poimirlem25 - Mathbox for Brendan Leahy

	Mathbox for Brendan Leahy		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > poimirlem25		Structured version Visualization version GIF version

Theorem poimirlem25 35729

Description: Lemma for poimir 35737 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 4276	. . 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 12282	. . . . . . . 8 ⊢ 2 ∈ ℤ
3		iddvds 15907	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∥ 2)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ 2 ∥ 2
5		df-2 11966	. . . . . . 7 ⊢ 2 = (1 + 1)
6	4, 5	breqtri 5095	. . . . . 6 ⊢ 2 ∥ (1 + 1)
7		vex 3426	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
8		fzfi 13620	. . . . . . . . . . . . 13 ⊢ (0...𝑁) ∈ Fin
9		rabfi 8973	. . . . . . . . . . . . 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 8979	. . . . . . . . . . . 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 4722	. . . . . . . . . . 11 ⊢ ¬ 𝑡 ∈ ({𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶} ∖ {𝑡})
14	12, 13	pm3.2i 470	. . . . . . . . . 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 14035	. . . . . . . . . 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 4740	. . . . . . . . . 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 6760	. . . . . . . . 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 2793	. . . . . . . 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 481	. . . . . . 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 4568	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → {𝑦} = {𝑡})
22	21	difeq2d 4053	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → ((0...𝑁) ∖ {𝑦}) = ((0...𝑁) ∖ {𝑡}))
23	22	rexeqdv 3340	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
24	23	ralbidv 3120	. . . . . . . . . 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 3617	. . . . . . . . 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 3107	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
28		nfcv 2906	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗𝑁
29		nfcsb1v 3853	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
30	28, 29	nfne 3044	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗 𝑁 ≠ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
31		csbeq1a 3842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑡 → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
32	31	neeq2d 3003	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑡 → (𝑁 ≠ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
33	30, 32	rspc 3539	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (0...𝑁) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑁 ≠ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
34	27, 33	mpan9 506	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → 𝑁 ≠ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
35		nesym 2999	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ≠ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)
36	34, 35	sylib 217	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ¬ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)
37		nfv 1918	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑡 ∈ (0...𝑁))
38	29	nfel1 2922	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁)
39	37, 38	nfim 1900	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁))
40		eleq1w 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑡 → (𝑗 ∈ (0...𝑁) ↔ 𝑡 ∈ (0...𝑁)))
41	40	anbi2d 628	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑡 → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ (0...𝑁))))
42	31	eleq1d 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑡 → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁) ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁)))
43	41, 42	imbi12d 344	. . . . . . . . . . . . . . . . 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 7288	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0..^𝐾) ∈ V
46		ovex 7288	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1...𝑁) ∈ V
47	45, 46	elmap 8617	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑇 ∈ ((0..^𝐾) ↑_m (1...𝑁)) ↔ 𝑇:(1...𝑁)⟶(0..^𝐾))
48	44, 47	sylibr 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑇 ∈ ((0..^𝐾) ↑_m (1...𝑁)))
49		poimirlem25.4	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
50		fzfi 13620	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1...𝑁) ∈ Fin
51		f1oexrnex 7748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ∧ (1...𝑁) ∈ Fin) → 𝑈 ∈ V)
52	50, 51	mpan2 687	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 ∈ V)
53		f1oeq1 6688	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = 𝑈 → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ 𝑈:(1...𝑁)–1-1-onto→(1...𝑁)))
54	53	elabg 3600	. . . . . . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . . . . 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 267	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
57	49, 56	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
58		opelxpi 5617	. . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨𝑇, 𝑈⟩ ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
60	59	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⟨𝑇, 𝑈⟩ ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
61		nfcv 2906	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠⟨𝑇, 𝑈⟩
62		nfv 1918	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑠(𝜑 ∧ 𝑗 ∈ (0...𝑁))
63		nfcsb1v 3853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑠⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
64	63	nfel1 2922	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑠⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁)
65	62, 64	nfim 1900	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁))
66		csbeq1a 3842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ⟨𝑇, 𝑈⟩ → 𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
67	66	eleq1d 2823	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = ⟨𝑇, 𝑈⟩ → (𝐶 ∈ (0...𝑁) ↔ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁)))
68	67	imbi2d 340	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ⟨𝑇, 𝑈⟩ → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝐶 ∈ (0...𝑁)) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁))))
69		elun 4079	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑝 ∈ ({1} ∪ {0}) ↔ (𝑝 ∈ {1} ∨ 𝑝 ∈ {0}))
70		fzofzp1 13412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑖 + 1) ∈ (0...𝐾))
71		elsni 4575	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑝 ∈ {1} → 𝑝 = 1)
72	71	oveq2d 7271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑝 ∈ {1} → (𝑖 + 𝑝) = (𝑖 + 1))
73	72	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑝 ∈ {1} → ((𝑖 + 𝑝) ∈ (0...𝐾) ↔ (𝑖 + 1) ∈ (0...𝐾)))
74	70, 73	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑝 ∈ {1} → (𝑖 + 𝑝) ∈ (0...𝐾)))
75		elfzonn0 13360	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑖 ∈ (0..^𝐾) → 𝑖 ∈ ℕ₀)
76	75	nn0cnd 12225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 ∈ (0..^𝐾) → 𝑖 ∈ ℂ)
77	76	addid1d 11105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑖 + 0) = 𝑖)
78		elfzofz 13331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝐾) → 𝑖 ∈ (0...𝐾))
79	77, 78	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑖 + 0) ∈ (0...𝐾))
80		elsni 4575	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑝 ∈ {0} → 𝑝 = 0)
81	80	oveq2d 7271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑝 ∈ {0} → (𝑖 + 𝑝) = (𝑖 + 0))
82	81	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑝 ∈ {0} → ((𝑖 + 𝑝) ∈ (0...𝐾) ↔ (𝑖 + 0) ∈ (0...𝐾)))
83	79, 82	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑝 ∈ {0} → (𝑖 + 𝑝) ∈ (0...𝐾)))
84	74, 83	jaod 855	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝐾) → ((𝑝 ∈ {1} ∨ 𝑝 ∈ {0}) → (𝑖 + 𝑝) ∈ (0...𝐾)))
85	69, 84	syl5bi 241	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (0..^𝐾) → (𝑝 ∈ ({1} ∪ {0}) → (𝑖 + 𝑝) ∈ (0...𝐾)))
86	85	imp 406	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝐾) ∧ 𝑝 ∈ ({1} ∪ {0})) → (𝑖 + 𝑝) ∈ (0...𝐾))
87	86	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 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 7836	. . . . . . . . . . . . . . . . . . . . . . . . 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 8595	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((1^st ‘𝑠) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘𝑠):(1...𝑁)⟶(0..^𝐾))
90	88, 89	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . 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 7837	. . . . . . . . . . . . . . . . . . . . . . . . 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 6769	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (2^nd ‘𝑠) ∈ V
94		f1oeq1 6688	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3602	. . . . . . . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . . . . . . 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 10902	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ V
98	97	fconst 6644	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘𝑠) “ (1...𝑗)) × {1}):((2^nd ‘𝑠) “ (1...𝑗))⟶{1}
99		c0ex 10900	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 ∈ V
100	99	fconst 6644	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}):((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁))⟶{0}
101	98, 100	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6706	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6503	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 504	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 13278	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ₀)
106	105	nn0red 12224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
107	106	ltp1d 11835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
108		fzdisj 13212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
109	107, 108	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
110	109	imaeq2d 5958	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘𝑠) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((2^nd ‘𝑠) “ ∅))
111		ima0 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2^nd ‘𝑠) “ ∅) = ∅
112	110, 111	eqtrdi 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘𝑠) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
113	104, 112	sylan9req 2800	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6620	. . . . . . . . . . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . . . . . . . . . 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 12202	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
117	105, 116	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
118		nnuz 12550	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ℕ = (ℤ_≥‘1)
119	117, 118	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ_≥‘1))
120		elfzuz3 13182	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑗))
121		fzsplit2 13210	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
122	119, 120, 121	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
123	122	imaeq2d 5958	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘𝑠) “ (1...𝑁)) = ((2^nd ‘𝑠) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
124		imaundi 6042	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2^nd ‘𝑠) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (((2^nd ‘𝑠) “ (1...𝑗)) ∪ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)))
125	123, 124	eqtr2di 2796	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑁) → (((2^nd ‘𝑠) “ (1...𝑗)) ∪ ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁))) = ((2^nd ‘𝑠) “ (1...𝑁)))
126		f1ofo 6707	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘𝑠):(1...𝑁)–onto→(1...𝑁))
127		foima 6677	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2^nd ‘𝑠):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘𝑠) “ (1...𝑁)) = (1...𝑁))
128	126, 127	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2^nd ‘𝑠):(1...𝑁)–1-1-onto→(1...𝑁) → ((2^nd ‘𝑠) “ (1...𝑁)) = (1...𝑁))
129	125, 128	sylan9eqr 2801	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6570	. . . . . . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . . . . . 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 579	. . . . . . . . . . . . . . . . . . . . . . 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 13621	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
134		inidm 4149	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
135	87, 91, 132, 133, 133, 134	off 7529	. . . . . . . . . . . . . . . . . . . . . 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 7288	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
137		feq1 6565	. . . . . . . . . . . . . . . . . . . . . . . . 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 628	. . . . . . . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . . . . . . 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 3488	. . . . . . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . . . . . 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 645	. . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . 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 3502	. . . . . . . . . . . . . . . . . 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 38	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁))
149	39, 43, 148	chvarfv 2236	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁))
150		poimir.0	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℕ)
151	150	nnnn0d 12223	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
152		nn0uz 12549	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ₀ = (ℤ_≥‘0)
153	151, 152	eleqtrdi 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
154		fzm1 13265	. . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁))
158	157	ord 860	. . . . . . . . . . . . . 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 148	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
160	159	adantrr 713	. . . . . . . . . . . 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 13620	. . . . . . . . . . . . . . 15 ⊢ (0...(𝑁 − 1)) ∈ Fin
162	150	nncnd 11919	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
163		1cnd 10901	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 ∈ ℂ)
164	162, 163, 163	addsubd 11283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 + 1) − 1) = ((𝑁 − 1) + 1))
165		hashfz0 14075	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(0...𝑁)) = (𝑁 + 1))
166	151, 165	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘(0...𝑁)) = (𝑁 + 1))
167	166	oveq1d 7270	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((♯‘(0...𝑁)) − 1) = ((𝑁 + 1) − 1))
168		nnm1nn0 12204	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
169		hashfz0 14075	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ ℕ₀ → (♯‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
170	150, 168, 169	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
171	164, 167, 170	3eqtr4rd 2789	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘(0...(𝑁 − 1))) = ((♯‘(0...𝑁)) − 1))
172		hashdifsn 14057	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0...𝑁) ∈ Fin ∧ 𝑡 ∈ (0...𝑁)) → (♯‘((0...𝑁) ∖ {𝑡})) = ((♯‘(0...𝑁)) − 1))
173	8, 172	mpan 686	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (0...𝑁) → (♯‘((0...𝑁) ∖ {𝑡})) = ((♯‘(0...𝑁)) − 1))
174	173	eqcomd 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (0...𝑁) → ((♯‘(0...𝑁)) − 1) = (♯‘((0...𝑁) ∖ {𝑡})))
175	171, 174	sylan9eq 2799	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (♯‘(0...(𝑁 − 1))) = (♯‘((0...𝑁) ∖ {𝑡})))
176		diffi 8979	. . . . . . . . . . . . . . . . . 18 ⊢ ((0...𝑁) ∈ Fin → ((0...𝑁) ∖ {𝑡}) ∈ Fin)
177	8, 176	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((0...𝑁) ∖ {𝑡}) ∈ Fin
178		hashen 13989	. . . . . . . . . . . . . . . . 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 688	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘(0...(𝑁 − 1))) = (♯‘((0...𝑁) ∖ {𝑡})) ↔ (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑡}))
180	175, 179	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑡}))
181		phpreu 35688	. . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . 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 228	. . . . . . . . . . . . 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 454	. . . . . . . . . . . 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 1918	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧 𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
186		nfcsb1v 3853	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
187	186	nfeq2 2923	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗 𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
188		csbeq1a 3842	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑧 → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
189	188	eqeq2d 2749	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑧 → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
190	185, 187, 189	cbvreuw 3365	. . . . . . . . . . . . . 14 ⊢ (∃!𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃!𝑧 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
191		eqeq1 2742	. . . . . . . . . . . . . . 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 3315	. . . . . . . . . . . . . 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	syl5bb 282	. . . . . . . . . . . . 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 3547	. . . . . . . . . . . 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 65	. . . . . . . . . . 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 642	. . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . 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 710	. . . . . . . . . . . . . 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 2750	. . . . . . . . . . . . . . . . . . . . . 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 4602	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑗 ∈ {𝑡}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ [𝑡 / 𝑗]𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
201	29	nfeq2 2923	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑗 𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
202	31	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑡 → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
203	201, 202	sbciegf 3750	. . . . . . . . . . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑗 ∈ {𝑡}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
206		rexsns 4602	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑗 ∈ {𝑧}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ [𝑧 / 𝑗]𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
207		vex 3426	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑧 ∈ V
208	187, 189	sbciegf 3750	. . . . . . . . . . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑗 ∈ {𝑧}𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
211	199, 205, 210	3bitr4g 313	. . . . . . . . . . . . . . . . . . . . 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 913	. . . . . . . . . . . . . . . . . . . 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 4120	. . . . . . . . . . . . . . . . . . . 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 4120	. . . . . . . . . . . . . . . . . . . 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 313	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 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 4720	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) → 𝑧 ≠ 𝑡)
218	217	necomd 2998	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) → 𝑡 ≠ 𝑧)
219		dif32 4223	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}) = (((0...𝑁) ∖ {𝑧}) ∖ {𝑡})
220	219	uneq2i 4090	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑡} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})) = ({𝑡} ∪ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡}))
221		snssi 4738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 ∈ ((0...𝑁) ∖ {𝑧}) → {𝑡} ⊆ ((0...𝑁) ∖ {𝑧}))
222		eldifsn 4717	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 ∈ ((0...𝑁) ∖ {𝑧}) ↔ (𝑡 ∈ (0...𝑁) ∧ 𝑡 ≠ 𝑧))
223		undif 4412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({𝑡} ⊆ ((0...𝑁) ∖ {𝑧}) ↔ ({𝑡} ∪ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡})) = ((0...𝑁) ∖ {𝑧}))
224	221, 222, 223	3imtr3i 290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 ∈ (0...𝑁) ∧ 𝑡 ≠ 𝑧) → ({𝑡} ∪ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡})) = ((0...𝑁) ∖ {𝑧}))
225	220, 224	syl5eq 2791	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑡 ∈ (0...𝑁) ∧ 𝑡 ≠ 𝑧) → ({𝑡} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})) = ((0...𝑁) ∖ {𝑧}))
226	218, 225	sylan2 592	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → ({𝑡} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})) = ((0...𝑁) ∖ {𝑧}))
227	226	rexeqdv 3340	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 4738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) → {𝑧} ⊆ ((0...𝑁) ∖ {𝑡}))
230		undif 4412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑧} ⊆ ((0...𝑁) ∖ {𝑡}) ↔ ({𝑧} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})) = ((0...𝑁) ∖ {𝑡}))
231	229, 230	sylib 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) → ({𝑧} ∪ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})) = ((0...𝑁) ∖ {𝑡}))
232	231	rexeqdv 3340	. . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . 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 308	. . . . . . . . . . . . . . . . 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 3120	. . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 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 648	. . . . . . . . . . . . . 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 579	. . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → 𝑡 ∈ (0...𝑁))
240	239	anim2i 616	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑡})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)) → (𝜑 ∧ 𝑡 ∈ (0...𝑁)))
241		necom 2996	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ≠ 𝑡 ↔ 𝑡 ≠ 𝑧)
242	241	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ≠ 𝑡 → 𝑡 ≠ 𝑧)
243	242	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ (0...𝑁) ∧ 𝑧 ≠ 𝑡) → 𝑡 ≠ 𝑧)
244	243	anim2i 616	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑡 ∈ (0...𝑁) ∧ (𝑧 ∈ (0...𝑁) ∧ 𝑧 ≠ 𝑡)) → (𝑡 ∈ (0...𝑁) ∧ 𝑡 ≠ 𝑧))
245		eldifsn 4717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) ↔ (𝑧 ∈ (0...𝑁) ∧ 𝑧 ≠ 𝑡))
246	245	anbi2i 622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) ↔ (𝑡 ∈ (0...𝑁) ∧ (𝑧 ∈ (0...𝑁) ∧ 𝑧 ≠ 𝑡)))
247	244, 246, 222	3imtr4i 291	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 ∈ (0...𝑁) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → 𝑡 ∈ ((0...𝑁) ∖ {𝑧}))
248	247	adantll 710	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → 𝑡 ∈ ((0...𝑁) ∖ {𝑧}))
249	248	adantr 480	. . . . . . . . . . . . . . . . . . . 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}))
250	29	nfel1 2922	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑗⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))
251	37, 250	nfim 1900	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
252	31	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑡 → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
253	41, 252	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . 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)))))
254	26	necomd 2998	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ≠ 𝑁)
255	254	neneqd 2947	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ¬ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)
256		fzm1 13265	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁) ↔ (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)))
257	153, 256	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁) ↔ (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)))
258	257	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...𝑁) ↔ (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁)))
259	148, 258	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ∨ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁))
260	259	ord 860	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (¬ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = 𝑁))
261	255, 260	mt3d 148	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
262	251, 253, 261	chvarfv 2236	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
263	262	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . 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)))
264		eldifi 4057	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ ((0...𝑁) ∖ {𝑡}) → 𝑧 ∈ (0...𝑁))
265		eleq1w 2821	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = 𝑧 → (𝑡 ∈ (0...𝑁) ↔ 𝑧 ∈ (0...𝑁)))
266	265	anbi2d 628	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑧 → ((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑧 ∈ (0...𝑁))))
267		sneq 4568	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑡 = 𝑧 → {𝑡} = {𝑧})
268	267	difeq2d 4053	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = 𝑧 → ((0...𝑁) ∖ {𝑡}) = ((0...𝑁) ∖ {𝑧}))
269	268	breq2d 5082	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑧 → ((0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑡}) ↔ (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧})))
270	266, 269	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑧 → (((𝜑 ∧ 𝑡 ∈ (0...𝑁)) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑡})) ↔ ((𝜑 ∧ 𝑧 ∈ (0...𝑁)) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧}))))
271	270, 180	chvarvv 2003	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑧 ∈ (0...𝑁)) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧}))
272	264, 271	sylan2 592	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧}))
273	272	adantlr 711	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝑁)) ∧ 𝑧 ∈ ((0...𝑁) ∖ {𝑡})) → (0...(𝑁 − 1)) ≈ ((0...𝑁) ∖ {𝑧}))
274		phpreu 35688	. . . . . . . . . . . . . . . . . . . . . . . . . 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))
275	161, 274	mpan 686	. . . . . . . . . . . . . . . . . . . . . . . . 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))
276	275	biimpa 476	. . . . . . . . . . . . . . . . . . . . . . . 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)
277	273, 276	sylan 579	. . . . . . . . . . . . . . . . . . . . . . 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)
278		eqeq1 2742	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
279	278	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 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))
280	201, 279	reubida 3313	. . . . . . . . . . . . . . . . . . . . . . . 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))
281	280	rspcv 3547	. . . . . . . . . . . . . . . . . . . . . . 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))
282	263, 277, 281	sylc 65	. . . . . . . . . . . . . . . . . . . . . 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)
283		reurmo 3354	. . . . . . . . . . . . . . . . . . . . . 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)
284	282, 283	syl 17	. . . . . . . . . . . . . . . . . . . . 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)
285		nfv 1918	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑖⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
286	285	rmo3 3818	. . . . . . . . . . . . . . . . . . . . 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))
287	284, 286	sylib 217	. . . . . . . . . . . . . . . . . . . 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))
288		equcomi 2021	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝑡 → 𝑡 = 𝑖)
289	288	csbeq1d 3832	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = 𝑡 → ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑖 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
290		sbsbc 3715	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)
291		vex 3426	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑖 ∈ V
292		sbceqg 4340	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
293	29	csbconstgf 3846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 ∈ V → ⦋𝑖 / 𝑗⦌⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
294	293	eqeq1d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
295	292, 294	bitrd 278	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
296	291, 295	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)
297	290, 296	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . 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)
298	289, 297	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑡 → [𝑖 / 𝑗]⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
299	298	biantrud 531	. . . . . . . . . . . . . . . . . . . . . . . 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)))
300	299	bicomd 222	. . . . . . . . . . . . . . . . . . . . . . 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))
301		eqeq2 2750	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑡 → (𝑗 = 𝑖 ↔ 𝑗 = 𝑡))
302	300, 301	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . 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)))
303	302	rspcv 3547	. . . . . . . . . . . . . . . . . . . . 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)))
304	303	ralimdv 3103	. . . . . . . . . . . . . . . . . . . 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)))
305	249, 287, 304	sylc 65	. . . . . . . . . . . . . . . . . . 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))
306		dif32 4223	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡}) = (((0...𝑁) ∖ {𝑡}) ∖ {𝑧})
307	306	eleq2i 2830	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡}) ↔ 𝑗 ∈ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}))
308		eldifsn 4717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (((0...𝑁) ∖ {𝑧}) ∖ {𝑡}) ↔ (𝑗 ∈ ((0...𝑁) ∖ {𝑧}) ∧ 𝑗 ≠ 𝑡))
309		eldifsn 4717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (((0...𝑁) ∖ {𝑡}) ∖ {𝑧}) ↔ (𝑗 ∈ ((0...𝑁) ∖ {𝑡}) ∧ 𝑗 ≠ 𝑧))
310	307, 308, 309	3bitr3ri 301	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ ((0...𝑁) ∖ {𝑡}) ∧ 𝑗 ≠ 𝑧) ↔ (𝑗 ∈ ((0...𝑁) ∖ {𝑧}) ∧ 𝑗 ≠ 𝑡))
311	310	imbi1i 349	. . . . . . . . . . . . . . . . . . . . . 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))
312		impexp 450	. . . . . . . . . . . . . . . . . . . . . 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)))
313		impexp 450	. . . . . . . . . . . . . . . . . . . . . 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)))
314	311, 312, 313	3bitr3ri 301	. . . . . . . . . . . . . . . . . . . . 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)))
315	314	albii 1823	. . . . . . . . . . . . . . . . . . . 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)))
316		con34b 315	. . . . . . . . . . . . . . . . . . . . . . 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))
317		df-ne 2943	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ≠ 𝑡 ↔ ¬ 𝑗 = 𝑡)
318	317	imbi1i 349	. . . . . . . . . . . . . . . . . . . . . . 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))
319	316, 318	bitr4i 277	. . . . . . . . . . . . . . . . . . . . . 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))
320	319	ralbii 3090	. . . . . . . . . . . . . . . . . . . . 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))
321		df-ral 3068	. . . . . . . . . . . . . . . . . . . . 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)))
322	320, 321	bitri 274	. . . . . . . . . . . . . . . . . . . 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)))
323		df-ral 3068	. . . . . . . . . . . . . . . . . . . 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)))
324	315, 322, 323	3bitr4i 302	. . . . . . . . . . . . . . . . . . 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))
325	305, 324	sylib 217	. . . . . . . . . . . . . . . . . 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))
326		df-ne 2943	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ≠ 𝑧 ↔ ¬ 𝑗 = 𝑧)
327	326	imbi1i 349	. . . . . . . . . . . . . . . . . . . . 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))
328		con34b 315	. . . . . . . . . . . . . . . . . . . . 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))
329	327, 328	bitr4i 277	. . . . . . . . . . . . . . . . . . . 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))
330		ancr 546	. . . . . . . . . . . . . . . . . . . 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)))
331	329, 330	sylbi 216	. . . . . . . . . . . . . . . . . . 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)))
332	331	ralimi 3086	. . . . . . . . . . . . . . . . . 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)))
333	325, 332	syl 17	. . . . . . . . . . . . . . . . 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)))
334	240, 333	sylanl1 676	. . . . . . . . . . . . . . . 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)))
335	201, 278	rexbid 3248	. . . . . . . . . . . . . . . . . . . 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))
336	335	rspcva 3550	. . . . . . . . . . . . . . . . . . 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)
337	262, 336	sylan 579	. . . . . . . . . . . . . . . . . 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)
338	337	anasss 466	. . . . . . . . . . . . . . . . 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)
339	338	ad2antrr 722	. . . . . . . . . . . . . . . 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)
340		rexim 3168	. . . . . . . . . . . . . . . 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)))
341	334, 339, 340	sylc 65	. . . . . . . . . . . . . . 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))
342		rexex 3167	. . . . . . . . . . . . . . 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))
343	341, 342	syl 17	. . . . . . . . . . . . . 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))
344	29, 186	nfeq 2919	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗⦋𝑡 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋𝑧 / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
345	188	eqeq2d 2749	. . . . . . . . . . . . . . 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))
346	344, 345	equsexv 2263	. . . . . . . . . . . . . 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)
347	343, 346	sylib 217	. . . . . . . . . . . . 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)
348	238, 347	impbida 797	. . . . . . . . . . . 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))
349	348	reubidva 3314	. . . . . . . . . . 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))
350	195, 349	mpbid 231	. . . . . . . . . 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)
351		an32 642	. . . . . . . . . . . . . 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))
352	245	anbi1i 623	. . . . . . . . . . . . . 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))
353		sneq 4568	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
354	353	difeq2d 4053	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((0...𝑁) ∖ {𝑦}) = ((0...𝑁) ∖ {𝑧}))
355	354	rexeqdv 3340	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑧})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
356	355	ralbidv 3120	. . . . . . . . . . . . . . . 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))
357	356	elrab 3617	. . . . . . . . . . . . . . 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))
358	357	anbi1i 623	. . . . . . . . . . . . . 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))
359	351, 352, 358	3bitr4i 302	. . . . . . . . . . . . 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))
360		eldifsn 4717	. . . . . . . . . . . . 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))
361	359, 360	bitr4i 277	. . . . . . . . . . . 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}))
362	361	eubii 2585	. . . . . . . . . . 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}))
363		df-reu 3070	. . . . . . . . . . 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))
364		euhash1 14063	. . . . . . . . . . . 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})))
365	12, 364	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}))
366	362, 363, 365	3bitr4i 302	. . . . . . . . . 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)
367	350, 366	sylib 217	. . . . . . . . 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)
368	25, 367	sylan2b 593	. . . . . . . 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)
369	368	oveq1d 7270	. . . . . . 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))
370	20, 369	eqtr3d 2780	. . . . . 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))
371	6, 370	breqtrrid 5108	. . . . 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}))
372	371	ex 412	. . . 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})))
373	372	exlimdv 1937	. . 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})))
374	1, 373	syl5bi 241	. 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})))
375		dvds0 15909	. . . . 5 ⊢ (2 ∈ ℤ → 2 ∥ 0)
376	2, 375	ax-mp 5	. . . 4 ⊢ 2 ∥ 0
377		hash0 14010	. . . 4 ⊢ (♯‘∅) = 0
378	376, 377	breqtrri 5097	. . 3 ⊢ 2 ∥ (♯‘∅)
379		fveq2 6756	. . 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}) = (♯‘∅))
380	378, 379	breqtrrid 5108	. 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}))
381	374, 380	pm2.61d2 181	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∀wal 1537 = wceq 1539 ∃wex 1783 [wsb 2068 ∈ wcel 2108 ∃!weu 2568 {cab 2715 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 ∃!wreu 3065 ∃*wrmo 3066 {crab 3067 Vcvv 3422 [wsbc 3711 ⦋csb 3828 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 {csn 4558 ⟨cop 4564 class class class wbr 5070 × cxp 5578 ^◡ccnv 5579 “ cima 5583 Fun wfun 6412 ⟶wf 6414 –onto→wfo 6416 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 1^st c1st 7802 2^nd c2nd 7803 ↑_m cmap 8573 ≈ cen 8688 Fincfn 8691 0cc0 10802 1c1 10803 + caddc 10805 < clt 10940 − cmin 11135 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ...cfz 13168 ..^cfzo 13311 ♯chash 13972 ∥ cdvds 15891

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-hash 13973 df-dvds 15892

This theorem is referenced by: poimirlem26 35730

W3C validator