poimirlem24 - Mathbox for Brendan Leahy

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Theorem poimirlem24 37170

Description: Lemma for poimir 37179, two ways of expressing that a simplex has an admissible face on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)

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

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

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

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

Step	Hyp	Ref	Expression
1		nfv 1909	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1)))
2		nfcsb1v 3911	. . . . . . . . . 10 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))
3		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑗(1...𝑁)
4		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑗(0...𝐾)
5	2, 3, 4	nff 6713	. . . . . . . . 9 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)
6	1, 5	nfim 1891	. . . . . . . 8 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
7		eleq1 2813	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝑗 ∈ (0...(𝑁 − 1)) ↔ 𝑦 ∈ (0...(𝑁 − 1))))
8	7	anbi2d 628	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) ↔ (𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1)))))
9		csbeq1a 3900	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
10	9	feq1d 6702	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾) ↔ ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)))
11	8, 10	imbi12d 343	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)) ↔ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))))
12		poimir.0	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
13	12	nncnd 12253	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
14		npcan1 11664	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
15	13, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
16	12	nnzd 12610	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
17		peano2zm 12630	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
18	16, 17	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
19		uzid 12862	. . . . . . . . . . . . 13 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
20		peano2uz 12910	. . . . . . . . . . . . 13 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
21	18, 19, 20	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
22	15, 21	eqeltrrd 2826	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
23		fzss2 13568	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
24	22, 23	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁))
25	24	sselda 3973	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ (0...𝑁))
26		elun 4142	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ({1} ∪ {0}) ↔ (𝑦 ∈ {1} ∨ 𝑦 ∈ {0}))
27		fzofzp1 13756	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑥 + 1) ∈ (0...𝐾))
28		elsni 4642	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {1} → 𝑦 = 1)
29	28	oveq2d 7429	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {1} → (𝑥 + 𝑦) = (𝑥 + 1))
30	29	eleq1d 2810	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {1} → ((𝑥 + 𝑦) ∈ (0...𝐾) ↔ (𝑥 + 1) ∈ (0...𝐾)))
31	27, 30	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑦 ∈ {1} → (𝑥 + 𝑦) ∈ (0...𝐾)))
32		elfzoelz 13659	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (0..^𝐾) → 𝑥 ∈ ℤ)
33	32	zcnd 12692	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (0..^𝐾) → 𝑥 ∈ ℂ)
34	33	addridd 11439	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑥 + 0) = 𝑥)
35		elfzofz 13675	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (0..^𝐾) → 𝑥 ∈ (0...𝐾))
36	34, 35	eqeltrd 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑥 + 0) ∈ (0...𝐾))
37		elsni 4642	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {0} → 𝑦 = 0)
38	37	oveq2d 7429	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {0} → (𝑥 + 𝑦) = (𝑥 + 0))
39	38	eleq1d 2810	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {0} → ((𝑥 + 𝑦) ∈ (0...𝐾) ↔ (𝑥 + 0) ∈ (0...𝐾)))
40	36, 39	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑦 ∈ {0} → (𝑥 + 𝑦) ∈ (0...𝐾)))
41	31, 40	jaod 857	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0..^𝐾) → ((𝑦 ∈ {1} ∨ 𝑦 ∈ {0}) → (𝑥 + 𝑦) ∈ (0...𝐾)))
42	26, 41	biimtrid 241	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^𝐾) → (𝑦 ∈ ({1} ∪ {0}) → (𝑥 + 𝑦) ∈ (0...𝐾)))
43	42	imp 405	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (0..^𝐾) ∧ 𝑦 ∈ ({1} ∪ {0})) → (𝑥 + 𝑦) ∈ (0...𝐾))
44	43	adantl 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑥 ∈ (0..^𝐾) ∧ 𝑦 ∈ ({1} ∪ {0}))) → (𝑥 + 𝑦) ∈ (0...𝐾))
45		poimirlem25.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾))
46	45	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑇:(1...𝑁)⟶(0..^𝐾))
47		1ex 11235	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
48	47	fconst 6777	. . . . . . . . . . . . 13 ⊢ ((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1}
49		c0ex 11233	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
50	49	fconst 6777	. . . . . . . . . . . . 13 ⊢ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0}
51	48, 50	pm3.2i 469	. . . . . . . . . . . 12 ⊢ (((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1} ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0})
52		poimirlem25.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
53		dff1o3 6838	. . . . . . . . . . . . . . 15 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (𝑈:(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡𝑈))
54	53	simprbi 495	. . . . . . . . . . . . . 14 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡𝑈)
55		imain 6633	. . . . . . . . . . . . . 14 ⊢ (Fun ^◡𝑈 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))))
56	52, 54, 55	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))))
57		elfznn0 13621	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ₀)
58	57	nn0red 12558	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
59	58	ltp1d 12169	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
60		fzdisj 13555	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
61	59, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
62	61	imaeq2d 6059	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (𝑈 “ ∅))
63		ima0 6076	. . . . . . . . . . . . . 14 ⊢ (𝑈 “ ∅) = ∅
64	62, 63	eqtrdi 2781	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
65	56, 64	sylan9req 2786	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅)
66		fun 6753	. . . . . . . . . . . 12 ⊢ (((((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1} ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0}) ∧ ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
67	51, 65, 66	sylancr 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
68		nn0p1nn 12536	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
69	57, 68	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
70		nnuz 12890	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ_≥‘1)
71	69, 70	eleqtrdi 2835	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ_≥‘1))
72		elfzuz3 13525	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑗))
73		fzsplit2 13553	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
74	71, 72, 73	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
75	74	imaeq2d 6059	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
76		imaundi 6150	. . . . . . . . . . . . . 14 ⊢ (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))
77	75, 76	eqtr2di 2782	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑁) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁))) = (𝑈 “ (1...𝑁)))
78		f1ofo 6839	. . . . . . . . . . . . . 14 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–onto→(1...𝑁))
79		foima 6809	. . . . . . . . . . . . . 14 ⊢ (𝑈:(1...𝑁)–onto→(1...𝑁) → (𝑈 “ (1...𝑁)) = (1...𝑁))
80	52, 78, 79	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑈 “ (1...𝑁)) = (1...𝑁))
81	77, 80	sylan9eqr 2787	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁))) = (1...𝑁))
82	81	feq2d 6703	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}) ↔ (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
83	67, 82	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
84		ovex 7446	. . . . . . . . . . 11 ⊢ (1...𝑁) ∈ V
85	84	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ V)
86		inidm 4214	. . . . . . . . . 10 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
87	44, 46, 83, 85, 85, 86	off 7697	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
88	25, 87	syldan 589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
89	6, 11, 88	chvarfv 2228	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
90		fzp1elp1 13581	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (0...((𝑁 − 1) + 1)))
91	15	oveq2d 7429	. . . . . . . . . . 11 ⊢ (𝜑 → (0...((𝑁 − 1) + 1)) = (0...𝑁))
92	91	eleq2d 2811	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 + 1) ∈ (0...((𝑁 − 1) + 1)) ↔ (𝑦 + 1) ∈ (0...𝑁)))
93	92	biimpa 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 + 1) ∈ (0...((𝑁 − 1) + 1))) → (𝑦 + 1) ∈ (0...𝑁))
94	90, 93	sylan2 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ (0...𝑁))
95		nfv 1909	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝜑 ∧ (𝑦 + 1) ∈ (0...𝑁))
96		nfcsb1v 3911	. . . . . . . . . . 11 ⊢ Ⅎ𝑗⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))
97	96, 3, 4	nff 6713	. . . . . . . . . 10 ⊢ Ⅎ𝑗⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)
98	95, 97	nfim 1891	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝜑 ∧ (𝑦 + 1) ∈ (0...𝑁)) → ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
99		ovex 7446	. . . . . . . . 9 ⊢ (𝑦 + 1) ∈ V
100		eleq1 2813	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → (𝑗 ∈ (0...𝑁) ↔ (𝑦 + 1) ∈ (0...𝑁)))
101	100	anbi2d 628	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑦 + 1) ∈ (0...𝑁))))
102		csbeq1a 3900	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
103	102	feq1d 6702	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → ((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾) ↔ ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)))
104	101, 103	imbi12d 343	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)) ↔ ((𝜑 ∧ (𝑦 + 1) ∈ (0...𝑁)) → ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))))
105	98, 99, 104, 87	vtoclf 3543	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 + 1) ∈ (0...𝑁)) → ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
106	94, 105	syldan 589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
107		csbeq1 3889	. . . . . . . . 9 ⊢ (𝑦 = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → ⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
108	107	feq1d 6702	. . . . . . . 8 ⊢ (𝑦 = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → (⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾) ↔ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)))
109		csbeq1 3889	. . . . . . . . 9 ⊢ ((𝑦 + 1) = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
110	109	feq1d 6702	. . . . . . . 8 ⊢ ((𝑦 + 1) = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → (⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾) ↔ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)))
111	108, 110	ifboth 4564	. . . . . . 7 ⊢ ((⦋𝑦 / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾) ∧ ⦋(𝑦 + 1) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾)) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
112	89, 106, 111	syl2anc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
113		ovex 7446	. . . . . . 7 ⊢ (0...𝐾) ∈ V
114	113, 84	elmap 8883	. . . . . 6 ⊢ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ ((0...𝐾) ↑_m (1...𝑁)) ↔ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝐾))
115	112, 114	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ ((0...𝐾) ↑_m (1...𝑁)))
116	115	fmpttd 7118	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))):(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
117		ovex 7446	. . . . 5 ⊢ ((0...𝐾) ↑_m (1...𝑁)) ∈ V
118		ovex 7446	. . . . 5 ⊢ (0...(𝑁 − 1)) ∈ V
119	117, 118	elmap 8883	. . . 4 ⊢ ((𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))):(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
120	116, 119	sylibr 233	. . 3 ⊢ (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1))))
121		rneq 5933	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → ran 𝑥 = ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
122	121	mpteq1d 5239	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → (𝑝 ∈ ran 𝑥 ↦ 𝐵) = (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵))
123	122	rneqd 5935	. . . . . 6 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) = ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵))
124	123	sseq2d 4006	. . . . 5 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ (0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵)))
125	121	rexeqdv 3316	. . . . 5 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0))
126	124, 125	anbi12d 630	. . . 4 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0)))
127	126	ceqsrexv 3635	. . 3 ⊢ ((𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1))) → (∃𝑥 ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0)))
128	120, 127	syl 17	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0)))
129		dfss3 3962	. . . 4 ⊢ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∀𝑖 ∈ (0...(𝑁 − 1))𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵))
130		ovex 7446	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
131		poimirlem28.1	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
132	130, 131	csbie 3922	. . . . . . . . . . . 12 ⊢ ⦋((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = 𝐶
133	132	csbeq2i 3894	. . . . . . . . . . 11 ⊢ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌⦋((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
134		opex 5461	. . . . . . . . . . . . 13 ⊢ ⟨𝑇, 𝑈⟩ ∈ V
135	134	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨𝑇, 𝑈⟩ ∈ V)
136		fveq2 6890	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = ⟨𝑇, 𝑈⟩ → (1^st ‘𝑠) = (1^st ‘⟨𝑇, 𝑈⟩))
137		fex 7232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇:(1...𝑁)⟶(0..^𝐾) ∧ (1...𝑁) ∈ V) → 𝑇 ∈ V)
138	45, 84, 137	sylancl 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ∈ V)
139		f1oexrnex 7929	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ∧ (1...𝑁) ∈ V) → 𝑈 ∈ V)
140	52, 84, 139	sylancl 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 ∈ V)
141		op1stg 7999	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ V ∧ 𝑈 ∈ V) → (1^st ‘⟨𝑇, 𝑈⟩) = 𝑇)
142	138, 140, 141	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1^st ‘⟨𝑇, 𝑈⟩) = 𝑇)
143	136, 142	sylan9eqr 2787	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 = ⟨𝑇, 𝑈⟩) → (1^st ‘𝑠) = 𝑇)
144		fveq2 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = ⟨𝑇, 𝑈⟩ → (2^nd ‘𝑠) = (2^nd ‘⟨𝑇, 𝑈⟩))
145		op2ndg 8000	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ V ∧ 𝑈 ∈ V) → (2^nd ‘⟨𝑇, 𝑈⟩) = 𝑈)
146	138, 140, 145	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘⟨𝑇, 𝑈⟩) = 𝑈)
147	144, 146	sylan9eqr 2787	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 = ⟨𝑇, 𝑈⟩) → (2^nd ‘𝑠) = 𝑈)
148		imaeq1 6054	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑠) = 𝑈 → ((2^nd ‘𝑠) “ (1...𝑗)) = (𝑈 “ (1...𝑗)))
149	148	xpeq1d 5702	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑠) = 𝑈 → (((2^nd ‘𝑠) “ (1...𝑗)) × {1}) = ((𝑈 “ (1...𝑗)) × {1}))
150		imaeq1 6054	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑠) = 𝑈 → ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) = (𝑈 “ ((𝑗 + 1)...𝑁)))
151	150	xpeq1d 5702	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑠) = 𝑈 → (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))
152	149, 151	uneq12d 4158	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑠) = 𝑈 → ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))
153	147, 152	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 = ⟨𝑇, 𝑈⟩) → ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))
154	143, 153	oveq12d 7431	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 = ⟨𝑇, 𝑈⟩) → ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
155	154	csbeq1d 3890	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 = ⟨𝑇, 𝑈⟩) → ⦋((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = ⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
156	135, 155	csbied 3924	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌⦋((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = ⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
157	133, 156	eqtr3id 2779	. . . . . . . . . 10 ⊢ (𝜑 → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
158	157	csbeq2dv 3893	. . . . . . . . 9 ⊢ (𝜑 → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
159	158	eqeq2d 2736	. . . . . . . 8 ⊢ (𝜑 → (𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵))
160	159	rexbidv 3169	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵))
161		vex 3467	. . . . . . . . 9 ⊢ 𝑖 ∈ V
162		eqid 2725	. . . . . . . . . 10 ⊢ (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) = (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵)
163	162	elrnmpt 5953	. . . . . . . . 9 ⊢ (𝑖 ∈ V → (𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = 𝐵))
164	161, 163	ax-mp 5	. . . . . . . 8 ⊢ (𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = 𝐵)
165		nfv 1909	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑖 = 𝐵
166		nfcsb1v 3911	. . . . . . . . . 10 ⊢ Ⅎ𝑝⦋𝑘 / 𝑝⦌𝐵
167	166	nfeq2 2910	. . . . . . . . 9 ⊢ Ⅎ𝑝 𝑖 = ⦋𝑘 / 𝑝⦌𝐵
168		csbeq1a 3900	. . . . . . . . . 10 ⊢ (𝑝 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑝⦌𝐵)
169	168	eqeq2d 2736	. . . . . . . . 9 ⊢ (𝑝 = 𝑘 → (𝑖 = 𝐵 ↔ 𝑖 = ⦋𝑘 / 𝑝⦌𝐵))
170	165, 167, 169	cbvrexw 3295	. . . . . . . 8 ⊢ (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = 𝐵 ↔ ∃𝑘 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = ⦋𝑘 / 𝑝⦌𝐵)
171		ovex 7446	. . . . . . . . . . 11 ⊢ (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
172	171	csbex 5307	. . . . . . . . . 10 ⊢ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
173	172	rgenw 3055	. . . . . . . . 9 ⊢ ∀𝑦 ∈ (0...(𝑁 − 1))⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
174		eqid 2725	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
175		csbeq1 3889	. . . . . . . . . . . 12 ⊢ (𝑘 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → ⦋𝑘 / 𝑝⦌𝐵 = ⦋⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
176		vex 3467	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
177	176, 99	ifex 4575	. . . . . . . . . . . . 13 ⊢ if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) ∈ V
178		csbnestgw 4418	. . . . . . . . . . . . 13 ⊢ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) ∈ V → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = ⦋⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
179	177, 178	ax-mp 5	. . . . . . . . . . . 12 ⊢ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = ⦋⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵
180	175, 179	eqtr4di 2783	. . . . . . . . . . 11 ⊢ (𝑘 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → ⦋𝑘 / 𝑝⦌𝐵 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
181	180	eqeq2d 2736	. . . . . . . . . 10 ⊢ (𝑘 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝑖 = ⦋𝑘 / 𝑝⦌𝐵 ↔ 𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵))
182	174, 181	rexrnmptw 7098	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (0...(𝑁 − 1))⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V → (∃𝑘 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = ⦋𝑘 / 𝑝⦌𝐵 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵))
183	173, 182	ax-mp 5	. . . . . . . 8 ⊢ (∃𝑘 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))𝑖 = ⦋𝑘 / 𝑝⦌𝐵 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
184	164, 170, 183	3bitri 296	. . . . . . 7 ⊢ (𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
185	160, 184	bitr4di 288	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵)))
186	24	sselda 3973	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ (0...𝑁))
187	186	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < 𝑉) → 𝑦 ∈ (0...𝑁))
188		elfzelz 13528	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
189	188	zred 12691	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
190	189	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
191		ltne 11336	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝑦 < 𝑉) → 𝑉 ≠ 𝑦)
192	191	necomd 2986	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ 𝑦 < 𝑉) → 𝑦 ≠ 𝑉)
193	190, 192	sylan 578	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < 𝑉) → 𝑦 ≠ 𝑉)
194		eldifsn 4787	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((0...𝑁) ∖ {𝑉}) ↔ (𝑦 ∈ (0...𝑁) ∧ 𝑦 ≠ 𝑉))
195	187, 193, 194	sylanbrc 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < 𝑉) → 𝑦 ∈ ((0...𝑁) ∖ {𝑉}))
196	94	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < 𝑉) → (𝑦 + 1) ∈ (0...𝑁))
197		poimirlem24.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ∈ (0...𝑁))
198	197	elfzelzd 13529	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ ℤ)
199	198	zred 12691	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉 ∈ ℝ)
200	199	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < 𝑉) → 𝑉 ∈ ℝ)
201		zre 12587	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ ℤ → 𝑉 ∈ ℝ)
202		zre 12587	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
203		lenlt 11317	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑉 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑉))
204	201, 202, 203	syl2an 594	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑉 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑉))
205		zleltp1 12638	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑉 ≤ 𝑦 ↔ 𝑉 < (𝑦 + 1)))
206	204, 205	bitr3d 280	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (¬ 𝑦 < 𝑉 ↔ 𝑉 < (𝑦 + 1)))
207	198, 188, 206	syl2an 594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (¬ 𝑦 < 𝑉 ↔ 𝑉 < (𝑦 + 1)))
208	207	biimpa 475	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < 𝑉) → 𝑉 < (𝑦 + 1))
209	200, 208	gtned 11374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < 𝑉) → (𝑦 + 1) ≠ 𝑉)
210		eldifsn 4787	. . . . . . . . . . 11 ⊢ ((𝑦 + 1) ∈ ((0...𝑁) ∖ {𝑉}) ↔ ((𝑦 + 1) ∈ (0...𝑁) ∧ (𝑦 + 1) ≠ 𝑉))
211	196, 209, 210	sylanbrc 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < 𝑉) → (𝑦 + 1) ∈ ((0...𝑁) ∖ {𝑉}))
212	195, 211	ifclda 4560	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) ∈ ((0...𝑁) ∖ {𝑉}))
213		nfcsb1v 3911	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
214	213	nfeq2 2910	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
215		csbeq1a 3900	. . . . . . . . . . . 12 ⊢ (𝑗 = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
216	215	eqeq2d 2736	. . . . . . . . . . 11 ⊢ (𝑗 = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
217	214, 216	rspce 3592	. . . . . . . . . 10 ⊢ ((if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) ∈ ((0...𝑁) ∖ {𝑉}) ∧ 𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
218	217	ex 411	. . . . . . . . 9 ⊢ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) ∈ ((0...𝑁) ∖ {𝑉}) → (𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
219	212, 218	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
220	219	rexlimdva 3145	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
221		nfv 1909	. . . . . . . 8 ⊢ Ⅎ𝑗𝜑
222		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑗(0...(𝑁 − 1))
223	222, 214	nfrexw 3301	. . . . . . . 8 ⊢ Ⅎ𝑗∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶
224		eldifi 4120	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ∈ (0...𝑁))
225	224, 57	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ∈ ℕ₀)
226	225	nn0ge0d 12560	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 0 ≤ 𝑗)
227	226	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 0 ≤ 𝑗)
228	225	nn0red 12558	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ∈ ℝ)
229	228	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 ∈ ℝ)
230	199	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑉 ∈ ℝ)
231	16	zred 12691	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
232	231	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑁 ∈ ℝ)
233		simpr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 < 𝑉)
234		elfzle2 13532	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ (0...𝑁) → 𝑉 ≤ 𝑁)
235	197, 234	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ≤ 𝑁)
236	235	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑉 ≤ 𝑁)
237	229, 230, 232, 233, 236	ltletrd 11399	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 < 𝑁)
238	224	elfzelzd 13529	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ∈ ℤ)
239		zltlem1 12640	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 < 𝑁 ↔ 𝑗 ≤ (𝑁 − 1)))
240	238, 16, 239	syl2anr 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 < 𝑁 ↔ 𝑗 ≤ (𝑁 − 1)))
241	240	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → (𝑗 < 𝑁 ↔ 𝑗 ≤ (𝑁 − 1)))
242	237, 241	mpbid 231	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 ≤ (𝑁 − 1))
243		0z 12594	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
244		elfz 13517	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑗 ∈ (0...(𝑁 − 1)) ↔ (0 ≤ 𝑗 ∧ 𝑗 ≤ (𝑁 − 1))))
245	243, 244	mp3an2 1445	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑗 ∈ (0...(𝑁 − 1)) ↔ (0 ≤ 𝑗 ∧ 𝑗 ≤ (𝑁 − 1))))
246	238, 18, 245	syl2anr 595	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 ∈ (0...(𝑁 − 1)) ↔ (0 ≤ 𝑗 ∧ 𝑗 ≤ (𝑁 − 1))))
247	246	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → (𝑗 ∈ (0...(𝑁 − 1)) ↔ (0 ≤ 𝑗 ∧ 𝑗 ≤ (𝑁 − 1))))
248	227, 242, 247	mpbir2and 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 ∈ (0...(𝑁 − 1)))
249		0red 11242	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 0 ∈ ℝ)
250	199	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑉 ∈ ℝ)
251	228	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑗 ∈ ℝ)
252		elfzle1 13531	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ (0...𝑁) → 0 ≤ 𝑉)
253	197, 252	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ 𝑉)
254	253	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 0 ≤ 𝑉)
255		lenlt 11317	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑉 ≤ 𝑗 ↔ ¬ 𝑗 < 𝑉))
256	199, 228, 255	syl2an 594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑉 ≤ 𝑗 ↔ ¬ 𝑗 < 𝑉))
257	256	biimpar 476	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑉 ≤ 𝑗)
258		eldifsni 4790	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ≠ 𝑉)
259	258	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑗 ≠ 𝑉)
260		ltlen 11340	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑉 < 𝑗 ↔ (𝑉 ≤ 𝑗 ∧ 𝑗 ≠ 𝑉)))
261	199, 228, 260	syl2an 594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑉 < 𝑗 ↔ (𝑉 ≤ 𝑗 ∧ 𝑗 ≠ 𝑉)))
262	261	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → (𝑉 < 𝑗 ↔ (𝑉 ≤ 𝑗 ∧ 𝑗 ≠ 𝑉)))
263	257, 259, 262	mpbir2and 711	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑉 < 𝑗)
264	249, 250, 251, 254, 263	lelttrd 11397	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 0 < 𝑗)
265		zgt0ge1 12641	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℤ → (0 < 𝑗 ↔ 1 ≤ 𝑗))
266	238, 265	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → (0 < 𝑗 ↔ 1 ≤ 𝑗))
267	266	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → (0 < 𝑗 ↔ 1 ≤ 𝑗))
268	264, 267	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 1 ≤ 𝑗)
269		elfzle2 13532	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ≤ 𝑁)
270	224, 269	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ≤ 𝑁)
271	270	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑗 ≤ 𝑁)
272		1z 12617	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
273		elfz 13517	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (1...𝑁) ↔ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
274	272, 273	mp3an2 1445	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (1...𝑁) ↔ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
275	238, 16, 274	syl2anr 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 ∈ (1...𝑁) ↔ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
276	275	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → (𝑗 ∈ (1...𝑁) ↔ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
277	268, 271, 276	mpbir2and 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑗 ∈ (1...𝑁))
278		elfzmlbm 13638	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑁) → (𝑗 − 1) ∈ (0...(𝑁 − 1)))
279	277, 278	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → (𝑗 − 1) ∈ (0...(𝑁 − 1)))
280	248, 279	ifclda 4560	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) ∈ (0...(𝑁 − 1)))
281		breq1 5147	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑗 < 𝑉 ↔ if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉))
282		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → 𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)))
283		oveq1 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑗 + 1) = (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1))
284	281, 282, 283	ifbieq12d 4553	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → if(𝑗 < 𝑉, 𝑗, (𝑗 + 1)) = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)))
285	284	eqeq2d 2736	. . . . . . . . . . . . . 14 ⊢ (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 + 1)) ↔ 𝑗 = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1))))
286		breq1 5147	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → ((𝑗 − 1) < 𝑉 ↔ if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉))
287		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)))
288		oveq1 7420	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → ((𝑗 − 1) + 1) = (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1))
289	286, 287, 288	ifbieq12d 4553	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → if((𝑗 − 1) < 𝑉, (𝑗 − 1), ((𝑗 − 1) + 1)) = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)))
290	289	eqeq2d 2736	. . . . . . . . . . . . . 14 ⊢ ((𝑗 − 1) = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑗 = if((𝑗 − 1) < 𝑉, (𝑗 − 1), ((𝑗 − 1) + 1)) ↔ 𝑗 = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1))))
291		iftrue 4531	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 < 𝑉 → if(𝑗 < 𝑉, 𝑗, (𝑗 + 1)) = 𝑗)
292	291	eqcomd 2731	. . . . . . . . . . . . . . 15 ⊢ (𝑗 < 𝑉 → 𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 + 1)))
293	292	adantl 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ 𝑗 < 𝑉) → 𝑗 = if(𝑗 < 𝑉, 𝑗, (𝑗 + 1)))
294		zlem1lt 12639	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ ℤ ∧ 𝑉 ∈ ℤ) → (𝑗 ≤ 𝑉 ↔ (𝑗 − 1) < 𝑉))
295	238, 198, 294	syl2anr 595	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 ≤ 𝑉 ↔ (𝑗 − 1) < 𝑉))
296	258	necomd 2986	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑉 ≠ 𝑗)
297	296	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → 𝑉 ≠ 𝑗)
298		ltlen 11340	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ ℝ ∧ 𝑉 ∈ ℝ) → (𝑗 < 𝑉 ↔ (𝑗 ≤ 𝑉 ∧ 𝑉 ≠ 𝑗)))
299	228, 199, 298	syl2anr 595	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 < 𝑉 ↔ (𝑗 ≤ 𝑉 ∧ 𝑉 ≠ 𝑗)))
300	299	biimprd 247	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → ((𝑗 ≤ 𝑉 ∧ 𝑉 ≠ 𝑗) → 𝑗 < 𝑉))
301	297, 300	mpan2d 692	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑗 ≤ 𝑉 → 𝑗 < 𝑉))
302	295, 301	sylbird 259	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → ((𝑗 − 1) < 𝑉 → 𝑗 < 𝑉))
303	302	con3dimp 407	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → ¬ (𝑗 − 1) < 𝑉)
304	303	iffalsed 4536	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → if((𝑗 − 1) < 𝑉, (𝑗 − 1), ((𝑗 − 1) + 1)) = ((𝑗 − 1) + 1))
305	225	nn0cnd 12559	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑗 ∈ ℂ)
306		npcan1 11664	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℂ → ((𝑗 − 1) + 1) = 𝑗)
307	305, 306	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → ((𝑗 − 1) + 1) = 𝑗)
308	307	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → ((𝑗 − 1) + 1) = 𝑗)
309	304, 308	eqtr2d 2766	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) ∧ ¬ 𝑗 < 𝑉) → 𝑗 = if((𝑗 − 1) < 𝑉, (𝑗 − 1), ((𝑗 − 1) + 1)))
310	285, 290, 293, 309	ifbothda 4563	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → 𝑗 = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)))
311		csbeq1a 3900	. . . . . . . . . . . . 13 ⊢ (𝑗 = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
312	310, 311	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
313	312	eqeq2d 2736	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
314	313	biimpd 228	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → 𝑖 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
315		breq1 5147	. . . . . . . . . . . . . 14 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑦 < 𝑉 ↔ if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉))
316		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → 𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)))
317		oveq1 7420	. . . . . . . . . . . . . 14 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑦 + 1) = (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1))
318	315, 316, 317	ifbieq12d 4553	. . . . . . . . . . . . 13 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)))
319	318	csbeq1d 3890	. . . . . . . . . . . 12 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
320	319	eqeq2d 2736	. . . . . . . . . . 11 ⊢ (𝑦 = if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) → (𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ 𝑖 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
321	320	rspcev 3603	. . . . . . . . . 10 ⊢ ((if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) ∈ (0...(𝑁 − 1)) ∧ 𝑖 = ⦋if(if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) < 𝑉, if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)), (if(𝑗 < 𝑉, 𝑗, (𝑗 − 1)) + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) → ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)
322	280, 314, 321	syl6an 682	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...𝑁) ∖ {𝑉})) → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
323	322	ex 411	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ ((0...𝑁) ∖ {𝑉}) → (𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)))
324	221, 223, 323	rexlimd 3254	. . . . . . 7 ⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 → ∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
325	220, 324	impbid 211	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ (0...(𝑁 − 1))𝑖 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
326	185, 325	bitr3d 280	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
327	326	ralbidv 3168	. . . 4 ⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))𝑖 ∈ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
328	129, 327	bitrid 282	. . 3 ⊢ (𝜑 → ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶))
329	328	anbi1d 629	. 2 ⊢ (𝜑 → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ↦ 𝐵) ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0)))
330	12, 45, 52, 197	poimirlem23 37169	. . 3 ⊢ (𝜑 → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁))))
331	330	anbi2d 628	. 2 ⊢ (𝜑 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ ∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))))
332	128, 329, 331	3bitrd 304	1 ⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑_m (1...𝑁)) ↑_m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 Vcvv 3463 ⦋csb 3886 ∖ cdif 3938 ∪ cun 3939 ∩ cin 3940 ⊆ wss 3941 ∅c0 4319 ifcif 4525 {csn 4625 ⟨cop 4631 class class class wbr 5144 ↦ cmpt 5227 × cxp 5671 ^◡ccnv 5672 ran crn 5674 “ cima 5676 Fun wfun 6537 ⟶wf 6539 –onto→wfo 6541 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7413 ∘_f cof 7677 1^st c1st 7985 2^nd c2nd 7986 ↑_m cmap 8838 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 < clt 11273 ≤ cle 11274 − cmin 11469 ℕcn 12237 ℕ₀cn0 12497 ℤcz 12583 ℤ_≥cuz 12847 ...cfz 13511 ..^cfzo 13654

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655

This theorem is referenced by: poimirlem27 37173

W3C validator