Theorem gsmsymgreqlem2 19473

Description: Lemma 2 for gsmsymgreq 19474. (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 14668	. . . . . . . . . 10 ⊢ (𝑋 ∈ Word 𝐵 → (♯‘(𝑋 ++ ⟨“𝐶”⟩)) = ((♯‘𝑋) + 1))
2	1	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑋 ∈ Word 𝐵 → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = (0..^((♯‘𝑋) + 1)))
3		lencl 14581	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Word 𝐵 → (♯‘𝑋) ∈ ℕ0)
4		elnn0uz 12948	. . . . . . . . . . 11 ⊢ ((♯‘𝑋) ∈ ℕ0 ↔ (♯‘𝑋) ∈ (ℤ≥‘0))
5	3, 4	sylib 218	. . . . . . . . . 10 ⊢ (𝑋 ∈ Word 𝐵 → (♯‘𝑋) ∈ (ℤ≥‘0))
6		fzosplitsn 13825	. . . . . . . . . 10 ⊢ ((♯‘𝑋) ∈ (ℤ≥‘0) → (0..^((♯‘𝑋) + 1)) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
7	5, 6	syl 17	. . . . . . . . 9 ⊢ (𝑋 ∈ Word 𝐵 → (0..^((♯‘𝑋) + 1)) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
8	2, 7	eqtrd 2780	. . . . . . . 8 ⊢ (𝑋 ∈ Word 𝐵 → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
9	8	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
10	9	3ad2ant1 1133	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
11	10	raleqdv 3334	. . . . 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)))
12	3	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (♯‘𝑋) ∈ ℕ0)
13	12	3ad2ant1 1133	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (♯‘𝑋) ∈ ℕ0)
14		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑖 = (♯‘𝑋) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = ((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋)))
15	14	fveq1d 6922	. . . . . . . . 9 ⊢ (𝑖 = (♯‘𝑋) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛))
16		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑖 = (♯‘𝑋) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = ((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋)))
17	16	fveq1d 6922	. . . . . . . . 9 ⊢ (𝑖 = (♯‘𝑋) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛))
18	15, 17	eqeq12d 2756	. . . . . . . 8 ⊢ (𝑖 = (♯‘𝑋) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛)))
19	18	ralbidv 3184	. . . . . . 7 ⊢ (𝑖 = (♯‘𝑋) → (∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛)))
20	19	ralunsn 4918	. . . . . 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))))
21	13, 20	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))))
22		simp1l 1197	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → 𝑋 ∈ Word 𝐵)
23		ccats1val1 14674	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋‘𝑖))
24	22, 23	sylan 579	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋‘𝑖))
25	24	fveq1d 6922	. . . . . . . . 9 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = ((𝑋‘𝑖)‘𝑛))
26		simp2l 1199	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → 𝑌 ∈ Word 𝑃)
27		oveq2 7456	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑋) = (♯‘𝑌) → (0..^(♯‘𝑋)) = (0..^(♯‘𝑌)))
28	27	eleq2d 2830	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑋) = (♯‘𝑌) → (𝑖 ∈ (0..^(♯‘𝑋)) ↔ 𝑖 ∈ (0..^(♯‘𝑌))))
29	28	biimpd 229	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑋) = (♯‘𝑌) → (𝑖 ∈ (0..^(♯‘𝑋)) → 𝑖 ∈ (0..^(♯‘𝑌))))
30	29	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (𝑖 ∈ (0..^(♯‘𝑋)) → 𝑖 ∈ (0..^(♯‘𝑌))))
31	30	imp 406	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → 𝑖 ∈ (0..^(♯‘𝑌)))
32		ccats1val1 14674	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑖 ∈ (0..^(♯‘𝑌))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌‘𝑖))
33	26, 31, 32	syl2an2r 684	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌‘𝑖))
34	33	fveq1d 6922	. . . . . . . . 9 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛))
35	25, 34	eqeq12d 2756	. . . . . . . 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)))
36	35	ralbidv 3184	. . . . . . 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)))
37	36	ralbidva 3182	. . . . . 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)))
38		eqidd 2741	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (♯‘𝑋) = (♯‘𝑋))
39		ccats1val2 14675	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (♯‘𝑋) = (♯‘𝑋)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋)) = 𝐶)
40	39	fveq1d 6922	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (♯‘𝑋) = (♯‘𝑋)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶‘𝑛))
41	38, 40	mpd3an3 1462	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶‘𝑛))
42	41	3ad2ant1 1133	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶‘𝑛))
43		ccats1val2 14675	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)) → ((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋)) = 𝑅)
44	43	fveq1d 6922	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅‘𝑛))
45	44	3expa 1118	. . . . . . . . 9 ⊢ (((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅‘𝑛))
46	45	3adant1 1130	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅‘𝑛))
47	42, 46	eqeq12d 2756	. . . . . . 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)))
48	47	ralbidv 3184	. . . . . 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)))
49	37, 48	anbi12d 631	. . . . 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))))
50	11, 21, 49	3bitrd 305	. . . 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))))
51	50	ad2antlr 726	. . 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))))
52		pm3.35 802	. . . . . . 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))
53		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ((𝑆 Σg 𝑋)‘𝑛) = ((𝑆 Σg 𝑋)‘𝑗))
54		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ((𝑍 Σg 𝑌)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑗))
55	53, 54	eqeq12d 2756	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)))
56	55	cbvralvw 3243	. . . . . . . . 9 ⊢ (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
57		simp-4l 782	. . . . . . . . . . . . . . 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)
58		simp-4r 783	. . . . . . . . . . . . . . 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)
59		simpr 484	. . . . . . . . . . . . . . 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)
60	57, 58, 59	3jca 1128	. . . . . . . . . . . . . 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))
61	60	adantr 480	. . . . . . . . . . . . 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))
62		simp-4r 783	. . . . . . . . . . . . 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)))
63		simplr 768	. . . . . . . . . . . . . 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))
64	63	anim1i 614	. . . . . . . . . . . . 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)))
65		gsmsymgrfix.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (SymGrp‘𝑁)
66		gsmsymgrfix.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (Base‘𝑆)
67		gsmsymgreq.z	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = (SymGrp‘𝑀)
68		gsmsymgreq.p	. . . . . . . . . . . . . . 15 ⊢ 𝑃 = (Base‘𝑍)
69		gsmsymgreq.i	. . . . . . . . . . . . . . 15 ⊢ 𝐼 = (𝑁 ∩ 𝑀)
70	65, 66, 67, 68, 69	gsmsymgreqlem1 19472	. . . . . . . . . . . . . 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)))
71	70	imp 406	. . . . . . . . . . . . 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))
72	61, 62, 64, 71	syl21anc 837	. . . . . . . . . . . 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))
73	72	ex 412	. . . . . . . . . . 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)))
74	73	ralimdva 3173	. . . . . . . . . 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)))
75	74	expcom 413	. . . . . . . . 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))))
76	56, 75	sylbi 217	. . . . . . . 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))))
77	76	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))))
78	52, 77	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))))
79	78	impancom 451	. . . . 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))))
80	79	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))))
81	80	imp 406	. . 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)))
82	51, 81	sylbid 240	. 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)))
83	82	ex 412	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∪ cun 3974   ∩ cin 3975  {csn 4648  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  0cc0 11184  1c1 11185   + caddc 11187  ℕ₀cn0 12553  ℤ_≥cuz 12903  ..^cfzo 13711  ♯chash 14379  Word cword 14562   ++ cconcat 14618  ⟨“cs1 14643  Basecbs 17258   Σ_g cgsu 17500  SymGrpcsymg 19410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-tset 17330  df-0g 17501  df-gsum 17502  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-efmnd 18904  df-grp 18976  df-symg 19411

This theorem is referenced by:  gsmsymgreq  19474