Theorem gsmsymgreqlem2 17851

Description: Lemma 2 for gsmsymgreq 17852. (Contributed by AV, 26-Jan-2019.)

Hypotheses

Ref	Expression
gsmsymgrfix.s	⊢ 𝑆 = (SymGrp‘𝑁)
gsmsymgrfix.b	⊢ 𝐵 = (Base‘𝑆)
gsmsymgreq.z	⊢ 𝑍 = (SymGrp‘𝑀)
gsmsymgreq.p	⊢ 𝑃 = (Base‘𝑍)
gsmsymgreq.i	⊢ 𝐼 = (𝑁 ∩ 𝑀)

Assertion

Ref	Expression
gsmsymgreqlem2	⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))

Distinct variable groups:   𝐵,𝑖   𝑖,𝑁   𝑃,𝑖   𝑛,𝐼   𝑛,𝑋   𝐶,𝑛   𝑅,𝑛   𝑆,𝑛   𝑛,𝑌   𝑛,𝑍   𝐵,𝑛   𝐶,𝑖,𝑛   𝑖,𝐼   𝑛,𝑀   𝑛,𝑁   𝑃,𝑛   𝑅,𝑖   𝑖,𝑋   𝑖,𝑌

Allowed substitution hints:   𝑆(𝑖)   𝑀(𝑖)   𝑍(𝑖)

Proof of Theorem gsmsymgreqlem2

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatws1len 13398	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (#‘(𝑋 ++ ⟨“𝐶”⟩)) = ((#‘𝑋) + 1))
2	1	3ad2ant1 1082	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (#‘(𝑋 ++ ⟨“𝐶”⟩)) = ((#‘𝑋) + 1))
3	2	oveq2d 6666	. . . . . . 7 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩))) = (0..^((#‘𝑋) + 1)))
4		lencl 13324	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Word 𝐵 → (#‘𝑋) ∈ ℕ0)
5		elnn0uz 11725	. . . . . . . . . . 11 ⊢ ((#‘𝑋) ∈ ℕ0 ↔ (#‘𝑋) ∈ (ℤ≥‘0))
6	4, 5	sylib 208	. . . . . . . . . 10 ⊢ (𝑋 ∈ Word 𝐵 → (#‘𝑋) ∈ (ℤ≥‘0))
7	6	adantr 481	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (#‘𝑋) ∈ (ℤ≥‘0))
8		fzosplitsn 12576	. . . . . . . . 9 ⊢ ((#‘𝑋) ∈ (ℤ≥‘0) → (0..^((#‘𝑋) + 1)) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
9	7, 8	syl 17	. . . . . . . 8 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (0..^((#‘𝑋) + 1)) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
10	9	3ad2ant1 1082	. . . . . . 7 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (0..^((#‘𝑋) + 1)) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
11	3, 10	eqtrd 2656	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
12	11	raleqdv 3144	. . . . 5 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ ((0..^(#‘𝑋)) ∪ {(#‘𝑋)})∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛)))
13	4	adantr 481	. . . . . . 7 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (#‘𝑋) ∈ ℕ0)
14	13	3ad2ant1 1082	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (#‘𝑋) ∈ ℕ0)
15		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑖 = (#‘𝑋) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = ((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋)))
16	15	fveq1d 6193	. . . . . . . . 9 ⊢ (𝑖 = (#‘𝑋) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛))
17		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑖 = (#‘𝑋) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = ((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋)))
18	17	fveq1d 6193	. . . . . . . . 9 ⊢ (𝑖 = (#‘𝑋) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛))
19	16, 18	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑖 = (#‘𝑋) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛)))
20	19	ralbidv 2986	. . . . . . 7 ⊢ (𝑖 = (#‘𝑋) → (∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛)))
21	20	ralunsn 4422	. . . . . 6 ⊢ ((#‘𝑋) ∈ ℕ0 → (∀𝑖 ∈ ((0..^(#‘𝑋)) ∪ {(#‘𝑋)})∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛))))
22	14, 21	syl 17	. . . . 5 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (∀𝑖 ∈ ((0..^(#‘𝑋)) ∪ {(#‘𝑋)})∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛))))
23		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → 𝑋 ∈ Word 𝐵)
24	23	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → 𝑋 ∈ Word 𝐵)
25	24	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑋 ∈ Word 𝐵)
26		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ 𝐵)
27	26	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → 𝐶 ∈ 𝐵)
28	27	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝐶 ∈ 𝐵)
29		simpr 477	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑖 ∈ (0..^(#‘𝑋)))
30		ccats1val1 13403	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑖 ∈ (0..^(#‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋‘𝑖))
31	25, 28, 29, 30	syl3anc 1326	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋‘𝑖))
32	31	fveq1d 6193	. . . . . . . . 9 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = ((𝑋‘𝑖)‘𝑛))
33		simpl2l 1114	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑌 ∈ Word 𝑃)
34		simpl2r 1115	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑅 ∈ 𝑃)
35		oveq2 6658	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑋) = (#‘𝑌) → (0..^(#‘𝑋)) = (0..^(#‘𝑌)))
36	35	eleq2d 2687	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑋) = (#‘𝑌) → (𝑖 ∈ (0..^(#‘𝑋)) ↔ 𝑖 ∈ (0..^(#‘𝑌))))
37	36	biimpd 219	. . . . . . . . . . . . 13 ⊢ ((#‘𝑋) = (#‘𝑌) → (𝑖 ∈ (0..^(#‘𝑋)) → 𝑖 ∈ (0..^(#‘𝑌))))
38	37	3ad2ant3 1084	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑖 ∈ (0..^(#‘𝑋)) → 𝑖 ∈ (0..^(#‘𝑌))))
39	38	imp 445	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑖 ∈ (0..^(#‘𝑌)))
40		ccats1val1 13403	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ 𝑖 ∈ (0..^(#‘𝑌))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌‘𝑖))
41	33, 34, 39, 40	syl3anc 1326	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌‘𝑖))
42	41	fveq1d 6193	. . . . . . . . 9 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛))
43	32, 42	eqeq12d 2637	. . . . . . . 8 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛)))
44	43	ralbidv 2986	. . . . . . 7 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → (∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛)))
45	44	ralbidva 2985	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛)))
46		eqidd 2623	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (#‘𝑋) = (#‘𝑋))
47	23, 26, 46	3jca 1242	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (#‘𝑋) = (#‘𝑋)))
48	47	3ad2ant1 1082	. . . . . . . . . 10 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (#‘𝑋) = (#‘𝑋)))
49		ccats1val2 13404	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (#‘𝑋) = (#‘𝑋)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋)) = 𝐶)
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋)) = 𝐶)
51	50	fveq1d 6193	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (𝐶‘𝑛))
52		df-3an 1039	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ (#‘𝑋) = (#‘𝑌)) ↔ ((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)))
53	52	biimpri 218	. . . . . . . . . 10 ⊢ (((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ (#‘𝑋) = (#‘𝑌)))
54	53	3adant1 1079	. . . . . . . . 9 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ (#‘𝑋) = (#‘𝑌)))
55		ccats1val2 13404	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ (#‘𝑋) = (#‘𝑌)) → ((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋)) = 𝑅)
56	55	fveq1d 6193	. . . . . . . . 9 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ (#‘𝑋) = (#‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) = (𝑅‘𝑛))
57	54, 56	syl 17	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) = (𝑅‘𝑛))
58	51, 57	eqeq12d 2637	. . . . . . 7 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → ((((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) ↔ (𝐶‘𝑛) = (𝑅‘𝑛)))
59	58	ralbidv 2986	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) ↔ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛)))
60	45, 59	anbi12d 747	. . . . 5 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛)) ↔ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛))))
61	12, 22, 60	3bitrd 294	. . . 4 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛))))
62	61	ad2antlr 763	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛))))
63		pm3.35 611	. . . . . . 7 ⊢ ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))
64		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ((𝑆 Σg 𝑋)‘𝑛) = ((𝑆 Σg 𝑋)‘𝑗))
65		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ((𝑍 Σg 𝑌)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑗))
66	64, 65	eqeq12d 2637	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)))
67	66	cbvralv 3171	. . . . . . . . 9 ⊢ (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
68		simp-4l 806	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → 𝑁 ∈ Fin)
69		simp-4r 807	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → 𝑀 ∈ Fin)
70		simpr 477	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → 𝑛 ∈ 𝐼)
71	68, 69, 70	3jca 1242	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛 ∈ 𝐼))
72	71	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)) → (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛 ∈ 𝐼))
73		simp-4r 807	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)) → ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌)))
74		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) → ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
75	74	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
76	75	anim1i 592	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)) → (∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)))
77		gsmsymgrfix.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (SymGrp‘𝑁)
78		gsmsymgrfix.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (Base‘𝑆)
79		gsmsymgreq.z	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = (SymGrp‘𝑀)
80		gsmsymgreq.p	. . . . . . . . . . . . . . 15 ⊢ 𝑃 = (Base‘𝑍)
81		gsmsymgreq.i	. . . . . . . . . . . . . . 15 ⊢ 𝐼 = (𝑁 ∩ 𝑀)
82	77, 78, 79, 80, 81	gsmsymgreqlem1 17850	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛 ∈ 𝐼) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ((∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
83	82	imp 445	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛 ∈ 𝐼) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ (∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶‘𝑛) = (𝑅‘𝑛))) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))
84	72, 73, 76, 83	syl21anc 1325	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))
85	84	ex 450	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → ((𝐶‘𝑛) = (𝑅‘𝑛) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
86	85	ralimdva 2962	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) → (∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
87	86	expcom 451	. . . . . . . . 9 ⊢ (∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → (∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
88	67, 87	sylbi 207	. . . . . . . 8 ⊢ (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → (∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
89	88	com23 86	. . . . . . 7 ⊢ (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) → (∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
90	63, 89	syl 17	. . . . . 6 ⊢ ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
91	90	impancom 456	. . . . 5 ⊢ ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛)) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
92	91	com13 88	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛)) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
93	92	imp 445	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛)) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
94	62, 93	sylbid 230	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
95	94	ex 450	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∪ cun 3572   ∩ cin 3573  {csn 4177  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  0cc0 9936  1c1 9937   + caddc 9939  ℕ₀cn0 11292  ℤ_≥cuz 11687  ..^cfzo 12465  #chash 13117  Word cword 13291   ++ cconcat 13293  ⟨“cs1 13294  Basecbs 15857   Σ_g cgsu 16101  SymGrpcsymg 17797

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-rmo 2920  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-uni 4437  df-int 4476  df-iun 4522  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-oadd 7564  df-er 7742  df-map 7859  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-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-tset 15960  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-symg 17798

This theorem is referenced by:  gsmsymgreq  17852