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Theorem efgsrel 18147

Description: The start and end of any extension sequence are related (i.e. evaluate to the same element of the quotient group to be created). (Contributed by Mario Carneiro, 1-Oct-2015.)

**Hypotheses**
Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
efgval.r	⊢ ∼ = ( ~_FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
efgred.d	⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
efgred.s	⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))

**Assertion**
Ref	Expression
efgsrel	⊢ (𝐹 ∈ dom 𝑆 → (𝐹‘0) ∼ (𝑆‘𝐹))

Distinct variable groups: 𝑦,𝑧 𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑚,𝑥 𝑚,𝑀 𝑥,𝑛,𝑀,𝑡,𝑣,𝑤 𝑘,𝑚,𝑡,𝑥,𝑇 𝑘,𝑛,𝑣,𝑤,𝑦,𝑧,𝑊,𝑚,𝑡,𝑥 ∼ ,𝑚,𝑡,𝑥,𝑦,𝑧 𝑚,𝐼,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧 𝐷,𝑚,𝑡 Allowed substitution hints: 𝐷(𝑥,𝑦,𝑧,𝑤,𝑣,𝑘,𝑛) ∼ (𝑤,𝑣,𝑘,𝑛) 𝑆(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝑇(𝑦,𝑧,𝑤,𝑣,𝑛) 𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝐼(𝑘) 𝑀(𝑦,𝑧,𝑘)

Proof of Theorem efgsrel Dummy variables 𝑎 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . 6 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
2		efgval.r	. . . . . 6 ⊢ ∼ = ( ~_FG ‘𝐼)
3		efgval2.m	. . . . . 6 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
4		efgval2.t	. . . . . 6 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5		efgred.d	. . . . . 6 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
6		efgred.s	. . . . . 6 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))
7	1, 2, 3, 4, 5, 6	efgsdm 18143	. . . . 5 ⊢ (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐹‘0) ∈ 𝐷 ∧ ∀𝑎 ∈ (1..^(#‘𝐹))(𝐹‘𝑎) ∈ ran (𝑇‘(𝐹‘(𝑎 − 1)))))
8	7	simp1bi 1076	. . . 4 ⊢ (𝐹 ∈ dom 𝑆 → 𝐹 ∈ (Word 𝑊 ∖ {∅}))
9		eldifsn 4317	. . . . 5 ⊢ (𝐹 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝐹 ∈ Word 𝑊 ∧ 𝐹 ≠ ∅))
10		lennncl 13325	. . . . 5 ⊢ ((𝐹 ∈ Word 𝑊 ∧ 𝐹 ≠ ∅) → (#‘𝐹) ∈ ℕ)
11	9, 10	sylbi 207	. . . 4 ⊢ (𝐹 ∈ (Word 𝑊 ∖ {∅}) → (#‘𝐹) ∈ ℕ)
12		fzo0end 12560	. . . 4 ⊢ ((#‘𝐹) ∈ ℕ → ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)))
13	8, 11, 12	3syl 18	. . 3 ⊢ (𝐹 ∈ dom 𝑆 → ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)))
14		nnm1nn0 11334	. . . . 5 ⊢ ((#‘𝐹) ∈ ℕ → ((#‘𝐹) − 1) ∈ ℕ₀)
15	8, 11, 14	3syl 18	. . . 4 ⊢ (𝐹 ∈ dom 𝑆 → ((#‘𝐹) − 1) ∈ ℕ₀)
16		eleq1 2689	. . . . . . 7 ⊢ (𝑎 = 0 → (𝑎 ∈ (0..^(#‘𝐹)) ↔ 0 ∈ (0..^(#‘𝐹))))
17		fveq2 6191	. . . . . . . 8 ⊢ (𝑎 = 0 → (𝐹‘𝑎) = (𝐹‘0))
18	17	breq2d 4665	. . . . . . 7 ⊢ (𝑎 = 0 → ((𝐹‘0) ∼ (𝐹‘𝑎) ↔ (𝐹‘0) ∼ (𝐹‘0)))
19	16, 18	imbi12d 334	. . . . . 6 ⊢ (𝑎 = 0 → ((𝑎 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑎)) ↔ (0 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘0))))
20	19	imbi2d 330	. . . . 5 ⊢ (𝑎 = 0 → ((𝐹 ∈ dom 𝑆 → (𝑎 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑎))) ↔ (𝐹 ∈ dom 𝑆 → (0 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘0)))))
21		eleq1 2689	. . . . . . 7 ⊢ (𝑎 = 𝑖 → (𝑎 ∈ (0..^(#‘𝐹)) ↔ 𝑖 ∈ (0..^(#‘𝐹))))
22		fveq2 6191	. . . . . . . 8 ⊢ (𝑎 = 𝑖 → (𝐹‘𝑎) = (𝐹‘𝑖))
23	22	breq2d 4665	. . . . . . 7 ⊢ (𝑎 = 𝑖 → ((𝐹‘0) ∼ (𝐹‘𝑎) ↔ (𝐹‘0) ∼ (𝐹‘𝑖)))
24	21, 23	imbi12d 334	. . . . . 6 ⊢ (𝑎 = 𝑖 → ((𝑎 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑎)) ↔ (𝑖 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑖))))
25	24	imbi2d 330	. . . . 5 ⊢ (𝑎 = 𝑖 → ((𝐹 ∈ dom 𝑆 → (𝑎 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑎))) ↔ (𝐹 ∈ dom 𝑆 → (𝑖 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑖)))))
26		eleq1 2689	. . . . . . 7 ⊢ (𝑎 = (𝑖 + 1) → (𝑎 ∈ (0..^(#‘𝐹)) ↔ (𝑖 + 1) ∈ (0..^(#‘𝐹))))
27		fveq2 6191	. . . . . . . 8 ⊢ (𝑎 = (𝑖 + 1) → (𝐹‘𝑎) = (𝐹‘(𝑖 + 1)))
28	27	breq2d 4665	. . . . . . 7 ⊢ (𝑎 = (𝑖 + 1) → ((𝐹‘0) ∼ (𝐹‘𝑎) ↔ (𝐹‘0) ∼ (𝐹‘(𝑖 + 1))))
29	26, 28	imbi12d 334	. . . . . 6 ⊢ (𝑎 = (𝑖 + 1) → ((𝑎 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑎)) ↔ ((𝑖 + 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘(𝑖 + 1)))))
30	29	imbi2d 330	. . . . 5 ⊢ (𝑎 = (𝑖 + 1) → ((𝐹 ∈ dom 𝑆 → (𝑎 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑎))) ↔ (𝐹 ∈ dom 𝑆 → ((𝑖 + 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘(𝑖 + 1))))))
31		eleq1 2689	. . . . . . 7 ⊢ (𝑎 = ((#‘𝐹) − 1) → (𝑎 ∈ (0..^(#‘𝐹)) ↔ ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹))))
32		fveq2 6191	. . . . . . . 8 ⊢ (𝑎 = ((#‘𝐹) − 1) → (𝐹‘𝑎) = (𝐹‘((#‘𝐹) − 1)))
33	32	breq2d 4665	. . . . . . 7 ⊢ (𝑎 = ((#‘𝐹) − 1) → ((𝐹‘0) ∼ (𝐹‘𝑎) ↔ (𝐹‘0) ∼ (𝐹‘((#‘𝐹) − 1))))
34	31, 33	imbi12d 334	. . . . . 6 ⊢ (𝑎 = ((#‘𝐹) − 1) → ((𝑎 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑎)) ↔ (((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘((#‘𝐹) − 1)))))
35	34	imbi2d 330	. . . . 5 ⊢ (𝑎 = ((#‘𝐹) − 1) → ((𝐹 ∈ dom 𝑆 → (𝑎 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑎))) ↔ (𝐹 ∈ dom 𝑆 → (((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘((#‘𝐹) − 1))))))
36	1, 2	efger 18131	. . . . . . . 8 ⊢ ∼ Er 𝑊
37	36	a1i 11	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 0 ∈ (0..^(#‘𝐹))) → ∼ Er 𝑊)
38		eldifi 3732	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word 𝑊 ∖ {∅}) → 𝐹 ∈ Word 𝑊)
39		wrdf 13310	. . . . . . . . 9 ⊢ (𝐹 ∈ Word 𝑊 → 𝐹:(0..^(#‘𝐹))⟶𝑊)
40	8, 38, 39	3syl 18	. . . . . . . 8 ⊢ (𝐹 ∈ dom 𝑆 → 𝐹:(0..^(#‘𝐹))⟶𝑊)
41	40	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 0 ∈ (0..^(#‘𝐹))) → (𝐹‘0) ∈ 𝑊)
42	37, 41	erref 7762	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 0 ∈ (0..^(#‘𝐹))) → (𝐹‘0) ∼ (𝐹‘0))
43	42	ex 450	. . . . 5 ⊢ (𝐹 ∈ dom 𝑆 → (0 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘0)))
44		elnn0uz 11725	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ₀ ↔ 𝑖 ∈ (ℤ_≥‘0))
45		peano2fzor 12575	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (ℤ_≥‘0) ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → 𝑖 ∈ (0..^(#‘𝐹)))
46	44, 45	sylanb 489	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → 𝑖 ∈ (0..^(#‘𝐹)))
47	46	3adant1 1079	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → 𝑖 ∈ (0..^(#‘𝐹)))
48	47	3expia 1267	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀) → ((𝑖 + 1) ∈ (0..^(#‘𝐹)) → 𝑖 ∈ (0..^(#‘𝐹))))
49	48	imim1d 82	. . . . . . . 8 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀) → ((𝑖 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑖)) → ((𝑖 + 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑖))))
50	40	3ad2ant1 1082	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → 𝐹:(0..^(#‘𝐹))⟶𝑊)
51	50, 47	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → (𝐹‘𝑖) ∈ 𝑊)
52		nn0p1nn 11332	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ₀ → (𝑖 + 1) ∈ ℕ)
53	52	3ad2ant2 1083	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → (𝑖 + 1) ∈ ℕ)
54		nnuz 11723	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
55	53, 54	syl6eleq 2711	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → (𝑖 + 1) ∈ (ℤ_≥‘1))
56		elfzolt2b 12481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 + 1) ∈ (0..^(#‘𝐹)) → (𝑖 + 1) ∈ ((𝑖 + 1)..^(#‘𝐹)))
57	56	3ad2ant3 1084	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → (𝑖 + 1) ∈ ((𝑖 + 1)..^(#‘𝐹)))
58		elfzo3 12486	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 + 1) ∈ (1..^(#‘𝐹)) ↔ ((𝑖 + 1) ∈ (ℤ_≥‘1) ∧ (𝑖 + 1) ∈ ((𝑖 + 1)..^(#‘𝐹))))
59	55, 57, 58	sylanbrc 698	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → (𝑖 + 1) ∈ (1..^(#‘𝐹)))
60	7	simp3bi 1078	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ dom 𝑆 → ∀𝑎 ∈ (1..^(#‘𝐹))(𝐹‘𝑎) ∈ ran (𝑇‘(𝐹‘(𝑎 − 1))))
61	60	3ad2ant1 1082	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → ∀𝑎 ∈ (1..^(#‘𝐹))(𝐹‘𝑎) ∈ ran (𝑇‘(𝐹‘(𝑎 − 1))))
62		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝑖 + 1) → (𝑎 − 1) = ((𝑖 + 1) − 1))
63	62	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝑖 + 1) → (𝐹‘(𝑎 − 1)) = (𝐹‘((𝑖 + 1) − 1)))
64	63	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑖 + 1) → (𝑇‘(𝐹‘(𝑎 − 1))) = (𝑇‘(𝐹‘((𝑖 + 1) − 1))))
65	64	rneqd 5353	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (𝑖 + 1) → ran (𝑇‘(𝐹‘(𝑎 − 1))) = ran (𝑇‘(𝐹‘((𝑖 + 1) − 1))))
66	27, 65	eleq12d 2695	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝑖 + 1) → ((𝐹‘𝑎) ∈ ran (𝑇‘(𝐹‘(𝑎 − 1))) ↔ (𝐹‘(𝑖 + 1)) ∈ ran (𝑇‘(𝐹‘((𝑖 + 1) − 1)))))
67	66	rspcv 3305	. . . . . . . . . . . . . 14 ⊢ ((𝑖 + 1) ∈ (1..^(#‘𝐹)) → (∀𝑎 ∈ (1..^(#‘𝐹))(𝐹‘𝑎) ∈ ran (𝑇‘(𝐹‘(𝑎 − 1))) → (𝐹‘(𝑖 + 1)) ∈ ran (𝑇‘(𝐹‘((𝑖 + 1) − 1)))))
68	59, 61, 67	sylc 65	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → (𝐹‘(𝑖 + 1)) ∈ ran (𝑇‘(𝐹‘((𝑖 + 1) − 1))))
69		nn0cn 11302	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℂ)
70	69	3ad2ant2 1083	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → 𝑖 ∈ ℂ)
71		ax-1cn 9994	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
72		pncan 10287	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑖 + 1) − 1) = 𝑖)
73	70, 71, 72	sylancl 694	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → ((𝑖 + 1) − 1) = 𝑖)
74	73	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → (𝐹‘((𝑖 + 1) − 1)) = (𝐹‘𝑖))
75	74	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → (𝑇‘(𝐹‘((𝑖 + 1) − 1))) = (𝑇‘(𝐹‘𝑖)))
76	75	rneqd 5353	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → ran (𝑇‘(𝐹‘((𝑖 + 1) − 1))) = ran (𝑇‘(𝐹‘𝑖)))
77	68, 76	eleqtrd 2703	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → (𝐹‘(𝑖 + 1)) ∈ ran (𝑇‘(𝐹‘𝑖)))
78	1, 2, 3, 4	efgi2 18138	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑖) ∈ 𝑊 ∧ (𝐹‘(𝑖 + 1)) ∈ ran (𝑇‘(𝐹‘𝑖))) → (𝐹‘𝑖) ∼ (𝐹‘(𝑖 + 1)))
79	51, 77, 78	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → (𝐹‘𝑖) ∼ (𝐹‘(𝑖 + 1)))
80	36	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → ∼ Er 𝑊)
81	80	ertr 7757	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → (((𝐹‘0) ∼ (𝐹‘𝑖) ∧ (𝐹‘𝑖) ∼ (𝐹‘(𝑖 + 1))) → (𝐹‘0) ∼ (𝐹‘(𝑖 + 1))))
82	79, 81	mpan2d 710	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀ ∧ (𝑖 + 1) ∈ (0..^(#‘𝐹))) → ((𝐹‘0) ∼ (𝐹‘𝑖) → (𝐹‘0) ∼ (𝐹‘(𝑖 + 1))))
83	82	3expia 1267	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀) → ((𝑖 + 1) ∈ (0..^(#‘𝐹)) → ((𝐹‘0) ∼ (𝐹‘𝑖) → (𝐹‘0) ∼ (𝐹‘(𝑖 + 1)))))
84	83	a2d 29	. . . . . . . 8 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀) → (((𝑖 + 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑖)) → ((𝑖 + 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘(𝑖 + 1)))))
85	49, 84	syld 47	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ₀) → ((𝑖 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑖)) → ((𝑖 + 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘(𝑖 + 1)))))
86	85	expcom 451	. . . . . 6 ⊢ (𝑖 ∈ ℕ₀ → (𝐹 ∈ dom 𝑆 → ((𝑖 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑖)) → ((𝑖 + 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘(𝑖 + 1))))))
87	86	a2d 29	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ → ((𝐹 ∈ dom 𝑆 → (𝑖 ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘𝑖))) → (𝐹 ∈ dom 𝑆 → ((𝑖 + 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘(𝑖 + 1))))))
88	20, 25, 30, 35, 43, 87	nn0ind 11472	. . . 4 ⊢ (((#‘𝐹) − 1) ∈ ℕ₀ → (𝐹 ∈ dom 𝑆 → (((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘((#‘𝐹) − 1)))))
89	15, 88	mpcom 38	. . 3 ⊢ (𝐹 ∈ dom 𝑆 → (((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)) → (𝐹‘0) ∼ (𝐹‘((#‘𝐹) − 1))))
90	13, 89	mpd 15	. 2 ⊢ (𝐹 ∈ dom 𝑆 → (𝐹‘0) ∼ (𝐹‘((#‘𝐹) − 1)))
91	1, 2, 3, 4, 5, 6	efgsval 18144	. 2 ⊢ (𝐹 ∈ dom 𝑆 → (𝑆‘𝐹) = (𝐹‘((#‘𝐹) − 1)))
92	90, 91	breqtrrd 4681	1 ⊢ (𝐹 ∈ dom 𝑆 → (𝐹‘0) ∼ (𝑆‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 ∖ cdif 3571 ∅c0 3915 {csn 4177 ⟨cop 4183 ⟨cotp 4185 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 dom cdm 5114 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1_𝑜c1o 7553 2_𝑜c2o 7554 Er wer 7739 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 splice csplice 13296 ⟨“cs2 13586 ~_FG cefg 18119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-s2 13593 df-efg 18122

This theorem is referenced by: efgredeu 18165 efgred2 18166

W3C validator