 + 1) → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))))
145		oveq1 6929	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑚 + 1) → (𝑘 + 1) = ((𝑚 + 1) + 1))
146	145	oveq1d 6937	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑚 + 1) → ((𝑘 + 1)...𝑁) = (((𝑚 + 1) + 1)...𝑁))
147	146	xpeq1d 5384	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑚 + 1) → (((𝑘 + 1)...𝑁) × {0}) = ((((𝑚 + 1) + 1)...𝑁) × {0}))
148	144, 147	uneq12d 3991	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑚 + 1) → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})))
149	148	csbeq1d 3758	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
150	149	eqeq2d 2788	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
151	139, 150	rexeqbidv 3327	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
152	139, 151	raleqbidv 3326	. . . . . . . . . 10 ⊢ (𝑘 = (𝑚 + 1) → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
153	138, 152	rabeqbidv 3392	. . . . . . . . 9 ⊢ (𝑘 = (𝑚 + 1) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})
154	153	fveq2d 6450	. . . . . . . 8 ⊢ (𝑘 = (𝑚 + 1) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))
155	154	breq2d 4898	. . . . . . 7 ⊢ (𝑘 = (𝑚 + 1) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
156	155	notbid 310	. . . . . 6 ⊢ (𝑘 = (𝑚 + 1) → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
157	156	imbi2d 332	. . . . 5 ⊢ (𝑘 = (𝑚 + 1) → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
158		oveq2 6930	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑁 → (1...𝑘) = (1...𝑁))
159	158	oveq2d 6938	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...𝑁)))
160		eqidd 2779	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑁 → 𝑓 = 𝑓)
161	160, 158, 158	f1oeq123d 6386	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑁 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)))
162	161	abbidv 2906	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
163	159, 162	xpeq12d 5386	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
164		oveq2 6930	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → (0...𝑘) = (0...𝑁))
165		oveq2 6930	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑁 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑁))
166	165	imaeq2d 5720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑁 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)))
167	166	xpeq1d 5384	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑁 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))
168	167	uneq2d 3990	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑁 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
169	168	oveq2d 6938	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑁 → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))))
170		oveq1 6929	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑁 → (𝑘 + 1) = (𝑁 + 1))
171	170	oveq1d 6937	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑁 → ((𝑘 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
172	171	xpeq1d 5384	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑁 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑁 + 1)...𝑁) × {0}))
173	169, 172	uneq12d 3991	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑁 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})))
174	173	csbeq1d 3758	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑁 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
175	174	eqeq2d 2788	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑁 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
176	164, 175	rexeqbidv 3327	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
177	164, 176	raleqbidv 3326	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
178	163, 177	rabeqbidv 3392	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})
179	178	fveq2d 6450	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))
180	179	breq2d 4898	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
181	180	notbid 310	. . . . . 6 ⊢ (𝑘 = 𝑁 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
182	181	imbi2d 332	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
183		n2dvds1 15496	. . . . . . 7 ⊢ ¬ 2 ∥ 1
184		opex 5164	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ∈ V
185		hashsng 13474	. . . . . . . . . 10 ⊢ (⟨∅, ∅⟩ ∈ V → (♯‘{⟨∅, ∅⟩}) = 1)
186	184, 185	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘{⟨∅, ∅⟩}) = 1
187		nnuz 12029	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ≥‘1)
188	1, 187	syl6eleq 2869	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
189		eluzfz1 12665	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁))
190	188, 189	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ (1...𝑁))
191		poimirlem28.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ ℕ)
192	191	nnnn0d 11702	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
193		0elfz 12755	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
194		fconst6g 6344	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0...𝐾) → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾))
195	192, 193, 194	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾))
196	70	fvconst2 6741	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (1...𝑁) → (((1...𝑁) × {0})‘1) = 0)
197	190, 196	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((1...𝑁) × {0})‘1) = 0)
198	190, 195, 197	3jca 1119	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
199		nfv 1957	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝(𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
200		nfcsb1v 3767	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑝⦋((1...𝑁) × {0}) / 𝑝⦌𝐵
201	200	nfeq1 2947	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0
202	199, 201	nfim 1943	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0)
203		ovex 6954	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑁) ∈ V
204		snex 5140	. . . . . . . . . . . . . . . 16 ⊢ {0} ∈ V
205	203, 204	xpex 7240	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑁) × {0}) ∈ V
206		feq1 6272	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ((1...𝑁) × {0}) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾)))
207		fveq1 6445	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = ((1...𝑁) × {0}) → (𝑝‘1) = (((1...𝑁) × {0})‘1))
208	207	eqeq1d 2780	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ((1...𝑁) × {0}) → ((𝑝‘1) = 0 ↔ (((1...𝑁) × {0})‘1) = 0))
209	206, 208	3anbi23d 1512	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((1...𝑁) × {0}) → ((1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0) ↔ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)))
210	209	anbi2d 622	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ((1...𝑁) × {0}) → ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) ↔ (𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))))
211		csbeq1a 3760	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((1...𝑁) × {0}) → 𝐵 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
212	211	eqeq1d 2780	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ((1...𝑁) × {0}) → (𝐵 = 0 ↔ ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0))
213	210, 212	imbi12d 336	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ((1...𝑁) × {0}) → (((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0) ↔ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0)))
214		1ex 10372	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
215		eleq1 2847	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → (𝑛 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁)))
216		fveqeq2 6455	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → ((𝑝‘𝑛) = 0 ↔ (𝑝‘1) = 0))
217	215, 216	3anbi13d 1511	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)))
218	217	anbi2d 622	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0))))
219		breq2 4890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (𝐵 < 𝑛 ↔ 𝐵 < 1))
220	218, 219	imbi12d 336	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 < 1)))
221		poimirlem28.3	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛)
222	214, 220, 221	vtocl 3460	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 < 1)
223		poimirlem28.2	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
224		elfznn0 12751	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℕ0)
225		nn0lt10b 11791	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℕ0 → (𝐵 < 1 ↔ 𝐵 = 0))
226	223, 224, 225	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝐵 < 1 ↔ 𝐵 = 0))
227	226	3ad2antr2 1197	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → (𝐵 < 1 ↔ 𝐵 = 0))
228	222, 227	mpbid 224	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0)
229	202, 205, 213, 228	vtoclf 3459	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0)
230	198, 229	mpdan 677	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0)
231	230	eqcomd 2784	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
232	231	ralrimivw 3149	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
233		rabid2 3305	. . . . . . . . . . 11 ⊢ ({⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵} ↔ ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
234	232, 233	sylibr 226	. . . . . . . . . 10 ⊢ (𝜑 → {⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})
235	234	fveq2d 6450	. . . . . . . . 9 ⊢ (𝜑 → (♯‘{⟨∅, ∅⟩}) = (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))
236	186, 235	syl5eqr 2828	. . . . . . . 8 ⊢ (𝜑 → 1 = (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))
237	236	breq2d 4898	. . . . . . 7 ⊢ (𝜑 → (2 ∥ 1 ↔ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})))
238	183, 237	mtbii 318	. . . . . 6 ⊢ (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))
239	238	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})))
240		2z 11761	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
241		fzfi 13090	. . . . . . . . . . . . . . . . 17 ⊢ (1...(𝑚 + 1)) ∈ Fin
242		mapfi 8550	. . . . . . . . . . . . . . . . 17 ⊢ (((0..^𝐾) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin)
243	10, 241, 242	mp2an 682	. . . . . . . . . . . . . . . 16 ⊢ ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin
244		ovex 6954	. . . . . . . . . . . . . . . . . . 19 ⊢ (1...(𝑚 + 1)) ∈ V
245	244, 244	mapval 8152	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) = {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
246		mapfi 8550	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1...(𝑚 + 1)) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin)
247	241, 241, 246	mp2an 682	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin
248	245, 247	eqeltrri 2856	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin
249		f1of 6391	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1)) → 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1)))
250	249	ss2abi 3895	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
251		ssfi 8468	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}) → {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin)
252	248, 250, 251	mp2an 682	. . . . . . . . . . . . . . . 16 ⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin
253		xpfi 8519	. . . . . . . . . . . . . . . 16 ⊢ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin) → (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin)
254	243, 252, 253	mp2an 682	. . . . . . . . . . . . . . 15 ⊢ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin
255		rabfi 8473	. . . . . . . . . . . . . . 15 ⊢ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin)
256		hashcl 13462	. . . . . . . . . . . . . . 15 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℕ0)
257	254, 255, 256	mp2b 10	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℕ0
258	257	nn0zi 11754	. . . . . . . . . . . . 13 ⊢ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ
259		rabfi 8473	. . . . . . . . . . . . . . 15 ⊢ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin)
260		hashcl 13462	. . . . . . . . . . . . . . 15 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℕ0)
261	254, 259, 260	mp2b 10	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℕ0
262	261	nn0zi 11754	. . . . . . . . . . . . 13 ⊢ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ
263	240, 258, 262	3pm3.2i 1395	. . . . . . . . . . . 12 ⊢ (2 ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ)
264		nn0p1nn 11683	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
265	264	ad2antrl 718	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑚 + 1) ∈ ℕ)
266		uneq1 3983	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})))
267	266	csbeq1d 3758	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
268	70	fconst 6341	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}
269	268	jctr 520	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}))
270	264	nnred 11391	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ)
271	270	ltp1d 11308	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) < ((𝑚 + 1) + 1))
272		fzdisj 12685	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 + 1) < ((𝑚 + 1) + 1) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
273	271, 272	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ0 → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
274		fun 6316	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}) ∧ ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
275	269, 273, 274	syl2anr 590	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
276	275	adantlr 705	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
277	276	adantl 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
278	264	peano2nnd 11393	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℕ)
279	278, 187	syl6eleq 2869	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ (ℤ≥‘1))
280	279	ad2antrl 718	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝑚 + 1) + 1) ∈ (ℤ≥‘1))
281		nn0z 11752	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
282	1	nnzd 11833	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑁 ∈ ℤ)
283		zltp1le 11779	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
284	281, 282, 283	syl2anr 590	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
285	284	biimpa 470	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → (𝑚 + 1) ≤ 𝑁)
286	285	anasss 460	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑚 + 1) ≤ 𝑁)
287	281	peano2zd 11837	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℤ)
288	287	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → (𝑚 + 1) ∈ ℤ)
289		eluz 12006	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
290	288, 282, 289	syl2anr 590	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑁 ∈ (ℤ≥‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
291	286, 290	mpbird 249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑁 ∈ (ℤ≥‘(𝑚 + 1)))
292		fzsplit2 12683	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑚 + 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑚 + 1))) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
293	280, 291, 292	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
294	293	eqcomd 2784	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)) = (1...𝑁))
295	192, 193	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 0 ∈ (0...𝐾))
296	295	snssd 4571	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → {0} ⊆ (0...𝐾))
297		ssequn2 4009	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({0} ⊆ (0...𝐾) ↔ ((0...𝐾) ∪ {0}) = (0...𝐾))
298	296, 297	sylib 210	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((0...𝐾) ∪ {0}) = (0...𝐾))
299	298	adantr 474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((0...𝐾) ∪ {0}) = (0...𝐾))
300	294, 299	feq23d 6286	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))
301	300	adantrr 707	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))
302	277, 301	mpbid 224	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
303		nfv 1957	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑝(𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
304		nfcsb1v 3767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵
305	304	nfel1 2948	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)
306	303, 305	nfim 1943	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑝((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
307		vex 3401	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑞 ∈ V
308		ovex 6954	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑚 + 1) + 1)...𝑁) ∈ V
309	308, 204	xpex 7240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}) ∈ V
310	307, 309	unex 7233	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) ∈ V
311		feq1 6272	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))
312	311	anbi2d 622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) ↔ (𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))))
313		csbeq1a 3760	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → 𝐵 = ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
314	313	eleq1d 2844	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ∈ (0...𝑁) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)))
315	312, 314	imbi12d 336	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) ↔ ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))))
316	306, 310, 315, 223	vtoclf 3459	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
317	302, 316	syldan 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
318	317	anassrs 461	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
319		elfznn0 12751	. . . . . . . . . . . . . . . . 17 ⊢ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℕ0)
320	318, 319	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℕ0)
321	264	nnnn0d 11702	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
322	321	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → (𝑚 + 1) ∈ ℕ0)
323	322	ad2antlr 717	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑚 + 1) ∈ ℕ0)
324		leloe 10463	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑚 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
325	270, 3, 324	syl2anr 590	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
326	284, 325	bitrd 271	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
327	326	biimpd 221	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 → ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
328	327	imdistani 564	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → ((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
329	328	anasss 460	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
330		simplll 765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝜑)
331	278	nnge1d 11423	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℕ0 → 1 ≤ ((𝑚 + 1) + 1))
332	331	ad2antlr 717	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 1 ≤ ((𝑚 + 1) + 1))
333		zltp1le 11779	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
334	287, 282, 333	syl2anr 590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
335	334	biimpa 470	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ≤ 𝑁)
336	287	peano2zd 11837	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℤ)
337		1z 11759	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ∈ ℤ
338		elfz 12649	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
339	337, 338	mp3an2 1522	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
340	336, 282, 339	syl2anr 590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
341	340	adantr 474	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
342	332, 335, 341	mpbir2and 703	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
343	342	adantlr 705	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
344		nn0re 11652	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℝ)
345	344	ad2antlr 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 ∈ ℝ)
346	270	ad2antlr 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℝ)
347	3	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ ℝ)
348	344	ltp1d 11308	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → 𝑚 < (𝑚 + 1))
349	348	ad2antlr 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < (𝑚 + 1))
350		simpr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) < 𝑁)
351	345, 346, 347, 349, 350	lttrd 10537	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
352	351	adantlr 705	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
353		anass 462	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ↔ (𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)))
354	302	anassrs 461	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
355	353, 354	sylanb 576	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
356	355	an32s 642	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
357	352, 356	syldan 585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
358		ffn 6291	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → 𝑞 Fn (1...(𝑚 + 1)))
359	358	ad2antlr 717	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑞 Fn (1...(𝑚 + 1)))
360	273	ad3antlr 721	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
361		eluz 12006	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
362	336, 282, 361	syl2anr 590	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
363	362	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
364	335, 363	mpbird 249	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)))
365		eluzfz1 12665	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
366	364, 365	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
367	366	adantlr 705	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
368		fnconstg 6343	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 ∈ V → ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁))
369	70, 368	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁)
370		fvun2 6530	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)))
371	369, 370	mp3an2 1522	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)))
372	359, 360, 367, 371	syl12anc 827	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)))
373	70	fvconst2 6741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0)
374	367, 373	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0)
375	372, 374	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)
376		nfv 1957	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑝(𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))
377		nfcv 2934	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑝 <
378		nfcv 2934	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑝((𝑚 + 1) + 1)
379	304, 377, 378	nfbr 4933	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)
380	376, 379	nfim 1943	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑝((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))
381		fveq1 6445	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘((𝑚 + 1) + 1)) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)))
382	381	eqeq1d 2780	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘((𝑚 + 1) + 1)) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))
383	311, 382	3anbi23d 1512	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0) ↔ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)))
384	383	anbi2d 622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) ↔ (𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))))
385	313	breq1d 4896	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < ((𝑚 + 1) + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)))
386	384, 385	imbi12d 336	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1)) ↔ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))))
387		ovex 6954	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 + 1) + 1) ∈ V
388		eleq1 2847	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝑛 ∈ (1...𝑁) ↔ ((𝑚 + 1) + 1) ∈ (1...𝑁)))
389		fveqeq2 6455	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝑝‘𝑛) = 0 ↔ (𝑝‘((𝑚 + 1) + 1)) = 0))
390	388, 389	3anbi13d 1511	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)))
391	390	anbi2d 622	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0))))
392		breq2 4890	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝐵 < 𝑛 ↔ 𝐵 < ((𝑚 + 1) + 1)))
393	391, 392	imbi12d 336	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = ((𝑚 + 1) + 1) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1))))
394	387, 393, 221	vtocl 3460	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1))
395	380, 310, 386, 394	vtoclf 3459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))
396	330, 343, 357, 375, 395	syl13anc 1440	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))
397	353, 318	sylanb 576	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
398	397	an32s 642	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
399		elfzelz 12659	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ)
400	398, 399	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ)
401	352, 400	syldan 585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ)
402	287	ad3antlr 721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℤ)
403		zleltp1 11780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ ∧ (𝑚 + 1) ∈ ℤ) → (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)))
404	401, 402, 403	syl2anc 579	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)))
405	396, 404	mpbird 249	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
406	348	ad2antlr 717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → 𝑚 < (𝑚 + 1))
407		breq2 4890	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 + 1) = 𝑁 → (𝑚 < (𝑚 + 1) ↔ 𝑚 < 𝑁))
408	407	biimpac 472	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 < (𝑚 + 1) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
409	406, 408	sylan 575	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
410		elfzle2 12662	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁)
411	398, 410	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁)
412	409, 411	syldan 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁)
413		simpr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑚 + 1) = 𝑁)
414	412, 413	breqtrrd 4914	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
415	405, 414	jaodan 943	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
416	415	an32s 642	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
417	329, 416	sylan 575	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
418		elfz2nn0 12749	. . . . . . . . . . . . . . . 16 ⊢ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...(𝑚 + 1)) ↔ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℕ0 ∧ (𝑚 + 1) ∈ ℕ0 ∧ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)))
419	320, 323, 417, 418	syl3anbrc 1400	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...(𝑚 + 1)))
420		fzss2 12698	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (ℤ≥‘(𝑚 + 1)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
421	291, 420	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
422	421	sselda 3821	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑛 ∈ (1...(𝑚 + 1))) → 𝑛 ∈ (1...𝑁))
423	422	3ad2antr1 1196	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → 𝑛 ∈ (1...𝑁))
424	354	3ad2antr2 1197	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
425	358	ad2antll 719	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑞 Fn (1...(𝑚 + 1)))
426	273	ad2antlr 717	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
427		simprl 761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑛 ∈ (1...(𝑚 + 1)))
428		fvun1 6529	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
429	369, 428	mp3an2 1522	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
430	425, 426, 427, 429	syl12anc 827	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
431	430	adantlrr 711	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
432	431	3adantr3 1173	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
433		simpr3 1209	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑞‘𝑛) = 0)
434	432, 433	eqtrd 2814	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)
435	423, 424, 434	3jca 1119	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
436		nfv 1957	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
437		nfcv 2934	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑝𝑛
438	304, 377, 437	nfbr 4933	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛
439	436, 438	nfim 1943	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑝((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)
440		fveq1 6445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘𝑛) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛))
441	440	eqeq1d 2780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘𝑛) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
442	311, 441	3anbi23d 1512	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)))
443	442	anbi2d 622	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))))
444	313	breq1d 4896	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < 𝑛 ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛))
445	443, 444	imbi12d 336	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)))
446	439, 310, 445, 221	vtoclf 3459	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)
447	446	adantlr 705	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)
448	435, 447	syldan 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)
449		simp1 1127	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾) → 𝑛 ∈ (1...(𝑚 + 1)))
450	422	anasss 460	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → 𝑛 ∈ (1...𝑁))
451	449, 450	sylanr2 673	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → 𝑛 ∈ (1...𝑁))
452		simp2 1128	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾) → 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))
453	452, 302	sylanr2 673	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
454	430	3adantr3 1173	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
455		simpr3 1209	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → (𝑞‘𝑛) = 𝐾)
456	454, 455	eqtrd 2814	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
457	456	anasss 460	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
458	457	adantrlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
459	451, 453, 458	3jca 1119	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
460		nfv 1957	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
461		nfcv 2934	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑝(𝑛 − 1)
462	304, 461	nfne 3072	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)
463	460, 462	nfim 1943	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑝((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))
464	440	eqeq1d 2780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘𝑛) = 𝐾 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
465	311, 464	3anbi23d 1512	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)))
466	465	anbi2d 622	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))))
467	313	neeq1d 3028	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ≠ (𝑛 − 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)))
468	466, 467	imbi12d 336	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))))
469		poimirlem28.4	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
470	463, 310, 468, 469	vtoclf 3459	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))
471	459, 470	syldan 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))
472	471	anassrs 461	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))
473	265, 267, 419, 448, 472	poimirlem27 34062	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
474	265, 267, 419	poimirlem26 34061	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
475		fzfi 13090	. . . . . . . . . . . . . . . . . . 19 ⊢ (0...(𝑚 + 1)) ∈ Fin
476		xpfi 8519	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin ∧ (0...(𝑚 + 1)) ∈ Fin) → ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin)
477	254, 475, 476	mp2an 682	. . . . . . . . . . . . . . . . . 18 ⊢ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin
478		rabfi 8473	. . . . . . . . . . . . . . . . . 18 ⊢ (((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin)
479		hashcl 13462	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℕ0)
480	477, 478, 479	mp2b 10	. . . . . . . . . . . . . . . . 17 ⊢ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℕ0
481	480	nn0zi 11754	. . . . . . . . . . . . . . . 16 ⊢ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ
482		zsubcl 11771	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ) → ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ)
483	481, 262, 482	mp2an 682	. . . . . . . . . . . . . . 15 ⊢ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ
484		zsubcl 11771	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ) → ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ∈ ℤ)
485	481, 258, 484	mp2an 682	. . . . . . . . . . . . . . 15 ⊢ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ∈ ℤ
486		dvds2sub 15423	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℤ ∧ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ ∧ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ∈ ℤ) → ((2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∧ 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) → 2 ∥ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))))
487	240, 483, 485, 486	mp3an 1534	. . . . . . . . . . . . . 14 ⊢ ((2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∧ 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) → 2 ∥ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
488	473, 474, 487	syl2anc 579	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
489	480	nn0cni 11655	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ
490	261	nn0cni 11655	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℂ
491	257	nn0cni 11655	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ
492		nnncan1 10659	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℂ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ) → (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) = ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
493	489, 490, 491, 492	mp3an 1534	. . . . . . . . . . . . 13 ⊢ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) = ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
494	488, 493	syl6breq 4927	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
495		dvdssub2 15430	. . . . . . . . . . . 12 ⊢ (((2 ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ) ∧ 2 ∥ ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
496	263, 494, 495	sylancr 581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
497		nn0cn 11653	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
498		pncan1 10799	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℂ → ((𝑚 + 1) − 1) = 𝑚)
499	497, 498	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) − 1) = 𝑚)
500	499	oveq2d 6938	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ0 → (0...((𝑚 + 1) − 1)) = (0...𝑚))
501	500	rexeqdv 3341	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ0 → (∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
502	500, 501	raleqbidv 3326	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ0 → (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
503	502	3anbi1d 1513	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → ((∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1)) ↔ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))))
504	503	rabbidv 3386	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})
505	504	fveq2d 6450	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
506	505	ad2antrl 718	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
507	1	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑁 ∈ ℕ)
508	191	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝐾 ∈ ℕ)
509		simprl 761	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑚 ∈ ℕ0)
510		simprr 763	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑚 < 𝑁)
511	507, 508, 509, 510	poimirlem4 34039	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})
512		fzfi 13090	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑚) ∈ Fin
513		mapfi 8550	. . . . . . . . . . . . . . . . . 18 ⊢ (((0..^𝐾) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((0..^𝐾) ↑𝑚 (1...𝑚)) ∈ Fin)
514	10, 512, 513	mp2an 682	. . . . . . . . . . . . . . . . 17 ⊢ ((0..^𝐾) ↑𝑚 (1...𝑚)) ∈ Fin
515		ovex 6954	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑚) ∈ V
516	515, 515	mapval 8152	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1...𝑚) ↑𝑚 (1...𝑚)) = {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)}
517		mapfi 8550	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1...𝑚) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((1...𝑚) ↑𝑚 (1...𝑚)) ∈ Fin)
518	512, 512, 517	mp2an 682	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1...𝑚) ↑𝑚 (1...𝑚)) ∈ Fin
519	516, 518	eqeltrri 2856	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin
520		f1of 6391	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:(1...𝑚)–1-1-onto→(1...𝑚) → 𝑓:(1...𝑚)⟶(1...𝑚))
521	520	ss2abi 3895	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)}
522		ssfi 8468	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)}) → {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin)
523	519, 521, 522	mp2an 682	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin
524		xpfi 8519	. . . . . . . . . . . . . . . . 17 ⊢ ((((0..^𝐾) ↑𝑚 (1...𝑚)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin) → (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin)
525	514, 523, 524	mp2an 682	. . . . . . . . . . . . . . . 16 ⊢ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin
526		rabfi 8473	. . . . . . . . . . . . . . . 16 ⊢ ((((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin)
527	525, 526	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin
528		rabfi 8473	. . . . . . . . . . . . . . . 16 ⊢ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin)
529	254, 528	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin
530		hashen 13452	. . . . . . . . . . . . . . 15 ⊢ (({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin ∧ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin) → ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
531	527, 529, 530	mp2an 682	. . . . . . . . . . . . . 14 ⊢ ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})
532	511, 531	sylibr 226	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
533	506, 532	eqtr4d 2817	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))
534	533	breq2d 4898	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
535	496, 534	bitrd 271	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
536	535	biimpd 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
537	536	con3d 150	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
538	537	expcom 404	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → (𝜑 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
539	538	a2d 29	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) → (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
540	539	3adant1 1121	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) → (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
541	107, 132, 157, 182, 239, 540	fnn0ind 11828	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 𝑁) → (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
542	5, 541	mpcom 38	. . 3 ⊢ (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))
543		dvds0 15404	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∥ 0)
544	240, 543	ax-mp 5	. . . . . . 7 ⊢ 2 ∥ 0
545		hash0 13473	. . . . . . 7 ⊢ (♯‘∅) = 0
546	544, 545	breqtrri 4913	. . . . . 6 ⊢ 2 ∥ (♯‘∅)
547		fveq2 6446	. . . . . 6 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) = (♯‘∅))
548	546, 547	syl5breqr 4924	. . . . 5 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
549	3	ltp1d 11308	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 < (𝑁 + 1))
550	282	peano2zd 11837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
551		fzn 12674	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
552	550, 282, 551	syl2anc 579	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
553	549, 552	mpbid 224	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
554	553	xpeq1d 5384	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = (∅ × {0}))
555	554, 86	syl6eq 2830	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = ∅)
556	555	uneq2d 3990	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ ∅))
557		un0 4193	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ ∅) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
558	556, 557	syl6eq 2830	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))))
559	558	csbeq1d 3758	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
560		ovex 6954	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
561		poimirlem28.1	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
562	560, 561	csbie 3777	. . . . . . . . . . . 12 ⊢ ⦋((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = 𝐶
563	559, 562	syl6eq 2830	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = 𝐶)
564	563	eqeq2d 2788	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = 𝐶))
565	564	rexbidv 3237	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
566	565	ralbidv 3168	. . . . . . . 8 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
567	566	rabbidv 3386	. . . . . . 7 ⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
568	567	fveq2d 6450	. . . . . 6 ⊢ (𝜑 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
569	568	breq2d 4898	. . . . 5 ⊢ (𝜑 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))
570	548, 569	syl5ibr 238	. . . 4 ⊢ (𝜑 → ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
571	570	necon3bd 2983	. . 3 ⊢ (𝜑 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅))
572	542, 571	mpd 15	. 2 ⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅)
573		rabn0 4188	. 2 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅ ↔ ∃𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
574	572, 573	sylib 210	1 ⊢ (𝜑 → ∃𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  {cab 2763   ≠ wne 2969  ∀wral 3090  ∃wrex 3091  {crab 3094  Vcvv 3398  ⦋csb 3751   ∖ cdif 3789   ∪ cun 3790   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  {csn 4398  ⟨cop 4404   class class class wbr 4886   ↦ cmpt 4965   × cxp 5353   “ cima 5358   Fn wfn 6130  ⟶wf 6131  –1-1-onto→wf1o 6134  ‘cfv 6135  (class class class)co 6922   ∘_𝑓 cof 7172  1^st c1st 7443  2^nd c2nd 7444  1_oc1o 7836   ↑_𝑚 cmap 8140   ≈ cen 8238  Fincfn 8241  ℂcc 10270  ℝcr 10271  0cc0 10272  1c1 10273   + caddc 10275   < clt 10411   ≤ cle 10412   − cmin 10606  ℕcn 11374  2c2 11430  ℕ₀cn0 11642  ℤcz 11728  ℤ_≥cuz 11992  ...cfz 12643  ..^cfzo 12784  ♯chash 13435   ∥ cdvds 15387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-disj 4855  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-oi 8704  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-rp 12138  df-fz 12644  df-fzo 12785  df-seq 13120  df-exp 13179  df-fac 13379  df-bc 13408  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-sum 14825  df-dvds 15388

This theorem is referenced by:  poimirlem32  34067