poimirlem25 - Mathbox for Brendan Leahy

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Theorem poimirlem25 38394

Description: Lemma for poimir 38402 stating that for a given simplex such that no vertex maps to 𝑁, the number of admissible faces is even. (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))
poimirlem25.5	⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑁 ≠ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)

**Assertion**
Ref	Expression
poimirlem25	⊢ (𝜑 → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}))

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

Proof of Theorem poimirlem25 Dummy variables 𝑓 𝑡 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 4299	. . 3 ⊢ (¬ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} = ∅ ↔ ∃𝑡 𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶})
2		2z 12650	. . . . . . . 8 ⊢ 2 ∈ ℤ
3		iddvds 16359	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∥ 2)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ 2 ∥ 2
5		df-2 12327	. . . . . . 7 ⊢ 2 = (1 + 1)
6	4, 5	breqtri 5130	. . . . . 6 ⊢ 2 ∥ (1 + 1)
7		vex 3454	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
8		fzfi 14036	. . . . . . . . . . . . 13 ⊢ (0...𝑁) ∈ Fin
9		rabfi 9241	. . . . . . . . . . . . 13 ⊢ ((0...𝑁) ∈ Fin → {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∈ Fin)
10	8, 9	ax-mp 5	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∈ Fin
11		diffi 9169	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∈ Fin → ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}) ∈ Fin)
12	10, 11	ax-mp 5	. . . . . . . . . . 11 ⊢ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}) ∈ Fin
13		neldifsn 4755	. . . . . . . . . . 11 ⊢ ¬ 𝑡 ∈ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})
14	12, 13	pm3.2i 476	. . . . . . . . . 10 ⊢ (({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}) ∈ Fin ∧ ¬ 𝑡 ∈ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}))
15		hashunsng 14456	. . . . . . . . . 10 ⊢ (𝑡 ∈ V → ((({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}) ∈ Fin ∧ ¬ 𝑡 ∈ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})) → (♯‘(({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}) ∪ {𝑡})) = ((♯‘({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})) + 1)))
16	7, 14, 15	mp2 9	. . . . . . . . 9 ⊢ (♯‘(({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}) ∪ {𝑡})) = ((♯‘({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})) + 1)
17		difsnid 4771	. . . . . . . . . 10 ⊢ (𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} → (({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}) ∪ {𝑡}) = {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶})
18	17	fveq2d 6882	. . . . . . . . 9 ⊢ (𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} → (♯‘(({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}) ∪ {𝑡})) = (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}))
19	16, 18	eqtr3id 2809	. . . . . . . 8 ⊢ (𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} → ((♯‘({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})) + 1) = (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}))
20	19	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}) → ((♯‘({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})) + 1) = (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}))
21		sneq 4594	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → {𝑦} = {𝑡})
22	21	difeq2d 4074	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → ((0...𝑁) ∖ {𝑦}) = ((0...𝑁) ∖ {𝑡}))
23	22	rexeqdv 3320	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
24	23	ralbidv 3185	. . . . . . . . . 10 ⊢ (𝑦 = 𝑡 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
25	24	elrab 3645	. . . . . . . . 9 ⊢ (𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ↔ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
26		poimirlem25.5	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑁 ≠ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
27	26	ralrimiva 3154	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
28		nfcv 2922	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗𝑁
29		nfcsb1v 3871	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
30	28, 29	nfne 3058	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗 𝑁 ≠ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
31		csbeq1a 3861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑡 → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
32	31	neeq2d 3015	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑡 → (𝑁 ≠ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
33	30, 32	rspc 3564	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (0...𝑁) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑁 ≠ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
34	27, 33	mpan9 516	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → 𝑁 ≠ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
35		nesym 3011	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ≠ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)
36	34, 35	sylib 221	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)
37		nfv 1947	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑡 ∈ (0...𝑁))
38	29	nfel1 2938	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁)
39	37, 38	nfim 1929	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁))
40		eleq1w 2843	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑡 → (𝑗 ∈ (0...𝑁) ↔ 𝑡 ∈ (0...𝑁)))
41	40	anbi2d 642	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑡 → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ (0...𝑁))))
42	31	eleq1d 2845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑡 → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁) ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁)))
43	41, 42	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑡 → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁)) ↔ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁))))
44		poimirlem25.3	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾))
45		ovex 7446	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0..^𝐾) ∈ V
46		ovex 7446	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1...𝑁) ∈ V
47	45, 46	elmap 8878	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑇 ∈ ((0..^𝐾) ↑_m (1...𝑁)) ↔ 𝑇:(1...𝑁)⟶(0..^𝐾))
48	44, 47	sylibr 237	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑇 ∈ ((0..^𝐾) ↑_m (1...𝑁)))
49		poimirlem25.4	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
50		fzfi 14036	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1...𝑁) ∈ Fin
51		f1oexrnex 7924	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ∧ (1...𝑁) ∈ Fin) → 𝑈 ∈ V)
52	50, 51	mpan2 704	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 ∈ V)
53		f1oeq1 6805	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = 𝑈 → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ 𝑈:(1...𝑁)–1-1-onto→(1...𝑁)))
54	53	elabg 3630	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑈 ∈ V → (𝑈 ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ 𝑈:(1...𝑁)–1-1-onto→(1...𝑁)))
55	52, 54	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → (𝑈 ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ 𝑈:(1...𝑁)–1-1-onto→(1...𝑁)))
56	55	ibir 271	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
57	49, 56	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
58		opelxpi 5692	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑇 ∈ ((0..^𝐾) ↑_m (1...𝑁)) ∧ 𝑈 ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⟨𝑇, 𝑈⟩ ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
59	48, 57, 58	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨𝑇, 𝑈⟩ ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
60	59	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⟨𝑇, 𝑈⟩ ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
61		nfcv 2922	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠⟨𝑇, 𝑈⟩
62		nfv 1947	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑠(𝜑 ∧ 𝑗 ∈ (0...𝑁))
63		nfcsb1v 3871	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑠⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
64	63	nfel1 2938	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑠⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁)
65	62, 64	nfim 1929	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁))
66		csbeq1a 3861	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ⟨𝑇, 𝑈⟩ → 𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
67	66	eleq1d 2845	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = ⟨𝑇, 𝑈⟩ → (𝐶 ∈ (0...𝑁) ↔ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁)))
68	67	imbi2d 343	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ⟨𝑇, 𝑈⟩ → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝐶 ∈ (0...𝑁)) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁))))
69		elun 4100	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑝 ∈ ({1} ∪ {0}) ↔ (𝑝 ∈ {1} ∨ 𝑝 ∈ {0}))
70		fzofzp1 13820	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑖 + 1) ∈ (0...𝐾))
71		elsni 4601	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑝 ∈ {1} → 𝑝 = 1)
72	71	oveq2d 7429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑝 ∈ {1} → (𝑖 + 𝑝) = (𝑖 + 1))
73	72	eleq1d 2845	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑝 ∈ {1} → ((𝑖 + 𝑝) ∈ (0...𝐾) ↔ (𝑖 + 1) ∈ (0...𝐾)))
74	70, 73	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑝 ∈ {1} → (𝑖 + 𝑝) ∈ (0...𝐾)))
75		elfzonn0 13763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑖 ∈ (0..^𝐾) → 𝑖 ∈ ℕ₀)
76	75	nn0cnd 12591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 ∈ (0..^𝐾) → 𝑖 ∈ ℂ)
77	76	addridd 11434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑖 + 0) = 𝑖)
78		elfzofz 13731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝐾) → 𝑖 ∈ (0...𝐾))
79	77, 78	eqeltrd 2860	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑖 + 0) ∈ (0...𝐾))
80		elsni 4601	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑝 ∈ {0} → 𝑝 = 0)
81	80	oveq2d 7429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑝 ∈ {0} → (𝑖 + 𝑝) = (𝑖 + 0))
82	81	eleq1d 2845	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑝 ∈ {0} → ((𝑖 + 𝑝) ∈ (0...𝐾) ↔ (𝑖 + 0) ∈ (0...𝐾)))
83	79, 82	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑝 ∈ {0} → (𝑖 + 𝑝) ∈ (0...𝐾)))
84	74, 83	jaod 873	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝐾) → ((𝑝 ∈ {1} ∨ 𝑝 ∈ {0}) → (𝑖 + 𝑝) ∈ (0...𝐾)))
85	69, 84	biimtrid 245	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑝 ∈ ({1} ∪ {0}) → (𝑖 + 𝑝) ∈ (0...𝐾)))
86	85	imp 412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝐾) ∧ 𝑝 ∈ ({1} ∪ {0})) → (𝑖 + 𝑝) ∈ (0...𝐾))
87	86	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑖 ∈ (0..^𝐾) ∧ 𝑝 ∈ ({1} ∪ {0}))) → (𝑖 + 𝑝) ∈ (0...𝐾))
88		xp1st 8018	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘𝑠) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
89		elmapi 8848	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((1^st ‘𝑠) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘𝑠):(1...𝑁)⟶(0..^𝐾))
90	88, 89	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘𝑠):(1...𝑁)⟶(0..^𝐾))
91	90	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑗 ∈ (0...𝑁)) → (1^st ‘𝑠):(1...𝑁)⟶(0..^𝐾))
92		xp2nd 8019	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘𝑠) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
93		fvex 6891	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (2^nd ‘𝑠) ∈ V
94		f1oeq1 6805	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = (2^nd ‘𝑠) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁)))
95	93, 94	elab 3633	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2^nd ‘𝑠) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁))
96	92, 95	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁))
97		1ex 11227	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ V
98	97	fconst 6761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘𝑠) “ (1...𝑗)) × {1}):((2^nd ‘𝑠) “ (1...𝑗))⟶{1}
99		c0ex 11224	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 ∈ V
100	99	fconst 6761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}):((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁))⟶{0}
101	98, 100	pm3.2i 476	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}):((2^nd ‘𝑠) “ (1...𝑗))⟶{1} ∧ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}):((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁))⟶{0})
102		dff1o3 6824	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘𝑠):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡(2^nd ‘𝑠)))
103		imain 6618	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun ^◡(2^nd ‘𝑠) → ((2^nd ‘𝑠) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (((2^nd ‘𝑠) “ (1...𝑗)) ∩ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁))))
104	102, 103	simplbiim 514	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁) → ((2^nd ‘𝑠) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (((2^nd ‘𝑠) “ (1...𝑗)) ∩ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁))))
105		elfznn0 13675	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ₀)
106	105	nn0red 12590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
107	106	ltp1d 12169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
108		fzdisj 13606	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
109	107, 108	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
110	109	imaeq2d 6056	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘𝑠) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((2^nd ‘𝑠) “ ∅))
111		ima0 6073	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2^nd ‘𝑠) “ ∅) = ∅
112	110, 111	eqtrdi 2811	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘𝑠) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
113	104, 112	sylan9req 2816	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (0...𝑁)) → (((2^nd ‘𝑠) “ (1...𝑗)) ∩ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁))) = ∅)
114		fun 6737	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((2^nd ‘𝑠) “ (1...𝑗)) × {1}):((2^nd ‘𝑠) “ (1...𝑗))⟶{1} ∧ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}):((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁))⟶{0}) ∧ (((2^nd ‘𝑠) “ (1...𝑗)) ∩ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁))) = ∅) → ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})):(((2^nd ‘𝑠) “ (1...𝑗)) ∪ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
115	101, 113, 114	sylancr 599	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (0...𝑁)) → ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})):(((2^nd ‘𝑠) “ (1...𝑗)) ∪ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
116		nn0p1nn 12567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
117	105, 116	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
118		nnuz 12926	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ℕ = (ℤ_≥‘1)
119	117, 118	eleqtrdi 2870	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ_≥‘1))
120		elfzuz3 13575	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑗))
121		fzsplit2 13604	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
122	119, 120, 121	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
123	122	imaeq2d 6056	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘𝑠) “ (1...𝑁)) = ((2^nd ‘𝑠) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
124		imaundi 6141	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2^nd ‘𝑠) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (((2^nd ‘𝑠) “ (1...𝑗)) ∪ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)))
125	123, 124	eqtr2di 2812	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑁) → (((2^nd ‘𝑠) “ (1...𝑗)) ∪ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁))) = ((2^nd ‘𝑠) “ (1...𝑁)))
126		f1ofo 6825	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘𝑠):(1...𝑁)–onto→(1...𝑁))
127		foima 6794	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2^nd ‘𝑠):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘𝑠) “ (1...𝑁)) = (1...𝑁))
128	126, 127	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁) → ((2^nd ‘𝑠) “ (1...𝑁)) = (1...𝑁))
129	125, 128	sylan9eqr 2817	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (0...𝑁)) → (((2^nd ‘𝑠) “ (1...𝑗)) ∪ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁))) = (1...𝑁))
130	129	feq2d 6686	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (0...𝑁)) → (((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})):(((2^nd ‘𝑠) “ (1...𝑗)) ∪ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}) ↔ ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
131	115, 130	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (0...𝑁)) → ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
132	96, 131	sylan 592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑗 ∈ (0...𝑁)) → ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
133		fzfid 14037	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
134		inidm 4172	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
135	87, 91, 132, 133, 133, 134	off 7696	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑗 ∈ (0...𝑁)) → ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
136		ovex 7446	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
137		feq1 6680	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)))
138	137	anbi2d 642	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) ↔ (𝜑 ∧ ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))))
139		poimirlem28.1	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
140	139	eleq1d 2845	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝐵 ∈ (0...𝑁) ↔ 𝐶 ∈ (0...𝑁)))
141	138, 140	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) ↔ ((𝜑 ∧ ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)) → 𝐶 ∈ (0...𝑁))))
142		poimirlem28.2	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
143	136, 141, 142	vtocl 3520	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)) → 𝐶 ∈ (0...𝑁))
144	135, 143	sylan2 605	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑗 ∈ (0...𝑁))) → 𝐶 ∈ (0...𝑁))
145	144	an12s 662	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (𝜑 ∧ 𝑗 ∈ (0...𝑁))) → 𝐶 ∈ (0...𝑁))
146	145	ex 418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝐶 ∈ (0...𝑁)))
147	61, 65, 68, 146	vtoclgaf 3535	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑇, 𝑈⟩ ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁)))
148	60, 147	mpcom 39	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁))
149	39, 43, 148	chvarfv 2276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁))
150		poimir.0	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℕ)
151	150	nnnn0d 12589	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
152		nn0uz 12925	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ₀ = (ℤ_≥‘0)
153	151, 152	eleqtrdi 2870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
154		fzm1 13662	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁) ↔ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)))
155	153, 154	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁) ↔ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)))
156	155	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁) ↔ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)))
157	149, 156	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁))
158	157	ord 878	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁))
159	36, 158	mt3d 149	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
160	159	adantrr 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
161		fzfi 14036	. . . . . . . . . . . . . . 15 ⊢ (0...(𝑁 − 1)) ∈ Fin
162	150	nncnd 12273	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
163		1cnd 11226	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 ∈ ℂ)
164	162, 163, 163	addsubd 11614	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 + 1) − 1) = ((𝑁 − 1) + 1))
165		hashfz0 14497	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(0...𝑁)) = (𝑁 + 1))
166	151, 165	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘(0...𝑁)) = (𝑁 + 1))
167	166	oveq1d 7428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((♯‘(0...𝑁)) − 1) = ((𝑁 + 1) − 1))
168		nnm1nn0 12569	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
169		hashfz0 14497	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ ℕ₀ → (♯‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
170	150, 168, 169	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
171	164, 167, 170	3eqtr4rd 2806	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘(0...(𝑁 − 1))) = ((♯‘(0...𝑁)) − 1))
172		hashdifsn 14479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0...𝑁) ∈ Fin ∧ 𝑡 ∈ (0...𝑁)) → (♯‘((0...𝑁) ∖ {𝑡})) = ((♯‘(0...𝑁)) − 1))
173	8, 172	mpan 703	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (0...𝑁) → (♯‘((0...𝑁) ∖ {𝑡})) = ((♯‘(0...𝑁)) − 1))
174	173	eqcomd 2766	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (0...𝑁) → ((♯‘(0...𝑁)) − 1) = (♯‘((0...𝑁) ∖ {𝑡})))
175	171, 174	sylan9eq 2815	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (♯‘(0...(𝑁 − 1))) = (♯‘((0...𝑁) ∖ {𝑡})))
176		diffi 9169	. . . . . . . . . . . . . . . . . 18 ⊢ ((0...𝑁) ∈ Fin → ((0...𝑁) ∖ {𝑡}) ∈ Fin)
177	8, 176	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((0...𝑁) ∖ {𝑡}) ∈ Fin
178		hashen 14411	. . . . . . . . . . . . . . . . 17 ⊢ (((0...(𝑁 − 1)) ∈ Fin ∧ ((0...𝑁) ∖ {𝑡}) ∈ Fin) → ((♯‘(0...(𝑁 − 1))) = (♯‘((0...𝑁) ∖ {𝑡})) ↔ (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑡})))
179	161, 177, 178	mp2an 705	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘(0...(𝑁 − 1))) = (♯‘((0...𝑁) ∖ {𝑡})) ↔ (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑡}))
180	175, 179	sylib 221	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑡}))
181		phpreu 38358	. . . . . . . . . . . . . . 15 ⊢ (((0...(𝑁 − 1)) ∈ Fin ∧ (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑡})) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃!𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
182	161, 180, 181	sylancr 599	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃!𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
183	182	biimpd 232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃!𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
184	183	impr 460	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃!𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
185		nfv 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧 𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
186		nfcsb1v 3871	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
187	186	nfeq2 2939	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗 𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
188		csbeq1a 3861	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑧 → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
189	188	eqeq2d 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑧 → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
190	185, 187, 189	cbvreuw 3391	. . . . . . . . . . . . . 14 ⊢ (∃!𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
191		eqeq1 2764	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
192	191	reubidv 3381	. . . . . . . . . . . . . 14 ⊢ (𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
193	190, 192	bitrid 286	. . . . . . . . . . . . 13 ⊢ (𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (∃!𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
194	193	rspcv 3572	. . . . . . . . . . . 12 ⊢ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) → (∀𝑖 ∈ (0...(𝑁 − 1))∃!𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
195	160, 184, 194	sylc 66	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) → ∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
196		an32 659	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ↔ ((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
197	196	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (((𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → ((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
198	197	adantll 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → ((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
199		eqeq2 2772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
200		rexsns 4632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑗 ∈ {𝑡}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ [𝑡 / 𝑗]𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
201	29	nfeq2 2939	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑗 𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
202	31	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑡 → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
203	201, 202	sbciegf 3777	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 ∈ V → ([𝑡 / 𝑗]𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
204	7, 203	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ([𝑡 / 𝑗]𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
205	200, 204	bitri 278	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑗 ∈ {𝑡}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
206		rexsns 4632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑗 ∈ {𝑧}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ [𝑧 / 𝑗]𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
207		vex 3454	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑧 ∈ V
208	187, 189	sbciegf 3777	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑗]𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
209	207, 208	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ([𝑧 / 𝑗]𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
210	206, 209	bitri 278	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑗 ∈ {𝑧}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
211	199, 205, 210	3bitr4g 317	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (∃𝑗 ∈ {𝑡}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ {𝑧}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
212	211	orbi1d 930	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ((∃𝑗 ∈ {𝑡}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∨ ∃𝑗 ∈ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ (∃𝑗 ∈ {𝑧}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∨ ∃𝑗 ∈ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
213		rexun 4142	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑗 ∈ ({𝑡} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ (∃𝑗 ∈ {𝑡}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∨ ∃𝑗 ∈ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
214		rexun 4142	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑗 ∈ ({𝑧} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ (∃𝑗 ∈ {𝑧}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∨ ∃𝑗 ∈ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
215	212, 213, 214	3bitr4g 317	. . . . . . . . . . . . . . . . . . 19 ⊢ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (∃𝑗 ∈ ({𝑡} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ({𝑧} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
216	215	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → (∃𝑗 ∈ ({𝑡} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ({𝑧} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
217		eldifsni 4753	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) → 𝑧 ≠ 𝑡)
218	217	necomd 3010	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) → 𝑡 ≠ 𝑧)
219		dif32 4248	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}) = (((0...𝑁) ∖ {𝑧}) ∖ {𝑡})
220	219	uneq2i 4112	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑡} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})) = ({𝑡} ∪ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡}))
221		snssi 4746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 ∈ ((0...𝑁) ∖ {𝑧}) → {𝑡} ⊆ ((0...𝑁) ∖ {𝑧}))
222		eldifsn 4748	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 ∈ ((0...𝑁) ∖ {𝑧}) ↔ (𝑡 ∈ (0...𝑁) ∧ 𝑡 ≠ 𝑧))
223		undif 4438	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({𝑡} ⊆ ((0...𝑁) ∖ {𝑧}) ↔ ({𝑡} ∪ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡})) = ((0...𝑁) ∖ {𝑧}))
224	221, 222, 223	3imtr3i 294	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 ∈ (0...𝑁) ∧ 𝑡 ≠ 𝑧) → ({𝑡} ∪ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡})) = ((0...𝑁) ∖ {𝑧}))
225	220, 224	eqtrid 2807	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑡 ∈ (0...𝑁) ∧ 𝑡 ≠ 𝑧) → ({𝑡} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})) = ((0...𝑁) ∖ {𝑧}))
226	218, 225	sylan2 605	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → ({𝑡} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})) = ((0...𝑁) ∖ {𝑧}))
227	226	rexeqdv 3320	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → (∃𝑗 ∈ ({𝑡} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
228	227	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → (∃𝑗 ∈ ({𝑡} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
229		snssi 4746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) → {𝑧} ⊆ ((0...𝑁) ∖ {𝑡}))
230		undif 4438	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑧} ⊆ ((0...𝑁) ∖ {𝑡}) ↔ ({𝑧} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})) = ((0...𝑁) ∖ {𝑡}))
231	229, 230	sylib 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) → ({𝑧} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})) = ((0...𝑁) ∖ {𝑡}))
232	231	rexeqdv 3320	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) → (∃𝑗 ∈ ({𝑧} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
233	232	ad2antlr 740	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → (∃𝑗 ∈ ({𝑧} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
234	216, 228, 233	3bitr3d 312	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
235	234	ralbidv 3185	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
236	235	biimpar 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
237	236	an32s 665	. . . . . . . . . . . . . 14 ⊢ ((((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
238	198, 237	sylan 592	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
239		simpl 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → 𝑡 ∈ (0...𝑁))
240	239	anim2i 629	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) → (𝜑 ∧ 𝑡 ∈ (0...𝑁)))
241		necom 3008	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ≠ 𝑡 ↔ 𝑡 ≠ 𝑧)
242	241	bilani 510	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ (0...𝑁) ∧ 𝑧 ≠ 𝑡) → 𝑡 ≠ 𝑧)
243	242	anim2i 629	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑡 ∈ (0...𝑁) ∧ (𝑧 ∈ (0...𝑁) ∧ 𝑧 ≠ 𝑡)) → (𝑡 ∈ (0...𝑁) ∧ 𝑡 ≠ 𝑧))
244		eldifsn 4748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) ↔ (𝑧 ∈ (0...𝑁) ∧ 𝑧 ≠ 𝑡))
245	244	anbi2i 635	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ↔ (𝑡 ∈ (0...𝑁) ∧ (𝑧 ∈ (0...𝑁) ∧ 𝑧 ≠ 𝑡)))
246	243, 245, 222	3imtr4i 295	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → 𝑡 ∈ ((0...𝑁) ∖ {𝑧}))
247	246	adantll 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → 𝑡 ∈ ((0...𝑁) ∖ {𝑧}))
248	247	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → 𝑡 ∈ ((0...𝑁) ∖ {𝑧}))
249	29	nfel1 2938	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑗⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))
250	37, 249	nfim 1929	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
251	31	eleq1d 2845	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑡 → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
252	41, 251	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑡 → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))) ↔ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))))
253	26	necomd 3010	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ≠ 𝑁)
254	253	neneqd 2960	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ¬ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)
255		fzm1 13662	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁) ↔ (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)))
256	153, 255	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁) ↔ (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)))
257	256	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁) ↔ (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)))
258	148, 257	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁))
259	258	ord 878	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (¬ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁))
260	254, 259	mt3d 149	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
261	250, 252, 260	chvarfv 2276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
262	261	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
263		eldifi 4078	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) → 𝑧 ∈ (0...𝑁))
264		eleq1w 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = 𝑧 → (𝑡 ∈ (0...𝑁) ↔ 𝑧 ∈ (0...𝑁)))
265	264	anbi2d 642	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑧 → ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑧 ∈ (0...𝑁))))
266		sneq 4594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑡 = 𝑧 → {𝑡} = {𝑧})
267	266	difeq2d 4074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = 𝑧 → ((0...𝑁) ∖ {𝑡}) = ((0...𝑁) ∖ {𝑧}))
268	267	breq2d 5115	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑧 → ((0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑡}) ↔ (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧})))
269	265, 268	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑧 → (((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑡})) ↔ ((𝜑 ∧ 𝑧 ∈ (0...𝑁)) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧}))))
270	269, 180	chvarvv 2022	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑧 ∈ (0...𝑁)) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧}))
271	263, 270	sylan2 605	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧}))
272	271	adantlr 728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧}))
273		phpreu 38358	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((0...(𝑁 − 1)) ∈ Fin ∧ (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧})) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃!𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
274	161, 273	mpan 703	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃!𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
275	274	biimpa 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧}) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃!𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
276	272, 275	sylan 592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃!𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
277		eqeq1 2764	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
278	277	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑧})) → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
279	201, 278	reubida 3389	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (∃!𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃!𝑗 ∈ ((0...𝑁) ∖ {𝑧})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
280	279	rspcv 3572	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) → (∀𝑖 ∈ (0...(𝑁 − 1))∃!𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃!𝑗 ∈ ((0...𝑁) ∖ {𝑧})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
281	262, 276, 280	sylc 66	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃!𝑗 ∈ ((0...𝑁) ∖ {𝑧})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
282		reurmo 3368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃!𝑗 ∈ ((0...𝑁) ∖ {𝑧})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃*𝑗 ∈ ((0...𝑁) ∖ {𝑧})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
283	281, 282	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃*𝑗 ∈ ((0...𝑁) ∖ {𝑧})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
284		nfv 1947	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑖⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
285	284	rmo3 3836	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃*𝑗 ∈ ((0...𝑁) ∖ {𝑧})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ ((0...𝑁) ∖ {𝑧})∀𝑖 ∈ ((0...𝑁) ∖ {𝑧})((⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ [𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → 𝑗 = 𝑖))
286	283, 285	sylib 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∀𝑗 ∈ ((0...𝑁) ∖ {𝑧})∀𝑖 ∈ ((0...𝑁) ∖ {𝑧})((⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ [𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → 𝑗 = 𝑖))
287		equcomi 2050	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝑡 → 𝑡 = 𝑖)
288	287	csbeq1d 3851	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = 𝑡 → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑖 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
289		sbsbc 3743	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ([𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ [𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
290		vex 3454	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑖 ∈ V
291		sbceqg 4370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ V → ([𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ⦋𝑖 / 𝑗⦌⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑖 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
292	29	csbconstgf 3865	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 ∈ V → ⦋𝑖 / 𝑗⦌⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
293	292	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ V → (⦋𝑖 / 𝑗⦌⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑖 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑖 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
294	291, 293	bitrd 282	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ V → ([𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑖 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
295	290, 294	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ([𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑖 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
296	289, 295	bitri 278	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ([𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑖 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
297	288, 296	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑡 → [𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
298	297	biantrud 541	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑡 → (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ [𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
299	298	bicomd 226	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑡 → ((⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ [𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
300		eqeq2 2772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑡 → (𝑗 = 𝑖 ↔ 𝑗 = 𝑡))
301	299, 300	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑡 → (((⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ [𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → 𝑗 = 𝑖) ↔ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑡)))
302	301	rspcv 3572	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ ((0...𝑁) ∖ {𝑧}) → (∀𝑖 ∈ ((0...𝑁) ∖ {𝑧})((⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ [𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → 𝑗 = 𝑖) → (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑡)))
303	302	ralimdv 3176	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ∈ ((0...𝑁) ∖ {𝑧}) → (∀𝑗 ∈ ((0...𝑁) ∖ {𝑧})∀𝑖 ∈ ((0...𝑁) ∖ {𝑧})((⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ [𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → 𝑗 = 𝑖) → ∀𝑗 ∈ ((0...𝑁) ∖ {𝑧})(⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑡)))
304	248, 286, 303	sylc 66	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∀𝑗 ∈ ((0...𝑁) ∖ {𝑧})(⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑡))
305		dif32 4248	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡}) = (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})
306	305	eleq2i 2852	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡}) ↔ 𝑗 ∈ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))
307		eldifsn 4748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡}) ↔ (𝑗 ∈ ((0...𝑁) ∖ {𝑧}) ∧ 𝑗 ≠ 𝑡))
308		eldifsn 4748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}) ↔ (𝑗 ∈ ((0...𝑁) ∖ {𝑡}) ∧ 𝑗 ≠ 𝑧))
309	306, 307, 308	3bitr3ri 305	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ ((0...𝑁) ∖ {𝑡}) ∧ 𝑗 ≠ 𝑧) ↔ (𝑗 ∈ ((0...𝑁) ∖ {𝑧}) ∧ 𝑗 ≠ 𝑡))
310	309	imbi1i 352	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑗 ∈ ((0...𝑁) ∖ {𝑡}) ∧ 𝑗 ≠ 𝑧) → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ ((𝑗 ∈ ((0...𝑁) ∖ {𝑧}) ∧ 𝑗 ≠ 𝑡) → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
311		impexp 456	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑗 ∈ ((0...𝑁) ∖ {𝑡}) ∧ 𝑗 ≠ 𝑧) → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ (𝑗 ∈ ((0...𝑁) ∖ {𝑡}) → (𝑗 ≠ 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
312		impexp 456	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑗 ∈ ((0...𝑁) ∖ {𝑧}) ∧ 𝑗 ≠ 𝑡) → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ (𝑗 ∈ ((0...𝑁) ∖ {𝑧}) → (𝑗 ≠ 𝑡 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
313	310, 311, 312	3bitr3ri 305	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ ((0...𝑁) ∖ {𝑧}) → (𝑗 ≠ 𝑡 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) ↔ (𝑗 ∈ ((0...𝑁) ∖ {𝑡}) → (𝑗 ≠ 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
314	313	albii 1852	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑗(𝑗 ∈ ((0...𝑁) ∖ {𝑧}) → (𝑗 ≠ 𝑡 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) ↔ ∀𝑗(𝑗 ∈ ((0...𝑁) ∖ {𝑡}) → (𝑗 ≠ 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
315		con34b 319	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑡) ↔ (¬ 𝑗 = 𝑡 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
316		df-ne 2956	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ≠ 𝑡 ↔ ¬ 𝑗 = 𝑡)
317	316	imbi1i 352	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ≠ 𝑡 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ (¬ 𝑗 = 𝑡 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
318	315, 317	bitr4i 281	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑡) ↔ (𝑗 ≠ 𝑡 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
319	318	ralbii 3108	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑗 ∈ ((0...𝑁) ∖ {𝑧})(⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑡) ↔ ∀𝑗 ∈ ((0...𝑁) ∖ {𝑧})(𝑗 ≠ 𝑡 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
320		df-ral 3077	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑗 ∈ ((0...𝑁) ∖ {𝑧})(𝑗 ≠ 𝑡 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ ∀𝑗(𝑗 ∈ ((0...𝑁) ∖ {𝑧}) → (𝑗 ≠ 𝑡 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
321	319, 320	bitri 278	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑗 ∈ ((0...𝑁) ∖ {𝑧})(⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑡) ↔ ∀𝑗(𝑗 ∈ ((0...𝑁) ∖ {𝑧}) → (𝑗 ≠ 𝑡 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
322		df-ral 3077	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑗 ∈ ((0...𝑁) ∖ {𝑡})(𝑗 ≠ 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ ∀𝑗(𝑗 ∈ ((0...𝑁) ∖ {𝑡}) → (𝑗 ≠ 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
323	314, 321, 322	3bitr4i 306	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑗 ∈ ((0...𝑁) ∖ {𝑧})(⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑡) ↔ ∀𝑗 ∈ ((0...𝑁) ∖ {𝑡})(𝑗 ≠ 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
324	304, 323	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∀𝑗 ∈ ((0...𝑁) ∖ {𝑡})(𝑗 ≠ 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
325		df-ne 2956	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ≠ 𝑧 ↔ ¬ 𝑗 = 𝑧)
326	325	imbi1i 352	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ≠ 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ (¬ 𝑗 = 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
327		con34b 319	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑧) ↔ (¬ 𝑗 = 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
328	326, 327	bitr4i 281	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ≠ 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑧))
329		ancr 556	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑗 = 𝑧) → (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
330	328, 329	sylbi 220	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ≠ 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
331	330	ralimi 3099	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑗 ∈ ((0...𝑁) ∖ {𝑡})(𝑗 ≠ 𝑧 → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∀𝑗 ∈ ((0...𝑁) ∖ {𝑡})(⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
332	324, 331	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∀𝑗 ∈ ((0...𝑁) ∖ {𝑡})(⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
333	240, 332	sylanl1 693	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∀𝑗 ∈ ((0...𝑁) ∖ {𝑡})(⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
334	201, 277	rexbid 3276	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
335	334	rspcva 3574	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
336	261, 335	sylan 592	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
337	336	anasss 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
338	337	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
339		rexim 3103	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑗 ∈ ((0...𝑁) ∖ {𝑡})(⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})(𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
340	333, 338, 339	sylc 66	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})(𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
341		rexex 3092	. . . . . . . . . . . . . . 15 ⊢ (∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})(𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃𝑗(𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
342	340, 341	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃𝑗(𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
343	29, 186	nfeq 2935	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
344	188	eqeq2d 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑧 → (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
345	343, 344	equsexv 2302	. . . . . . . . . . . . . 14 ⊢ (∃𝑗(𝑗 = 𝑧 ∧ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
346	342, 345	sylib 221	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
347	238, 346	impbida 813	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
348	347	reubidva 3379	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) → (∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
349	195, 348	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) → ∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
350		an32 659	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ (0...𝑁) ∧ 𝑧 ≠ 𝑡) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ ((𝑧 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ∧ 𝑧 ≠ 𝑡))
351	244	anbi1i 636	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ((0...𝑁) ∖ {𝑡}) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ ((𝑧 ∈ (0...𝑁) ∧ 𝑧 ≠ 𝑡) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
352		sneq 4594	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
353	352	difeq2d 4074	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((0...𝑁) ∖ {𝑦}) = ((0...𝑁) ∖ {𝑧}))
354	353	rexeqdv 3320	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
355	354	ralbidv 3185	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
356	355	elrab 3645	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ↔ (𝑧 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
357	356	anbi1i 636	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∧ 𝑧 ≠ 𝑡) ↔ ((𝑧 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ∧ 𝑧 ≠ 𝑡))
358	350, 351, 357	3bitr4i 306	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ((0...𝑁) ∖ {𝑡}) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ (𝑧 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∧ 𝑧 ≠ 𝑡))
359		eldifsn 4748	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}) ↔ (𝑧 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∧ 𝑧 ≠ 𝑡))
360	358, 359	bitr4i 281	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ((0...𝑁) ∖ {𝑡}) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ 𝑧 ∈ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}))
361	360	eubii 2610	. . . . . . . . . . 11 ⊢ (∃!𝑧(𝑧 ∈ ((0...𝑁) ∖ {𝑡}) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ↔ ∃!𝑧 𝑧 ∈ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}))
362		df-reu 3366	. . . . . . . . . . 11 ⊢ (∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃!𝑧(𝑧 ∈ ((0...𝑁) ∖ {𝑡}) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
363		euhash1 14485	. . . . . . . . . . . 12 ⊢ (({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}) ∈ Fin → ((♯‘({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})))
364	12, 363	ax-mp 5	. . . . . . . . . . 11 ⊢ ((♯‘({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡}))
365	361, 362, 364	3bitr4i 306	. . . . . . . . . 10 ⊢ (∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ (♯‘({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})) = 1)
366	349, 365	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) → (♯‘({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})) = 1)
367	25, 366	sylan2b 606	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}) → (♯‘({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})) = 1)
368	367	oveq1d 7428	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}) → ((♯‘({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})) + 1) = (1 + 1))
369	20, 368	eqtr3d 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}) → (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}) = (1 + 1))
370	6, 369	breqtrrid 5143	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}) → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}))
371	370	ex 418	. . . 4 ⊢ (𝜑 → (𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶})))
372	371	exlimdv 1966	. . 3 ⊢ (𝜑 → (∃𝑡 𝑡 ∈ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶})))
373	1, 372	biimtrid 245	. 2 ⊢ (𝜑 → (¬ {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} = ∅ → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶})))
374		dvds0 16361	. . . . 5 ⊢ (2 ∈ ℤ → 2 ∥ 0)
375	2, 374	ax-mp 5	. . . 4 ⊢ 2 ∥ 0
376		hash0 14431	. . . 4 ⊢ (♯‘∅) = 0
377	375, 376	breqtrri 5132	. . 3 ⊢ 2 ∥ (♯‘∅)
378		fveq2 6878	. . 3 ⊢ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} = ∅ → (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}) = (♯‘∅))
379	377, 378	breqtrrid 5143	. 2 ⊢ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} = ∅ → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}))
380	373, 379	pm2.61d2 183	1 ⊢ (𝜑 → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∀wal 1568 = wceq 1570 ∃wex 1812 [wsb 2099 ∈ wcel 2145 ∃!weu 2593 {cab 2738 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ∃!wreu 3363 ∃*wrmo 3364 {crab 3412 Vcvv 3450 [wsbc 3739 ⦋csb 3847 ∖ cdif 3896 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {csn 4584 ⟨cop 4590 class class class wbr 5103 × cxp 5653 ^◡ccnv 5654 “ cima 5658 Fun wfun 6527 ⟶wf 6529 –onto→wfo 6531 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7413 ∘_f cof 7676 1^st c1st 7984 2^nd c2nd 7985 ↑_m cmap 8826 ≈ cen 8949 Fincfn 8952 0cc0 11124 1c1 11125 + caddc 11127 < clt 11267 − cmin 11465 ℕcn 12257 2c2 12319 ℕ₀cn0 12528 ℤcz 12615 ℤ_≥cuz 12887 ...cfz 13561 ..^cfzo 13709 ♯chash 14394 ∥ cdvds 16342

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 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 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-oadd 8459 df-er 8696 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-dju 9906 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-n0 12529 df-z 12616 df-uz 12888 df-fz 13562 df-fzo 13710 df-hash 14395 df-dvds 16343

This theorem is used by: poimirlem26 38395

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