Theorem poimirlem28 38757

Description: Lemma for poimir 38762, 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)⟶1-1onto(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 12691	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
3	1	nnred 12374	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
4	3	leidd 11906	. . . . 5 ⊢ (𝜑 → 𝑁 ≤ 𝑁)
5	2, 2, 4	3jca 1146	. . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 𝑁))
6		oveq2 7448	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (1...𝑘) = (1...0))
7		fz10 13702	. . . . . . . . . . . . . . . 16 ⊢ (1...0) = ∅
8	6, 7	eqtrdi 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (1...𝑘) = ∅)
9	8	oveq2d 7456	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m ∅))
10		fzofi 14141	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐾) ∈ Fin
11		map0e 8935	. . . . . . . . . . . . . . . 16 ⊢ ((0..^𝐾) ∈ Fin → ((0..^𝐾) ↑m ∅) = 1o)
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((0..^𝐾) ↑m ∅) = 1o
13		df1o2 8508	. . . . . . . . . . . . . . 15 ⊢ 1o = {∅}
14	12, 13	eqtri 2786	. . . . . . . . . . . . . 14 ⊢ ((0..^𝐾) ↑m ∅) = {∅}
15	9, 14	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((0..^𝐾) ↑m (1...𝑘)) = {∅})
16		eqidd 2764	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → 𝑓 = 𝑓)
17	16, 8, 8	f1oeq123d 6837	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (𝑓:(1...𝑘)⟶1-1onto(1...𝑘) ↔ 𝑓:∅⟶1-1onto∅))
18		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ ∅ = ∅
19		f1o00 6880	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:∅⟶1-1onto∅ ↔ (𝑓 = ∅ ∧ ∅ = ∅))
20	18, 19	mpbiran2 723	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:∅⟶1-1onto∅ ↔ 𝑓 = ∅)
21	17, 20	bitrdi 290	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (𝑓:(1...𝑘)⟶1-1onto(1...𝑘) ↔ 𝑓 = ∅))
22	21	abbidv 2829	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(1...𝑘)} = {𝑓 ∣ 𝑓 = ∅})
23		df-sn 4589	. . . . . . . . . . . . . 14 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅}
24	22, 23	eqtr4di 2816	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(1...𝑘)} = {∅})
25	15, 24	xpeq12d 5686	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(1...𝑘)}) = ({∅} × {∅}))
26		0ex 5264	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
27	26, 26	xpsn 7162	. . . . . . . . . . . 12 ⊢ ({∅} × {∅}) = {⟨∅, ∅⟩}
28	25, 27	eqtr2di 2815	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → {⟨∅, ∅⟩} = (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(1...𝑘)}))
29		elsni 4601	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → 𝑠 = ⟨∅, ∅⟩)
30	26, 26	op1std 8032	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ⟨∅, ∅⟩ → (1st ‘𝑠) = ∅)
31	29, 30	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (1st ‘𝑠) = ∅)
32	31	oveq1d 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ((1st ‘𝑠) ∘f + ∅) = (∅ ∘f + ∅))
33		f0 6782	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅:∅⟶∅
34		ffn 6728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
35	33, 34	mp1i 14	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ∅ Fn ∅)
36	26	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ∅ ∈ V)
37		inidm 4126	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∩ ∅) = ∅
38		0fv 6946	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅‘𝑛) = ∅
39	38	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ {⟨∅, ∅⟩} ∧ 𝑛 ∈ ∅) → (∅‘𝑛) = ∅)
40	35, 35, 36, 36, 37, 39, 39	offval 7723	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘f + ∅) = (𝑛 ∈ ∅ ↦ (∅ + ∅)))
41		mpt0 6700	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ∅ ↦ (∅ + ∅)) = ∅
42	40, 41	eqtrdi 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘f + ∅) = ∅)
43	32, 42	eqtrd 2798	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ((1st ‘𝑠) ∘f + ∅) = ∅)
44	43	uneq1d 4169	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (((1st ‘𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) = (∅ ∪ ((1...𝑁) × {0})))
45		uncom 4160	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∪ ((1...𝑁) × {0})) = (((1...𝑁) × {0}) ∪ ∅)
46		un0 4347	. . . . . . . . . . . . . . . 16 ⊢ (((1...𝑁) × {0}) ∪ ∅) = ((1...𝑁) × {0})
47	45, 46	eqtri 2786	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ ((1...𝑁) × {0})) = ((1...𝑁) × {0})
48	44, 47	eqtr2di 2815	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ((1...𝑁) × {0}) = (((1st ‘𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})))
49	48	csbeq1d 3855	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
50	49	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st ‘𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
51		oveq2 7448	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (0...𝑘) = (0...0))
52		0z 12728	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
53		fzsn 13724	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ℤ → (0...0) = {0})
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (0...0) = {0}
55	51, 54	eqtrdi 2814	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (0...𝑘) = {0})
56		oveq2 7448	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...0))
57	56	imaeq2d 6063	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...0)))
58	57	xpeq1d 5684	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))
59	58	uneq2d 4170	. . . . . . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
62		0p1e1 12487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 + 1) = 1
63	61, 62	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → (𝑘 + 1) = 1)
64	63	oveq1d 7455	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 0 → ((𝑘 + 1)...𝑁) = (1...𝑁))
65	64	xpeq1d 5684	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (((𝑘 + 1)...𝑁) × {0}) = ((1...𝑁) × {0}))
66	60, 65	uneq12d 4171	. . . . . . . . . . . . . . . . . 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 3855	. . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . 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 3339	. . . . . . . . . . . . . . 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 11324	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
71		oveq2 7448	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 = 0 → (1...𝑗) = (1...0))
72	71, 7	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 0 → (1...𝑗) = ∅)
73	72	imaeq2d 6063	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd ‘𝑠) “ ∅))
74		ima0 6086	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2nd ‘𝑠) “ ∅) = ∅
75	73, 74	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ (1...𝑗)) = ∅)
76	75	xpeq1d 5684	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = (∅ × {1}))
77		0xp 5755	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ × {1}) = ∅
78	76, 77	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = ∅)
79		oveq1 7447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
80	79, 62	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 = 0 → (𝑗 + 1) = 1)
81	80	oveq1d 7455	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 = 0 → ((𝑗 + 1)...0) = (1...0))
82	81, 7	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 0 → ((𝑗 + 1)...0) = ∅)
83	82	imaeq2d 6063	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ ((𝑗 + 1)...0)) = ((2nd ‘𝑠) “ ∅))
84	83, 74	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ ((𝑗 + 1)...0)) = ∅)
85	84	xpeq1d 5684	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}) = (∅ × {0}))
86		0xp 5755	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ × {0}) = ∅
87	85, 86	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}) = ∅)
88	78, 87	uneq12d 4171	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 0 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0})) = (∅ ∪ ∅))
89		un0 4347	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∪ ∅) = ∅
90	88, 89	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 0 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0})) = ∅)
91	90	oveq2d 7456	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 0 → ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) = ((1st ‘𝑠) ∘f + ∅))
92	91	uneq1d 4169	. . . . . . . . . . . . . . . . . 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 3855	. . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . 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 4643	. . . . . . . . . . . . . . 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	bitrdi 290	. . . . . . . . . . . . . 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 3338	. . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . 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 4642	. . . . . . . . . . . . 13 ⊢ (∀𝑖 ∈ {0}𝑖 = ⦋(((1st ‘𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st ‘𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
100	97, 99	bitr2di 291	. . . . . . . . . . . 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 520	. . . . . . . . . . 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 3432	. . . . . . . . . 10 ⊢ (𝑘 = 0 → {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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 2769	. . . . . . . . 9 ⊢ (𝑘 = 0 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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 6909	. . . . . . . 8 ⊢ (𝑘 = 0 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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 5118	. . . . . . 7 ⊢ (𝑘 = 0 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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 321	. . . . . 6 ⊢ (𝑘 = 0 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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 343	. . . . 5 ⊢ (𝑘 = 0 → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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 7448	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (1...𝑘) = (1...𝑚))
109	108	oveq2d 7456	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m (1...𝑚)))
110		eqidd 2764	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → 𝑓 = 𝑓)
111	110, 108, 108	f1oeq123d 6837	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (𝑓:(1...𝑘)⟶1-1onto(1...𝑘) ↔ 𝑓:(1...𝑚)⟶1-1onto(1...𝑚)))
112	111	abbidv 2829	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(1...𝑘)} = {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(1...𝑚)})
113	109, 112	xpeq12d 5686	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(1...𝑘)}) = (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(1...𝑚)}))
114		oveq2 7448	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (0...𝑘) = (0...𝑚))
115		oveq2 7448	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑚 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑚))
116	115	imaeq2d 6063	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑚 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)))
117	116	xpeq1d 5684	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))
118	117	uneq2d 4170	. . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
121	120	oveq1d 7455	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → ((𝑘 + 1)...𝑁) = ((𝑚 + 1)...𝑁))
122	121	xpeq1d 5684	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑚 + 1)...𝑁) × {0}))
123	119, 122	uneq12d 4171	. . . . . . . . . . . . . 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 3855	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 3339	. . . . . . . . . . 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 3338	. . . . . . . . . 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 3433	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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)⟶1-1onto(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 6909	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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)⟶1-1onto(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 5118	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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)⟶1-1onto(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 321	. . . . . 6 ⊢ (𝑘 = 𝑚 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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)⟶1-1onto(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 343	. . . . 5 ⊢ (𝑘 = 𝑚 → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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)⟶1-1onto(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 7448	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (1...𝑘) = (1...(𝑚 + 1)))
134	133	oveq2d 7456	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m (1...(𝑚 + 1))))
135		eqidd 2764	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → 𝑓 = 𝑓)
136	135, 133, 133	f1oeq123d 6837	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (𝑓:(1...𝑘)⟶1-1onto(1...𝑘) ↔ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))))
137	136	abbidv 2829	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(1...𝑘)} = {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))})
138	134, 137	xpeq12d 5686	. . . . . . . . . 10 ⊢ (𝑘 = (𝑚 + 1) → (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(1...𝑘)}) = (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))}))
139		oveq2 7448	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → (0...𝑘) = (0...(𝑚 + 1)))
140		oveq2 7448	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑚 + 1) → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...(𝑚 + 1)))
141	140	imaeq2d 6063	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑚 + 1) → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))))
142	141	xpeq1d 5684	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑚 + 1) → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))
143	142	uneq2d 4170	. . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑚 + 1) → (𝑘 + 1) = ((𝑚 + 1) + 1))
146	145	oveq1d 7455	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑚 + 1) → ((𝑘 + 1)...𝑁) = (((𝑚 + 1) + 1)...𝑁))
147	146	xpeq1d 5684	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑚 + 1) → (((𝑘 + 1)...𝑁) × {0}) = ((((𝑚 + 1) + 1)...𝑁) × {0}))
148	144, 147	uneq12d 4171	. . . . . . . . . . . . . 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 3855	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 3339	. . . . . . . . . . 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 3338	. . . . . . . . . 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 3433	. . . . . . . . 9 ⊢ (𝑘 = (𝑚 + 1) → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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))⟶1-1onto(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 6909	. . . . . . . 8 ⊢ (𝑘 = (𝑚 + 1) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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))⟶1-1onto(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 5118	. . . . . . 7 ⊢ (𝑘 = (𝑚 + 1) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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))⟶1-1onto(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 321	. . . . . 6 ⊢ (𝑘 = (𝑚 + 1) → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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))⟶1-1onto(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 343	. . . . 5 ⊢ (𝑘 = (𝑚 + 1) → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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))⟶1-1onto(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 7448	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑁 → (1...𝑘) = (1...𝑁))
159	158	oveq2d 7456	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m (1...𝑁)))
160		eqidd 2764	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑁 → 𝑓 = 𝑓)
161	160, 158, 158	f1oeq123d 6837	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑁 → (𝑓:(1...𝑘)⟶1-1onto(1...𝑘) ↔ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)))
162	161	abbidv 2829	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(1...𝑘)} = {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})
163	159, 162	xpeq12d 5686	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(1...𝑘)}) = (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}))
164		oveq2 7448	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → (0...𝑘) = (0...𝑁))
165		oveq2 7448	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑁 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑁))
166	165	imaeq2d 6063	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑁 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)))
167	166	xpeq1d 5684	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑁 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))
168	167	uneq2d 4170	. . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑁 → (𝑘 + 1) = (𝑁 + 1))
171	170	oveq1d 7455	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑁 → ((𝑘 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
172	171	xpeq1d 5684	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑁 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑁 + 1)...𝑁) × {0}))
173	169, 172	uneq12d 4171	. . . . . . . . . . . . . 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 3855	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 3339	. . . . . . . . . . 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 3338	. . . . . . . . . 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 3433	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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)⟶1-1onto(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 6909	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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)⟶1-1onto(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 5118	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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)⟶1-1onto(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 321	. . . . . 6 ⊢ (𝑘 = 𝑁 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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)⟶1-1onto(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 343	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)⟶1-1onto(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)⟶1-1onto(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 16564	. . . . . . 7 ⊢ ¬ 2 ∥ 1
184		opex 5436	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ∈ V
185		hashsng 14537	. . . . . . . . . 10 ⊢ (⟨∅, ∅⟩ ∈ V → (♯‘{⟨∅, ∅⟩}) = 1)
186	184, 185	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘{⟨∅, ∅⟩}) = 1
187		nnuz 13028	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ≥‘1)
188	1, 187	eleqtrdi 2874	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
189		eluzfz1 13688	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁))
190	188, 189	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ (1...𝑁))
191		poimirlem28.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ ℕ)
192	191	nnnn0d 12691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
193		0elfz 13782	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
194		fconst6g 6790	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0...𝐾) → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾))
195	192, 193, 194	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾))
196	70	fvconst2 7230	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (1...𝑁) → (((1...𝑁) × {0})‘1) = 0)
197	190, 196	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((1...𝑁) × {0})‘1) = 0)
198	190, 195, 197	3jca 1146	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
199		nfv 1949	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝(𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
200		nfcsb1v 3875	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑝⦋((1...𝑁) × {0}) / 𝑝⦌𝐵
201	200	nfeq1 2941	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0
202	199, 201	nfim 1931	. . . . . . . . . . . . . . 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 7473	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑁) ∈ V
204		snex 5401	. . . . . . . . . . . . . . . 16 ⊢ {0} ∈ V
205	203, 204	xpex 7788	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑁) × {0}) ∈ V
206		feq1 6706	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ((1...𝑁) × {0}) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾)))
207		fveq1 6904	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = ((1...𝑁) × {0}) → (𝑝‘1) = (((1...𝑁) × {0})‘1))
208	207	eqeq1d 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ((1...𝑁) × {0}) → ((𝑝‘1) = 0 ↔ (((1...𝑁) × {0})‘1) = 0))
209	206, 208	3anbi23d 1468	. . . . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . . . 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 3865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((1...𝑁) × {0}) → 𝐵 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
212	211	eqeq1d 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ((1...𝑁) × {0}) → (𝐵 = 0 ↔ ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0))
213	210, 212	imbi12d 347	. . . . . . . . . . . . . . 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 11327	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
215		eleq1 2852	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → (𝑛 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁)))
216		fveqeq2 6914	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → ((𝑝‘𝑛) = 0 ↔ (𝑝‘1) = 0))
217	215, 216	3anbi13d 1467	. . . . . . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (𝐵 < 𝑛 ↔ 𝐵 < 1))
220	218, 219	imbi12d 347	. . . . . . . . . . . . . . . . 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 3524	. . . . . . . . . . . . . . . 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 13778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℕ0)
225		nn0lt10b 12785	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℕ0 → (𝐵 < 1 ↔ 𝐵 = 0))
226	223, 224, 225	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝐵 < 1 ↔ 𝐵 = 0))
227	226	3ad2antr2 1208	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → (𝐵 < 1 ↔ 𝐵 = 0))
228	222, 227	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0)
229	202, 205, 213, 228	vtoclf 3529	. . . . . . . . . . . . . 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 700	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0)
231	230	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
232	231	ralrimivw 3162	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
233		rabid2 3448	. . . . . . . . . . 11 ⊢ ({⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵} ↔ ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
234	232, 233	sylibr 237	. . . . . . . . . 10 ⊢ (𝜑 → {⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})
235	234	fveq2d 6909	. . . . . . . . 9 ⊢ (𝜑 → (♯‘{⟨∅, ∅⟩}) = (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))
236	186, 235	eqtr3id 2812	. . . . . . . 8 ⊢ (𝜑 → 1 = (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))
237	236	breq2d 5118	. . . . . . 7 ⊢ (𝜑 → (2 ∥ 1 ↔ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})))
238	183, 237	mtbii 329	. . . . . 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 12752	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
241		fzfi 14139	. . . . . . . . . . . . . . . . 17 ⊢ (1...(𝑚 + 1)) ∈ Fin
242		mapfi 9362	. . . . . . . . . . . . . . . . 17 ⊢ (((0..^𝐾) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((0..^𝐾) ↑m (1...(𝑚 + 1))) ∈ Fin)
243	10, 241, 242	mp2an 705	. . . . . . . . . . . . . . . 16 ⊢ ((0..^𝐾) ↑m (1...(𝑚 + 1))) ∈ Fin
244		ovex 7473	. . . . . . . . . . . . . . . . . . 19 ⊢ (1...(𝑚 + 1)) ∈ V
245	244, 244	mapval 8883	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...(𝑚 + 1)) ↑m (1...(𝑚 + 1))) = {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
246		mapfi 9362	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1...(𝑚 + 1)) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((1...(𝑚 + 1)) ↑m (1...(𝑚 + 1))) ∈ Fin)
247	241, 241, 246	mp2an 705	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...(𝑚 + 1)) ↑m (1...(𝑚 + 1))) ∈ Fin
248	245, 247	eqeltrri 2861	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin
249		f1of 6843	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1)) → 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1)))
250	249	ss2abi 3994	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))} ⊆ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
251		ssfi 9213	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))} ⊆ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}) → {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))} ∈ Fin)
252	248, 250, 251	mp2an 705	. . . . . . . . . . . . . . . 16 ⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))} ∈ Fin
253		xpfi 9336	. . . . . . . . . . . . . . . 16 ⊢ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))} ∈ Fin) → (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))}) ∈ Fin)
254	243, 252, 253	mp2an 705	. . . . . . . . . . . . . . 15 ⊢ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))}) ∈ Fin
255		rabfi 9287	. . . . . . . . . . . . . . 15 ⊢ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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 14524	. . . . . . . . . . . . . . 15 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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 12745	. . . . . . . . . . . . 13 ⊢ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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 9287	. . . . . . . . . . . . . . 15 ⊢ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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 14524	. . . . . . . . . . . . . . 15 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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 12745	. . . . . . . . . . . . 13 ⊢ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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 1358	. . . . . . . . . . . 12 ⊢ (2 ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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 12669	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
265	264	ad2antrl 741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑚 + 1) ∈ ℕ)
266		uneq1 4163	. . . . . . . . . . . . . . . 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 3855	. . . . . . . . . . . . . . 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 6787	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}
269	268	jctr 534	. . . . . . . . . . . . . . . . . . . . . . 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 12374	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ)
271	270	ltp1d 12271	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) < ((𝑚 + 1) + 1))
272		fzdisj 13709	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 + 1) < ((𝑚 + 1) + 1) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
273	271, 272	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ0 → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
274		fun 6763	. . . . . . . . . . . . . . . . . . . . . . 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 609	. . . . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . 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 12376	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℕ)
279	278, 187	eleqtrdi 2874	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ (ℤ≥‘1))
280	279	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝑚 + 1) + 1) ∈ (ℤ≥‘1))
281		nn0z 12741	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
282	1	nnzd 12743	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑁 ∈ ℤ)
283		zltp1le 12770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
284	281, 282, 283	syl2anr 609	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
285	284	biimpa 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → (𝑚 + 1) ≤ 𝑁)
286	285	anasss 472	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑚 + 1) ≤ 𝑁)
287	281	peano2zd 12830	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℤ)
288	287	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → (𝑚 + 1) ∈ ℤ)
289		eluz 13003	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
290	288, 282, 289	syl2anr 609	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑁 ∈ (ℤ≥‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
291	286, 290	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑁 ∈ (ℤ≥‘(𝑚 + 1)))
292		fzsplit2 13707	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑚 + 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑚 + 1))) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
293	280, 291, 292	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
294	293	eqcomd 2769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)) = (1...𝑁))
295	192, 193	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 0 ∈ (0...𝐾))
296	295	snssd 4747	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → {0} ⊆ (0...𝐾))
297		ssequn2 4190	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({0} ⊆ (0...𝐾) ↔ ((0...𝐾) ∪ {0}) = (0...𝐾))
298	296, 297	sylib 221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((0...𝐾) ∪ {0}) = (0...𝐾))
299	298	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((0...𝐾) ∪ {0}) = (0...𝐾))
300	294, 299	feq23d 6723	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
303		nfv 1949	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑝(𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
304		nfcsb1v 3875	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵
305	304	nfel1 2942	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)
306	303, 305	nfim 1931	. . . . . . . . . . . . . . . . . . . 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 3458	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑞 ∈ V
308		ovex 7473	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑚 + 1) + 1)...𝑁) ∈ V
309	308, 204	xpex 7788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}) ∈ V
310	307, 309	unex 7782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) ∈ V
311		feq1 6706	. . . . . . . . . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . . . . . . . . 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 3865	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → 𝐵 = ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
314	313	eleq1d 2849	. . . . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . . . 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 3529	. . . . . . . . . . . . . . . . . . 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 603	. . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
319		elfznn0 13778	. . . . . . . . . . . . . . . . 17 ⊢ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℕ0)
320	318, 319	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℕ0)
321	264	nnnn0d 12691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
322	321	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → (𝑚 + 1) ∈ ℕ0)
323	322	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑚 + 1) ∈ ℕ0)
324		leloe 11420	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑚 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
325	270, 3, 324	syl2anr 609	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
326	284, 325	bitrd 282	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
327	326	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 → ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
328	327	imdistani 579	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → ((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
329	328	anasss 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
330		simplll 787	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝜑)
331	278	nnge1d 12410	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℕ0 → 1 ≤ ((𝑚 + 1) + 1))
332	331	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 1 ≤ ((𝑚 + 1) + 1))
333		zltp1le 12770	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
334	287, 282, 333	syl2anr 609	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
335	334	biimpa 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ≤ 𝑁)
336	287	peano2zd 12830	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℤ)
337		1z 12750	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ∈ ℤ
338		elfz 13669	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1479	. . . . . . . . . . . . . . . . . . . . . . . . 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 609	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
341	340	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
343	342	adantlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
344		nn0re 12639	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℝ)
345	344	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 ∈ ℝ)
346	270	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℝ)
347	3	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ ℝ)
348	344	ltp1d 12271	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → 𝑚 < (𝑚 + 1))
349	348	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < (𝑚 + 1))
350		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) < 𝑁)
351	345, 346, 347, 349, 350	lttrd 11495	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
352	351	adantlr 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
353		anass 474	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ↔ (𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)))
354	302	anassrs 473	. . . . . . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . . . . . . 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 665	. . . . . . . . . . . . . . . . . . . . . 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 603	. . . . . . . . . . . . . . . . . . . . 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 6728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → 𝑞 Fn (1...(𝑚 + 1)))
359	358	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑞 Fn (1...(𝑚 + 1)))
360	273	ad3antlr 744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
361		eluz 13003	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
362	336, 282, 361	syl2anr 609	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
363	362	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
364	335, 363	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)))
365		eluzfz1 13688	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
366	364, 365	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
367	366	adantlr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
368		fnconstg 6789	. . . . . . . . . . . . . . . . . . . . . . . . 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 6997	. . . . . . . . . . . . . . . . . . . . . . . 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 1479	. . . . . . . . . . . . . . . . . . . . . . 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 850	. . . . . . . . . . . . . . . . . . . . . 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 7230	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0)
374	367, 373	syl 18	. . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . 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 1949	. . . . . . . . . . . . . . . . . . . . . . 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 2926	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑝 <
378		nfcv 2926	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑝((𝑚 + 1) + 1)
379	304, 377, 378	nfbr 5155	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)
380	376, 379	nfim 1931	. . . . . . . . . . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘((𝑚 + 1) + 1)) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)))
382	381	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . . . . . 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 1468	. . . . . . . . . . . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . . . . . . . . . . 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 5116	. . . . . . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . . . . . 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 7473	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 + 1) + 1) ∈ V
388		eleq1 2852	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝑛 ∈ (1...𝑁) ↔ ((𝑚 + 1) + 1) ∈ (1...𝑁)))
389		fveqeq2 6914	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝑝‘𝑛) = 0 ↔ (𝑝‘((𝑚 + 1) + 1)) = 0))
390	388, 389	3anbi13d 1467	. . . . . . . . . . . . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝐵 < 𝑛 ↔ 𝐵 < ((𝑚 + 1) + 1)))
393	391, 392	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 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 3524	. . . . . . . . . . . . . . . . . . . . . 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 3529	. . . . . . . . . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . . . . . . . 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 665	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
399	398	elfzelzd 13681	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ)
400	352, 399	syldan 603	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ)
401	287	ad3antlr 744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℤ)
402		zleltp1 12771	. . . . . . . . . . . . . . . . . . . . 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)))
403	400, 401, 402	syl2anc 596	. . . . . . . . . . . . . . . . . . . 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)))
404	396, 403	mpbird 260	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
405	348	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → 𝑚 < (𝑚 + 1))
406		breq2 5110	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 + 1) = 𝑁 → (𝑚 < (𝑚 + 1) ↔ 𝑚 < 𝑁))
407	406	biimpac 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 < (𝑚 + 1) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
408	405, 407	sylan 592	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
409		elfzle2 13685	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁)
410	398, 409	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁)
411	408, 410	syldan 603	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁)
412		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑚 + 1) = 𝑁)
413	411, 412	breqtrrd 5136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
414	404, 413	jaodan 972	. . . . . . . . . . . . . . . . . 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))
415	414	an32s 665	. . . . . . . . . . . . . . . . 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))
416	329, 415	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
417		elfz2nn0 13776	. . . . . . . . . . . . . . . 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)))
418	320, 323, 416, 417	syl3anbrc 1362	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...(𝑚 + 1)))
419		fzss2 13722	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (ℤ≥‘(𝑚 + 1)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
420	291, 419	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
421	420	sselda 3913	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑛 ∈ (1...(𝑚 + 1))) → 𝑛 ∈ (1...𝑁))
422	421	3ad2antr1 1207	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → 𝑛 ∈ (1...𝑁))
423	354	3ad2antr2 1208	. . . . . . . . . . . . . . . . 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))
424	358	ad2antll 742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑞 Fn (1...(𝑚 + 1)))
425	273	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
426		simprl 783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑛 ∈ (1...(𝑚 + 1)))
427		fvun1 6996	. . . . . . . . . . . . . . . . . . . . . 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))
428	369, 427	mp3an2 1479	. . . . . . . . . . . . . . . . . . . . 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))
429	424, 425, 426, 428	syl12anc 850	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
430	429	adantlrr 734	. . . . . . . . . . . . . . . . . . 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))
431	430	3adantr3 1190	. . . . . . . . . . . . . . . . . 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))
432		simpr3 1215	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑞‘𝑛) = 0)
433	431, 432	eqtrd 2798	. . . . . . . . . . . . . . . . 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)
434	422, 423, 433	3jca 1146	. . . . . . . . . . . . . . . 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))
435		nfv 1949	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
436		nfcv 2926	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑝𝑛
437	304, 377, 436	nfbr 5155	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛
438	435, 437	nfim 1931	. . . . . . . . . . . . . . . . . 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)
439		fveq1 6904	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘𝑛) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛))
440	439	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘𝑛) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
441	311, 440	3anbi23d 1468	. . . . . . . . . . . . . . . . . . . 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)))
442	441	anbi2d 642	. . . . . . . . . . . . . . . . . . 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))))
443	313	breq1d 5116	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < 𝑛 ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛))
444	442, 443	imbi12d 347	. . . . . . . . . . . . . . . . . 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)))
445	438, 310, 444, 221	vtoclf 3529	. . . . . . . . . . . . . . . . 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)
446	445	adantlr 728	. . . . . . . . . . . . . . . 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)
447	434, 446	syldan 603	. . . . . . . . . . . . . . 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)
448		simp1 1154	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾) → 𝑛 ∈ (1...(𝑚 + 1)))
449	421	anasss 472	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → 𝑛 ∈ (1...𝑁))
450	448, 449	sylanr2 696	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → 𝑛 ∈ (1...𝑁))
451		simp2 1155	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾) → 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))
452	451, 302	sylanr2 696	. . . . . . . . . . . . . . . . . 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))
453	429	3adantr3 1190	. . . . . . . . . . . . . . . . . . . . 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))
454		simpr3 1215	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → (𝑞‘𝑛) = 𝐾)
455	453, 454	eqtrd 2798	. . . . . . . . . . . . . . . . . . . 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)
456	455	anasss 472	. . . . . . . . . . . . . . . . . . 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)
457	456	adantrlr 736	. . . . . . . . . . . . . . . . . 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)
458	450, 452, 457	3jca 1146	. . . . . . . . . . . . . . . . 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))
459		nfv 1949	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
460		nfcv 2926	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑝(𝑛 − 1)
461	304, 460	nfne 3062	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)
462	459, 461	nfim 1931	. . . . . . . . . . . . . . . . . 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))
463	439	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘𝑛) = 𝐾 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
464	311, 463	3anbi23d 1468	. . . . . . . . . . . . . . . . . . . 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)))
465	464	anbi2d 642	. . . . . . . . . . . . . . . . . . 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))))
466	313	neeq1d 3018	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ≠ (𝑛 − 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)))
467	465, 466	imbi12d 347	. . . . . . . . . . . . . . . . . 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))))
468		poimirlem28.4	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
469	462, 310, 467, 468	vtoclf 3529	. . . . . . . . . . . . . . . . 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))
470	458, 469	syldan 603	. . . . . . . . . . . . . . . 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))
471	470	anassrs 473	. . . . . . . . . . . . . . 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))
472	265, 267, 418, 447, 471	poimirlem27 38756	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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))})))
473	265, 267, 418	poimirlem26 38755	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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})))
474		fzfi 14139	. . . . . . . . . . . . . . . . . . 19 ⊢ (0...(𝑚 + 1)) ∈ Fin
475		xpfi 9336	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))}) ∈ Fin ∧ (0...(𝑚 + 1)) ∈ Fin) → ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin)
476	254, 474, 475	mp2an 705	. . . . . . . . . . . . . . . . . 18 ⊢ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin
477		rabfi 9287	. . . . . . . . . . . . . . . . . 18 ⊢ (((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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)
478		hashcl 14524	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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)
479	476, 477, 478	mp2b 10	. . . . . . . . . . . . . . . . 17 ⊢ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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	479	nn0zi 12745	. . . . . . . . . . . . . . . 16 ⊢ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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
481		zsubcl 12762	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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)
482	480, 262, 481	mp2an 705	. . . . . . . . . . . . . . 15 ⊢ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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		zsubcl 12762	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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)
484	480, 258, 483	mp2an 705	. . . . . . . . . . . . . . 15 ⊢ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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		dvds2sub 16487	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℤ ∧ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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})))))
486	240, 482, 484, 485	mp3an 1491	. . . . . . . . . . . . . 14 ⊢ ((2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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	472, 473, 486	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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	479	nn0cni 12642	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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
489	261	nn0cni 12642	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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
490	257	nn0cni 12642	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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
491		nnncan1 11618	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))})))
492	488, 489, 490, 491	mp3an 1491	. . . . . . . . . . . . 13 ⊢ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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	487, 492	breqtrdi 5149	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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		dvdssub2 16497	. . . . . . . . . . . 12 ⊢ (((2 ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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))⟶1-1onto(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	263, 493, 494	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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		nn0cn 12640	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
497		pncan1 11764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℂ → ((𝑚 + 1) − 1) = 𝑚)
498	496, 497	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) − 1) = 𝑚)
499	498	oveq2d 7456	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ0 → (0...((𝑚 + 1) − 1)) = (0...𝑚))
500	499	rexeqdv 3324	. . . . . . . . . . . . . . . . . 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))
501	499, 500	raleqbidv 3338	. . . . . . . . . . . . . . . . 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))
502	501	3anbi1d 1469	. . . . . . . . . . . . . . . 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))))
503	502	rabbidv 3423	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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	fveq2d 6909	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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))⟶1-1onto(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	1	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑁 ∈ ℕ)
507	191	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝐾 ∈ ℕ)
508		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑚 ∈ ℕ0)
509		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑚 < 𝑁)
510	506, 507, 508, 509	poimirlem4 38733	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(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))⟶1-1onto(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))})
511		fzfi 14139	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑚) ∈ Fin
512		mapfi 9362	. . . . . . . . . . . . . . . . . 18 ⊢ (((0..^𝐾) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((0..^𝐾) ↑m (1...𝑚)) ∈ Fin)
513	10, 511, 512	mp2an 705	. . . . . . . . . . . . . . . . 17 ⊢ ((0..^𝐾) ↑m (1...𝑚)) ∈ Fin
514		ovex 7473	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑚) ∈ V
515	514, 514	mapval 8883	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1...𝑚) ↑m (1...𝑚)) = {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)}
516		mapfi 9362	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1...𝑚) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((1...𝑚) ↑m (1...𝑚)) ∈ Fin)
517	511, 511, 516	mp2an 705	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1...𝑚) ↑m (1...𝑚)) ∈ Fin
518	515, 517	eqeltrri 2861	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin
519		f1of 6843	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:(1...𝑚)⟶1-1onto(1...𝑚) → 𝑓:(1...𝑚)⟶(1...𝑚))
520	519	ss2abi 3994	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(1...𝑚)} ⊆ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)}
521		ssfi 9213	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(1...𝑚)} ⊆ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)}) → {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(1...𝑚)} ∈ Fin)
522	518, 520, 521	mp2an 705	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(1...𝑚)} ∈ Fin
523		xpfi 9336	. . . . . . . . . . . . . . . . 17 ⊢ ((((0..^𝐾) ↑m (1...𝑚)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(1...𝑚)} ∈ Fin) → (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(1...𝑚)}) ∈ Fin)
524	513, 522, 523	mp2an 705	. . . . . . . . . . . . . . . 16 ⊢ (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(1...𝑚)}) ∈ Fin
525		rabfi 9287	. . . . . . . . . . . . . . . 16 ⊢ ((((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(1...𝑚)}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(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)
526	524, 525	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(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		rabfi 9287	. . . . . . . . . . . . . . . 16 ⊢ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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)
528	254, 527	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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		hashen 14515	. . . . . . . . . . . . . . 15 ⊢ (({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(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))⟶1-1onto(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)⟶1-1onto(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))⟶1-1onto(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)⟶1-1onto(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))⟶1-1onto(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))}))
530	526, 528, 529	mp2an 705	. . . . . . . . . . . . . 14 ⊢ ((♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(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))⟶1-1onto(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)⟶1-1onto(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))⟶1-1onto(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	510, 530	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(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))⟶1-1onto(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	505, 531	eqtr4d 2801	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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)⟶1-1onto(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}))
533	532	breq2d 5118	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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)⟶1-1onto(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	495, 533	bitrd 282	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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)⟶1-1onto(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	534	biimpd 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶1-1onto(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)⟶1-1onto(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	con3d 153	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(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))⟶1-1onto(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})))
537	536	expcom 419	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → (𝜑 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(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))⟶1-1onto(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	a2d 30	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(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))⟶1-1onto(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	3adant1 1148	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)⟶1-1onto(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))⟶1-1onto(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	107, 132, 157, 182, 239, 539	fnn0ind 12822	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 𝑁) → (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(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})))
541	5, 540	mpcom 39	. . 3 ⊢ (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(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		dvds0 16467	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∥ 0)
543	240, 542	ax-mp 5	. . . . . . 7 ⊢ 2 ∥ 0
544		hash0 14535	. . . . . . 7 ⊢ (♯‘∅) = 0
545	543, 544	breqtrri 5135	. . . . . 6 ⊢ 2 ∥ (♯‘∅)
546		fveq2 6905	. . . . . 6 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) = (♯‘∅))
547	545, 546	breqtrrid 5146	. . . . 5 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
548	3	ltp1d 12271	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 < (𝑁 + 1))
549	282	peano2zd 12830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
550		fzn 13697	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
551	549, 282, 550	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
552	548, 551	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
553	552	xpeq1d 5684	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = (∅ × {0}))
554	553, 86	eqtrdi 2814	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = ∅)
555	554	uneq2d 4170	. . . . . . . . . . . . . 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}))) ∪ ∅))
556		un0 4347	. . . . . . . . . . . . . 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})))
557	555, 556	eqtrdi 2814	. . . . . . . . . . . . 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}))))
558	557	csbeq1d 3855	. . . . . . . . . . . 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)
559		ovex 7473	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
560		poimirlem28.1	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
561	559, 560	csbie 3886	. . . . . . . . . . . 12 ⊢ ⦋((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = 𝐶
562	558, 561	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋(((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = 𝐶)
563	562	eqeq2d 2774	. . . . . . . . . 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))
564	563	rexbidv 3190	. . . . . . . . 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))
565	564	ralbidv 3189	. . . . . . . 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))
566	565	rabbidv 3423	. . . . . . 7 ⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(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)⟶1-1onto(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
567	566	fveq2d 6909	. . . . . 6 ⊢ (𝜑 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(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)⟶1-1onto(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
568	567	breq2d 5118	. . . . 5 ⊢ (𝜑 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(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)⟶1-1onto(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))
569	547, 568	imbitrrid 249	. . . 4 ⊢ (𝜑 → ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(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})))
570	569	necon3bd 2973	. . 3 ⊢ (𝜑 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(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)⟶1-1onto(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅))
571	541, 570	mpd 16	. 2 ⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅)
572		rabn0 4342	. 2 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅ ↔ ∃𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
573	571, 572	sylib 221	1 ⊢ (𝜑 → ∃𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454  ⦋csb 3851   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4838   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653   “ cima 5658   Fn wfn 6550  ⟶wf 6551  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440   ∘_f cof 7712  1^st c1st 8020  2^nd c2nd 8021  1_oc1o 8494   ↑_m cmap 8872   ≈ cen 8995  Fincfn 8998  ℂcc 11222  ℝcr 11223  0cc0 11224  1c1 11225   + caddc 11227   < clt 11367   ≤ cle 11368   − cmin 11565  ℕcn 12359  2c2 12421  ℕ₀cn0 12630  ℤcz 12717  ℤ_≥cuz 12989  ...cfz 13663  ..^cfzo 13812  ♯chash 14498   ∥ cdvds 16448

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-inf2 9666  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-disj 5074  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-er 8742  df-map 8874  df-pm 8875  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-sup 9458  df-oi 9528  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-xnn0 12704  df-z 12718  df-uz 12990  df-rp 13145  df-fz 13664  df-fzo 13813  df-seq 14169  df-exp 14229  df-fac 14442  df-bc 14471  df-hash 14499  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-abs 15429  df-clim 15681  df-sum 15880  df-dvds 16449

This theorem is used by:  poimirlem32  38761