poimirlem24 - Mathbox for Brendan Leahy

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Theorem poimirlem24 34931

Description: Lemma for poimir 34940, two ways of expressing that a simplex has an admissible face on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)

**Hypotheses**
Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem28.1	⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
poimirlem28.2	⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
poimirlem25.3	⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾))
poimirlem25.4	⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
poimirlem24.5	⊢ (𝜑 → 𝑉 ∈ (0...𝑁))

**Assertion**
Ref	Expression
poimirlem24	⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))))

Distinct variable groups: 𝑖,𝑗,𝑝,𝑠,𝑥,𝑦,𝜑 𝑗,𝑁,𝑦 𝑇,𝑗,𝑦 𝑈,𝑗,𝑦 𝑗,𝑉,𝑦 𝜑,𝑖,𝑝,𝑠 𝐵,𝑖,𝑗,𝑠 𝑖,𝐾,𝑗,𝑝,𝑠 𝑖,𝑁,𝑝,𝑠 𝑇,𝑖,𝑝 𝑈,𝑖,𝑝 𝑇,𝑠 𝜑,𝑥 𝑥,𝐵,𝑦 𝐶,𝑖,𝑝,𝑥,𝑦 𝑥,𝐾,𝑦 𝑥,𝑁 𝑥,𝑇 𝑈,𝑠,𝑥 𝑖,𝑉,𝑝,𝑠,𝑥 Allowed substitution hints: 𝐵(𝑝) 𝐶(𝑗,𝑠)

Proof of Theorem poimirlem24 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1)))
2		nfcsb1v 3907	. . . . . . . . . 10 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))
3		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑗(1...𝑁)
4		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑗(0...𝐾)
5	2, 3, 4	nff 6510	. . . . . . . . 9 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)
6	1, 5	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
7		eleq1 2900	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝑗 ∈ (0...(𝑁 − 1)) ↔ 𝑦 ∈ (0...(𝑁 − 1))))
8	7	anbi2d 630	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) ↔ (𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1)))))
9		csbeq1a 3897	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
10	9	feq1d 6499	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾) ↔ ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)))
11	8, 10	imbi12d 347	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)) ↔ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))))
12		poimir.0	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
13	12	nncnd 11654	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
14		npcan1 11065	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
15	13, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
16	12	nnzd 12087	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
17		peano2zm 12026	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
18	16, 17	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
19		uzid 12259	. . . . . . . . . . . . 13 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
20		peano2uz 12302	. . . . . . . . . . . . 13 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
21	18, 19, 20	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
22	15, 21	eqeltrrd 2914	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
23		fzss2 12948	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
24	22, 23	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁))
25	24	sselda 3967	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ (0...𝑁))
26		elun 4125	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ({1} ∪ {0}) ↔ (𝑦 ∈ {1} ∨ 𝑦 ∈ {0}))
27		fzofzp1 13135	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑥 + 1) ∈ (0...𝐾))
28		elsni 4584	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {1} → 𝑦 = 1)
29	28	oveq2d 7172	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {1} → (𝑥 + 𝑦) = (𝑥 + 1))
30	29	eleq1d 2897	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {1} → ((𝑥 + 𝑦) ∈ (0...𝐾) ↔ (𝑥 + 1) ∈ (0...𝐾)))
31	27, 30	syl5ibrcom 249	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑦 ∈ {1} → (𝑥 + 𝑦) ∈ (0...𝐾)))
32		elfzoelz 13039	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (0..^𝐾) → 𝑥 ∈ ℤ)
33	32	zcnd 12089	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (0..^𝐾) → 𝑥 ∈ ℂ)
34	33	addid1d 10840	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑥 + 0) = 𝑥)
35		elfzofz 13054	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (0..^𝐾) → 𝑥 ∈ (0...𝐾))
36	34, 35	eqeltrd 2913	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑥 + 0) ∈ (0...𝐾))
37		elsni 4584	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {0} → 𝑦 = 0)
38	37	oveq2d 7172	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {0} → (𝑥 + 𝑦) = (𝑥 + 0))
39	38	eleq1d 2897	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {0} → ((𝑥 + 𝑦) ∈ (0...𝐾) ↔ (𝑥 + 0) ∈ (0...𝐾)))
40	36, 39	syl5ibrcom 249	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑦 ∈ {0} → (𝑥 + 𝑦) ∈ (0...𝐾)))
41	31, 40	jaod 855	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0..^𝐾) → ((𝑦 ∈ {1} ∨ 𝑦 ∈ {0}) → (𝑥 + 𝑦) ∈ (0...𝐾)))
42	26, 41	syl5bi 244	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑦 ∈ ({1} ∪ {0}) → (𝑥 + 𝑦) ∈ (0...𝐾)))
43	42	imp 409	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (0..^𝐾) ∧ 𝑦 ∈ ({1} ∪ {0})) → (𝑥 + 𝑦) ∈ (0...𝐾))
44	43	adantl 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑥 ∈ (0..^𝐾) ∧ 𝑦 ∈ ({1} ∪ {0}))) → (𝑥 + 𝑦) ∈ (0...𝐾))
45		poimirlem25.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾))
46	45	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑇:(1...𝑁)⟶(0..^𝐾))
47		1ex 10637	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
48	47	fconst 6565	. . . . . . . . . . . . 13 ⊢ ((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1}
49		c0ex 10635	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
50	49	fconst 6565	. . . . . . . . . . . . 13 ⊢ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0}
51	48, 50	pm3.2i 473	. . . . . . . . . . . 12 ⊢ (((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1} ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0})
52		poimirlem25.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
53		dff1o3 6621	. . . . . . . . . . . . . . 15 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (𝑈:(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡𝑈))
54	53	simprbi 499	. . . . . . . . . . . . . 14 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡𝑈)
55		imain 6439	. . . . . . . . . . . . . 14 ⊢ (Fun ^◡𝑈 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))))
56	52, 54, 55	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))))
57		elfznn0 13001	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ₀)
58	57	nn0red 11957	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
59	58	ltp1d 11570	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
60		fzdisj 12935	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
61	59, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
62	61	imaeq2d 5929	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (𝑈 “ ∅))
63		ima0 5945	. . . . . . . . . . . . . 14 ⊢ (𝑈 “ ∅) = ∅
64	62, 63	syl6eq 2872	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
65	56, 64	sylan9req 2877	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅)
66		fun 6540	. . . . . . . . . . . 12 ⊢ (((((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1} ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0}) ∧ ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
67	51, 65, 66	sylancr 589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
68		nn0p1nn 11937	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
69	57, 68	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
70		nnuz 12282	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ_≥‘1)
71	69, 70	eleqtrdi 2923	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ_≥‘1))
72		elfzuz3 12906	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑗))
73		fzsplit2 12933	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
74	71, 72, 73	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
75	74	imaeq2d 5929	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
76		imaundi 6008	. . . . . . . . . . . . . 14 ⊢ (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))
77	75, 76	syl6req 2873	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑁) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁))) = (𝑈 “ (1...𝑁)))
78		f1ofo 6622	. . . . . . . . . . . . . 14 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–onto→(1...𝑁))
79		foima 6595	. . . . . . . . . . . . . 14 ⊢ (𝑈:(1...𝑁)–onto→(1...𝑁) → (𝑈 “ (1...𝑁)) = (1...𝑁))
80	52, 78, 79	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑈 “ (1...𝑁)) = (1...𝑁))
81	77, 80	sylan9eqr 2878	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁))) = (1...𝑁))
82	81	feq2d 6500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}) ↔ (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
83	67, 82	mpbid 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
84		ovex 7189	. . . . . . . . . . 11 ⊢ (1...𝑁) ∈ V
85	84	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ V)
86		inidm 4195	. . . . . . . . . 10 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
87	44, 46, 83, 85, 85, 86	off 7424	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
88	25, 87	syldan 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
89	6, 11, 88	chvarfv 2242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
90		fzp1elp1 12961	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (0...((𝑁 − 1) + 1)))
91	15	oveq2d 7172	. . . . . . . . . . 11 ⊢ (𝜑 → (0...((𝑁 − 1) + 1)) = (0...𝑁))
92	91	eleq2d 2898	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 + 1) ∈ (0...((𝑁 − 1) + 1)) ↔ (𝑦 + 1) ∈ (0...𝑁)))
93	92	biimpa 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 + 1) ∈ (0...((𝑁 − 1) + 1))) → (𝑦 + 1) ∈ (0...𝑁))
94	90, 93	sylan2 594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ (0...𝑁))
95		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝜑 ∧ (𝑦 + 1) ∈ (0...𝑁))
96		nfcsb1v 3907	. . . . . . . . . . 11 ⊢ Ⅎ𝑗⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))
97	96, 3, 4	nff 6510	. . . . . . . . . 10 ⊢ Ⅎ𝑗⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)
98	95, 97	nfim 1897	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝜑 ∧ (𝑦 + 1) ∈ (0...𝑁)) → ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
99		ovex 7189	. . . . . . . . 9 ⊢ (𝑦 + 1) ∈ V
100		eleq1 2900	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → (𝑗 ∈ (0...𝑁) ↔ (𝑦 + 1) ∈ (0...𝑁)))
101	100	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑦 + 1) ∈ (0...𝑁))))
102		csbeq1a 3897	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
103	102	feq1d 6499	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → ((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾) ↔ ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)))
104	101, 103	imbi12d 347	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)) ↔ ((𝜑 ∧ (𝑦 + 1) ∈ (0...𝑁)) → ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))))
105	98, 99, 104, 87	vtoclf 3558	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 + 1) ∈ (0...𝑁)) → ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
106	94, 105	syldan 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
107		csbeq1 3886	. . . . . . . . 9 ⊢ (𝑦 = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
108	107	feq1d 6499	. . . . . . . 8 ⊢ (𝑦 = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → (⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾) ↔ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)))
109		csbeq1 3886	. . . . . . . . 9 ⊢ ((𝑦 + 1) = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
110	109	feq1d 6499	. . . . . . . 8 ⊢ ((𝑦 + 1) = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → (⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾) ↔ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)))
111	108, 110	ifboth 4505	. . . . . . 7 ⊢ ((⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾) ∧ ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
112	89, 106, 111	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
113		ovex 7189	. . . . . . 7 ⊢ (0...𝐾) ∈ V
114	113, 84	elmap 8435	. . . . . 6 ⊢ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ ((0...𝐾) ↑_m (1...𝑁)) ↔ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
115	112, 114	sylibr 236	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ ((0...𝐾) ↑_m (1...𝑁)))
116	115	fmpttd 6879	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))):(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
117		ovex 7189	. . . . 5 ⊢ ((0...𝐾) ↑_m (1...𝑁)) ∈ V
118		ovex 7189	. . . . 5 ⊢ (0...(𝑁 − 1)) ∈ V
119	117, 118	elmap 8435	. . . 4 ⊢ ((𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))):(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
120	116, 119	sylibr 236	. . 3 ⊢ (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1))))
121		rneq 5806	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → ran 𝑥 = ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
122	121	mpteq1d 5155	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → (𝑝 ∈ ran 𝑥 ↦ 𝐵) = (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵))
123	122	rneqd 5808	. . . . . 6 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) = ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵))
124	123	sseq2d 3999	. . . . 5 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ (0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵)))
125	121	rexeqdv 3416	. . . . 5 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0))
126	124, 125	anbi12d 632	. . . 4 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0)))
127	126	ceqsrexv 3649	. . 3 ⊢ ((𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1))) → (∃𝑥 ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0)))
128	120, 127	syl 17	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0)))
129		dfss3 3956	. . . 4 ⊢ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∀𝑖 ∈ (0...(𝑁 − 1))𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵))
130		ovex 7189	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
131		poimirlem28.1	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
132	130, 131	csbie 3918	. . . . . . . . . . . 12 ⊢ ⦋((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = 𝐶
133	132	csbeq2i 3891	. . . . . . . . . . 11 ⊢ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌⦋((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
134		opex 5356	. . . . . . . . . . . . 13 ⊢ ⟨𝑇, 𝑈⟩ ∈ V
135	134	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨𝑇, 𝑈⟩ ∈ V)
136		fveq2 6670	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = ⟨𝑇, 𝑈⟩ → (1^st ‘𝑠) = (1^st ‘⟨𝑇, 𝑈⟩))
137		fex 6989	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇:(1...𝑁)⟶(0..^𝐾) ∧ (1...𝑁) ∈ V) → 𝑇 ∈ V)
138	45, 84, 137	sylancl 588	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ∈ V)
139		f1oexrnex 7632	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ∧ (1...𝑁) ∈ V) → 𝑈 ∈ V)
140	52, 84, 139	sylancl 588	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 ∈ V)
141		op1stg 7701	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ V ∧ 𝑈 ∈ V) → (1^st ‘⟨𝑇, 𝑈⟩) = 𝑇)
142	138, 140, 141	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1^st ‘⟨𝑇, 𝑈⟩) = 𝑇)
143	136, 142	sylan9eqr 2878	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 = ⟨𝑇, 𝑈⟩) → (1^st ‘𝑠) = 𝑇)
144		fveq2 6670	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = ⟨𝑇, 𝑈⟩ → (2^nd ‘𝑠) = (2^nd ‘⟨𝑇, 𝑈⟩))
145		op2ndg 7702	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ V ∧ 𝑈 ∈ V) → (2^nd ‘⟨𝑇, 𝑈⟩) = 𝑈)
146	138, 140, 145	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘⟨𝑇, 𝑈⟩) = 𝑈)
147	144, 146	sylan9eqr 2878	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 = ⟨𝑇, 𝑈⟩) → (2^nd ‘𝑠) = 𝑈)
148		imaeq1 5924	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑠) = 𝑈 → ((2^nd ‘𝑠) “ (1...𝑗)) = (𝑈 “ (1...𝑗)))
149	148	xpeq1d 5584	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑠) = 𝑈 → (((2^nd ‘𝑠) “ (1...𝑗)) × {1}) = ((𝑈 “ (1...𝑗)) × {1}))
150		imaeq1 5924	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑠) = 𝑈 → ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) = (𝑈 “ ((𝑗 + 1)...𝑁)))
151	150	xpeq1d 5584	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑠) = 𝑈 → (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))
152	149, 151	uneq12d 4140	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑠) = 𝑈 → ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))
153	147, 152	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 = ⟨𝑇, 𝑈⟩) → ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))
154	143, 153	oveq12d 7174	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 = ⟨𝑇, 𝑈⟩) → ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
155	154	csbeq1d 3887	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 = ⟨𝑇, 𝑈⟩) → ⦋((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = ⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
156	135, 155	csbied 3919	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌⦋((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = ⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
157	133, 156	syl5eqr 2870	. . . . . . . . . 10 ⊢ (𝜑 → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
158	157	csbeq2dv 3890	. . . . . . . . 9 ⊢ (𝜑 → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
159	158	eqeq2d 2832	. . . . . . . 8 ⊢ (𝜑 → (𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵))
160	159	rexbidv 3297	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵))
161		vex 3497	. . . . . . . . 9 ⊢ 𝑖 ∈ V
162		eqid 2821	. . . . . . . . . 10 ⊢ (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) = (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵)
163	162	elrnmpt 5828	. . . . . . . . 9 ⊢ (𝑖 ∈ V → (𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = 𝐵))
164	161, 163	ax-mp 5	. . . . . . . 8 ⊢ (𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = 𝐵)
165		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑖 = 𝐵
166		nfcsb1v 3907	. . . . . . . . . 10 ⊢ Ⅎ𝑝⦋𝑘 / 𝑝⦌𝐵
167	166	nfeq2 2995	. . . . . . . . 9 ⊢ Ⅎ𝑝 𝑖 = ⦋𝑘 / 𝑝⦌𝐵
168		csbeq1a 3897	. . . . . . . . . 10 ⊢ (𝑝 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑝⦌𝐵)
169	168	eqeq2d 2832	. . . . . . . . 9 ⊢ (𝑝 = 𝑘 → (𝑖 = 𝐵 ↔ 𝑖 = ⦋𝑘 / 𝑝⦌𝐵))
170	165, 167, 169	cbvrexw 3442	. . . . . . . 8 ⊢ (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = 𝐵 ↔ ∃𝑘 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = ⦋𝑘 / 𝑝⦌𝐵)
171		ovex 7189	. . . . . . . . . . 11 ⊢ (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
172	171	csbex 5215	. . . . . . . . . 10 ⊢ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
173	172	rgenw 3150	. . . . . . . . 9 ⊢ ∀𝑦 ∈ (0...(𝑁 − 1))⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
174		eqid 2821	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
175		csbeq1 3886	. . . . . . . . . . . 12 ⊢ (𝑘 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → ⦋𝑘 / 𝑝⦌𝐵 = ⦋⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
176		vex 3497	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
177	176, 99	ifex 4515	. . . . . . . . . . . . 13 ⊢ if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) ∈ V
178		csbnestgw 4373	. . . . . . . . . . . . 13 ⊢ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) ∈ V → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = ⦋⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
179	177, 178	ax-mp 5	. . . . . . . . . . . 12 ⊢ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = ⦋⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵
180	175, 179	syl6eqr 2874	. . . . . . . . . . 11 ⊢ (𝑘 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → ⦋𝑘 / 𝑝⦌𝐵 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
181	180	eqeq2d 2832	. . . . . . . . . 10 ⊢ (𝑘 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝑖 = ⦋𝑘 / 𝑝⦌𝐵 ↔ 𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵))
182	174, 181	rexrnmptw 6861	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (0...(𝑁 − 1))⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V → (∃𝑘 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = ⦋𝑘 / 𝑝⦌𝐵 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵))
183	173, 182	ax-mp 5	. . . . . . . 8 ⊢ (∃𝑘 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = ⦋𝑘 / 𝑝⦌𝐵 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
184	164, 170, 183	3bitri 299	. . . . . . 7 ⊢ (𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
185	160, 184	syl6bbr 291	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵)))
186	24	sselda 3967	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ (0...𝑁))
187	186	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < 𝑉) → 𝑦 ∈ (0...𝑁))
188		elfzelz 12909	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
189	188	zred 12088	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
190	189	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
191		ltne 10737	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝑦 < 𝑉) → 𝑉 ≠ 𝑦)
192	191	necomd 3071	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ 𝑦 < 𝑉) → 𝑦 ≠ 𝑉)
193	190, 192	sylan 582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < 𝑉) → 𝑦 ≠ 𝑉)
194		eldifsn 4719	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((0...𝑁) ∖ {𝑉}) ↔ (𝑦 ∈ (0...𝑁) ∧ 𝑦 ≠ 𝑉))
195	187, 193, 194	sylanbrc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < 𝑉) → 𝑦 ∈ ((0...𝑁) ∖ {𝑉}))
196	94	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < 𝑉) → (𝑦 + 1) ∈ (0...𝑁))
197		poimirlem24.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ∈ (0...𝑁))
198		elfzelz 12909	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (0...𝑁) → 𝑉 ∈ ℤ)
199	197, 198	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ ℤ)
200	199	zred 12088	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉 ∈ ℝ)
201	200	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < 𝑉) → 𝑉 ∈ ℝ)
202		zre 11986	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ ℤ → 𝑉 ∈ ℝ)
203		zre 11986	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
204		lenlt 10719	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑉 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑉))
205	202, 203, 204	syl2an 597	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑉 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑉))
206		zleltp1 12034	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑉 ≤ 𝑦 ↔ 𝑉 < (𝑦 + 1)))
207	205, 206	bitr3d 283	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (¬ 𝑦 < 𝑉 ↔ 𝑉 < (𝑦 + 1)))
208	199, 188, 207	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (¬ 𝑦 < 𝑉 ↔ 𝑉 < (𝑦 + 1)))
209	208	biimpa 479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < 𝑉) → 𝑉 < (𝑦 + 1))
210	201, 209	gtned 10775	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < 𝑉) → (𝑦 + 1) ≠ 𝑉)
211		eldifsn 4719	. . . . . . . . . . 11 ⊢ ((𝑦 + 1) ∈ ((0...𝑁) ∖ {𝑉}) ↔ ((𝑦 + 1) ∈ (0...𝑁) ∧ (𝑦 + 1) ≠ 𝑉))
212	196, 210, 211	sylanbrc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < 𝑉) → (𝑦 + 1) ∈ ((0...𝑁) ∖ {𝑉}))
213	195, 212	ifclda 4501	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) ∈ ((0...𝑁) ∖ {𝑉}))
214		nfcsb1v 3907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
215	214	nfeq2 2995	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
216		csbeq1a 3897	. . . . . . . . . . . 12 ⊢ (𝑗 = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
217	216	eqeq2d 2832	. . . . . . . . . . 11 ⊢ (𝑗 = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
218	215, 217	rspce 3612	. . . . . . . . . 10 ⊢ ((if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) ∈ ((0...𝑁) ∖ {𝑉}) ∧ 𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
219	218	ex 415	. . . . . . . . 9 ⊢ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) ∈ ((0...𝑁) ∖ {𝑉}) → (𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
220	213, 219	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
221	220	rexlimdva 3284	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
222		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑗𝜑
223		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑗(0...(𝑁 − 1))
224	223, 215	nfrex 3309	. . . . . . . 8 ⊢ Ⅎ𝑗∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
225		eldifi 4103	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ∈ (0...𝑁))
226	225, 57	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ∈ ℕ₀)
227	226	nn0ge0d 11959	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 0 ≤ 𝑗)
228	227	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 0 ≤ 𝑗)
229	226	nn0red 11957	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ∈ ℝ)
230	229	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 ∈ ℝ)
231	200	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑉 ∈ ℝ)
232	16	zred 12088	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
233	232	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑁 ∈ ℝ)
234		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 < 𝑉)
235		elfzle2 12912	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ (0...𝑁) → 𝑉 ≤ 𝑁)
236	197, 235	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ≤ 𝑁)
237	236	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑉 ≤ 𝑁)
238	230, 231, 233, 234, 237	ltletrd 10800	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 < 𝑁)
239	226	nn0zd 12086	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ∈ ℤ)
240		zltlem1 12036	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 < 𝑁 ↔ 𝑗 ≤ (𝑁 − 1)))
241	239, 16, 240	syl2anr 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 < 𝑁 ↔ 𝑗 ≤ (𝑁 − 1)))
242	241	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → (𝑗 < 𝑁 ↔ 𝑗 ≤ (𝑁 − 1)))
243	238, 242	mpbid 234	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 ≤ (𝑁 − 1))
244		0z 11993	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
245		elfz 12899	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑗 ∈ (0...(𝑁 − 1)) ↔ (0 ≤ 𝑗 ∧ 𝑗 ≤ (𝑁 − 1))))
246	244, 245	mp3an2 1445	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑗 ∈ (0...(𝑁 − 1)) ↔ (0 ≤ 𝑗 ∧ 𝑗 ≤ (𝑁 − 1))))
247	239, 18, 246	syl2anr 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 ∈ (0...(𝑁 − 1)) ↔ (0 ≤ 𝑗 ∧ 𝑗 ≤ (𝑁 − 1))))
248	247	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → (𝑗 ∈ (0...(𝑁 − 1)) ↔ (0 ≤ 𝑗 ∧ 𝑗 ≤ (𝑁 − 1))))
249	228, 243, 248	mpbir2and 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 ∈ (0...(𝑁 − 1)))
250		0red 10644	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 0 ∈ ℝ)
251	200	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑉 ∈ ℝ)
252	229	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑗 ∈ ℝ)
253		elfzle1 12911	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ (0...𝑁) → 0 ≤ 𝑉)
254	197, 253	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ 𝑉)
255	254	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 0 ≤ 𝑉)
256		lenlt 10719	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑉 ≤ 𝑗 ↔ ¬ 𝑗 < 𝑉))
257	200, 229, 256	syl2an 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑉 ≤ 𝑗 ↔ ¬ 𝑗 < 𝑉))
258	257	biimpar 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑉 ≤ 𝑗)
259		eldifsni 4722	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ≠ 𝑉)
260	259	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑗 ≠ 𝑉)
261		ltlen 10741	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑉 < 𝑗 ↔ (𝑉 ≤ 𝑗 ∧ 𝑗 ≠ 𝑉)))
262	200, 229, 261	syl2an 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑉 < 𝑗 ↔ (𝑉 ≤ 𝑗 ∧ 𝑗 ≠ 𝑉)))
263	262	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → (𝑉 < 𝑗 ↔ (𝑉 ≤ 𝑗 ∧ 𝑗 ≠ 𝑉)))
264	258, 260, 263	mpbir2and 711	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑉 < 𝑗)
265	250, 251, 252, 255, 264	lelttrd 10798	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 0 < 𝑗)
266		zgt0ge1 12037	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℤ → (0 < 𝑗 ↔ 1 ≤ 𝑗))
267	239, 266	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → (0 < 𝑗 ↔ 1 ≤ 𝑗))
268	267	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → (0 < 𝑗 ↔ 1 ≤ 𝑗))
269	265, 268	mpbid 234	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 1 ≤ 𝑗)
270		elfzle2 12912	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ≤ 𝑁)
271	225, 270	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ≤ 𝑁)
272	271	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑗 ≤ 𝑁)
273		1z 12013	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
274		elfz 12899	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (1...𝑁) ↔ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
275	273, 274	mp3an2 1445	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (1...𝑁) ↔ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
276	239, 16, 275	syl2anr 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 ∈ (1...𝑁) ↔ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
277	276	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → (𝑗 ∈ (1...𝑁) ↔ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
278	269, 272, 277	mpbir2and 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑗 ∈ (1...𝑁))
279		elfzmlbm 13018	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑁) → (𝑗 − 1) ∈ (0...(𝑁 − 1)))
280	278, 279	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → (𝑗 − 1) ∈ (0...(𝑁 − 1)))
281	249, 280	ifclda 4501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) ∈ (0...(𝑁 − 1)))
282		breq1 5069	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑗 < 𝑉 ↔ if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉))
283		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → 𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)))
284		oveq1 7163	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑗 + 1) = (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1))
285	282, 283, 284	ifbieq12d 4494	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → if(𝑗 < 𝑉, 𝑗, (𝑗 + 1)) = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)))
286	285	eqeq2d 2832	. . . . . . . . . . . . . 14 ⊢ (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 + 1)) ↔ 𝑗 = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1))))
287		breq1 5069	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → ((𝑗 − 1) < 𝑉 ↔ if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉))
288		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)))
289		oveq1 7163	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → ((𝑗 − 1) + 1) = (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1))
290	287, 288, 289	ifbieq12d 4494	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → if((𝑗 − 1) < 𝑉, (𝑗 − 1), ((𝑗 − 1) + 1)) = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)))
291	290	eqeq2d 2832	. . . . . . . . . . . . . 14 ⊢ ((𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑗 = if((𝑗 − 1) < 𝑉, (𝑗 − 1), ((𝑗 − 1) + 1)) ↔ 𝑗 = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1))))
292		iftrue 4473	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 < 𝑉 → if(𝑗 < 𝑉, 𝑗, (𝑗 + 1)) = 𝑗)
293	292	eqcomd 2827	. . . . . . . . . . . . . . 15 ⊢ (𝑗 < 𝑉 → 𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 + 1)))
294	293	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 + 1)))
295		zlem1lt 12035	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ ℤ ∧ 𝑉 ∈ ℤ) → (𝑗 ≤ 𝑉 ↔ (𝑗 − 1) < 𝑉))
296	239, 199, 295	syl2anr 598	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 ≤ 𝑉 ↔ (𝑗 − 1) < 𝑉))
297	259	necomd 3071	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑉 ≠ 𝑗)
298	297	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → 𝑉 ≠ 𝑗)
299		ltlen 10741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ ℝ ∧ 𝑉 ∈ ℝ) → (𝑗 < 𝑉 ↔ (𝑗 ≤ 𝑉 ∧ 𝑉 ≠ 𝑗)))
300	229, 200, 299	syl2anr 598	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 < 𝑉 ↔ (𝑗 ≤ 𝑉 ∧ 𝑉 ≠ 𝑗)))
301	300	biimprd 250	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → ((𝑗 ≤ 𝑉 ∧ 𝑉 ≠ 𝑗) → 𝑗 < 𝑉))
302	298, 301	mpan2d 692	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 ≤ 𝑉 → 𝑗 < 𝑉))
303	296, 302	sylbird 262	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → ((𝑗 − 1) < 𝑉 → 𝑗 < 𝑉))
304	303	con3dimp 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → ¬ (𝑗 − 1) < 𝑉)
305	304	iffalsed 4478	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → if((𝑗 − 1) < 𝑉, (𝑗 − 1), ((𝑗 − 1) + 1)) = ((𝑗 − 1) + 1))
306	226	nn0cnd 11958	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ∈ ℂ)
307		npcan1 11065	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℂ → ((𝑗 − 1) + 1) = 𝑗)
308	306, 307	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → ((𝑗 − 1) + 1) = 𝑗)
309	308	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → ((𝑗 − 1) + 1) = 𝑗)
310	305, 309	eqtr2d 2857	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑗 = if((𝑗 − 1) < 𝑉, (𝑗 − 1), ((𝑗 − 1) + 1)))
311	286, 291, 294, 310	ifbothda 4504	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → 𝑗 = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)))
312		csbeq1a 3897	. . . . . . . . . . . . 13 ⊢ (𝑗 = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
313	311, 312	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
314	313	eqeq2d 2832	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
315	314	biimpd 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑖 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
316		breq1 5069	. . . . . . . . . . . . . 14 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑦 < 𝑉 ↔ if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉))
317		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → 𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)))
318		oveq1 7163	. . . . . . . . . . . . . 14 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑦 + 1) = (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1))
319	316, 317, 318	ifbieq12d 4494	. . . . . . . . . . . . 13 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)))
320	319	csbeq1d 3887	. . . . . . . . . . . 12 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
321	320	eqeq2d 2832	. . . . . . . . . . 11 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
322	321	rspcev 3623	. . . . . . . . . 10 ⊢ ((if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) ∈ (0...(𝑁 − 1)) ∧ 𝑖 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
323	281, 315, 322	syl6an 682	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
324	323	ex 415	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
325	222, 224, 324	rexlimd 3317	. . . . . . 7 ⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
326	221, 325	impbid 214	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
327	185, 326	bitr3d 283	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
328	327	ralbidv 3197	. . . 4 ⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
329	129, 328	syl5bb 285	. . 3 ⊢ (𝜑 → ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
330	329	anbi1d 631	. 2 ⊢ (𝜑 → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0)))
331	12, 45, 52, 197	poimirlem23 34930	. . 3 ⊢ (𝜑 → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁))))
332	331	anbi2d 630	. 2 ⊢ (𝜑 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))))
333	128, 330, 332	3bitrd 307	1 ⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⦋csb 3883 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ifcif 4467 {csn 4567 ⟨cop 4573 class class class wbr 5066 ↦ cmpt 5146 × cxp 5553 ^◡ccnv 5554 ran crn 5556 “ cima 5558 Fun wfun 6349 ⟶wf 6351 –onto→wfo 6353 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ∘_f cof 7407 1^st c1st 7687 2^nd c2nd 7688 ↑_m cmap 8406 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 − cmin 10870 ℕcn 11638 ℕ₀cn0 11898 ℤcz 11982 ℤ_≥cuz 12244 ...cfz 12893 ..^cfzo 13034

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035

This theorem is referenced by: poimirlem27 34934

W3C validator