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Theorem gsmsymgreqlem2 17897
 Description: Lemma 2 for gsmsymgreq 17898. (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 = (𝑁𝑀)
Assertion
Ref Expression
gsmsymgreqlem2 (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛𝐼 ((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.
StepHypRef Expression
1 ccatws1len 13437 . . . . . . . . . 10 (𝑋 ∈ Word 𝐵 → (#‘(𝑋 ++ ⟨“𝐶”⟩)) = ((#‘𝑋) + 1))
21adantr 480 . . . . . . . . 9 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (#‘(𝑋 ++ ⟨“𝐶”⟩)) = ((#‘𝑋) + 1))
323ad2ant1 1102 . . . . . . . 8 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (#‘(𝑋 ++ ⟨“𝐶”⟩)) = ((#‘𝑋) + 1))
43oveq2d 6706 . . . . . . 7 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩))) = (0..^((#‘𝑋) + 1)))
5 lencl 13356 . . . . . . . . . . 11 (𝑋 ∈ Word 𝐵 → (#‘𝑋) ∈ ℕ0)
6 elnn0uz 11763 . . . . . . . . . . 11 ((#‘𝑋) ∈ ℕ0 ↔ (#‘𝑋) ∈ (ℤ‘0))
75, 6sylib 208 . . . . . . . . . 10 (𝑋 ∈ Word 𝐵 → (#‘𝑋) ∈ (ℤ‘0))
87adantr 480 . . . . . . . . 9 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (#‘𝑋) ∈ (ℤ‘0))
9 fzosplitsn 12616 . . . . . . . . 9 ((#‘𝑋) ∈ (ℤ‘0) → (0..^((#‘𝑋) + 1)) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
108, 9syl 17 . . . . . . . 8 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (0..^((#‘𝑋) + 1)) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
11103ad2ant1 1102 . . . . . . 7 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (0..^((#‘𝑋) + 1)) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
124, 11eqtrd 2685 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
1312raleqdv 3174 . . . . 5 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ ((0..^(#‘𝑋)) ∪ {(#‘𝑋)})∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛)))
145adantr 480 . . . . . . 7 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (#‘𝑋) ∈ ℕ0)
15143ad2ant1 1102 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (#‘𝑋) ∈ ℕ0)
16 fveq2 6229 . . . . . . . . . 10 (𝑖 = (#‘𝑋) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = ((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋)))
1716fveq1d 6231 . . . . . . . . 9 (𝑖 = (#‘𝑋) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛))
18 fveq2 6229 . . . . . . . . . 10 (𝑖 = (#‘𝑋) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = ((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋)))
1918fveq1d 6231 . . . . . . . . 9 (𝑖 = (#‘𝑋) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛))
2017, 19eqeq12d 2666 . . . . . . . 8 (𝑖 = (#‘𝑋) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛)))
2120ralbidv 3015 . . . . . . 7 (𝑖 = (#‘𝑋) → (∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛)))
2221ralunsn 4454 . . . . . 6 ((#‘𝑋) ∈ ℕ0 → (∀𝑖 ∈ ((0..^(#‘𝑋)) ∪ {(#‘𝑋)})∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛))))
2315, 22syl 17 . . . . 5 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (∀𝑖 ∈ ((0..^(#‘𝑋)) ∪ {(#‘𝑋)})∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛))))
24 simpl 472 . . . . . . . . . . . . 13 ((𝑋 ∈ Word 𝐵𝐶𝐵) → 𝑋 ∈ Word 𝐵)
25243ad2ant1 1102 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → 𝑋 ∈ Word 𝐵)
2625adantr 480 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑋 ∈ Word 𝐵)
27 simpr 476 . . . . . . . . . . . . 13 ((𝑋 ∈ Word 𝐵𝐶𝐵) → 𝐶𝐵)
28273ad2ant1 1102 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → 𝐶𝐵)
2928adantr 480 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝐶𝐵)
30 simpr 476 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑖 ∈ (0..^(#‘𝑋)))
31 ccats1val1 13446 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝐵𝐶𝐵𝑖 ∈ (0..^(#‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋𝑖))
3226, 29, 30, 31syl3anc 1366 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋𝑖))
3332fveq1d 6231 . . . . . . . . 9 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = ((𝑋𝑖)‘𝑛))
34 simpl2l 1134 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑌 ∈ Word 𝑃)
35 simpl2r 1135 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑅𝑃)
36 oveq2 6698 . . . . . . . . . . . . . . 15 ((#‘𝑋) = (#‘𝑌) → (0..^(#‘𝑋)) = (0..^(#‘𝑌)))
3736eleq2d 2716 . . . . . . . . . . . . . 14 ((#‘𝑋) = (#‘𝑌) → (𝑖 ∈ (0..^(#‘𝑋)) ↔ 𝑖 ∈ (0..^(#‘𝑌))))
3837biimpd 219 . . . . . . . . . . . . 13 ((#‘𝑋) = (#‘𝑌) → (𝑖 ∈ (0..^(#‘𝑋)) → 𝑖 ∈ (0..^(#‘𝑌))))
39383ad2ant3 1104 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑖 ∈ (0..^(#‘𝑋)) → 𝑖 ∈ (0..^(#‘𝑌))))
4039imp 444 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑖 ∈ (0..^(#‘𝑌)))
41 ccats1val1 13446 . . . . . . . . . . 11 ((𝑌 ∈ Word 𝑃𝑅𝑃𝑖 ∈ (0..^(#‘𝑌))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌𝑖))
4234, 35, 40, 41syl3anc 1366 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌𝑖))
4342fveq1d 6231 . . . . . . . . 9 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛))
4433, 43eqeq12d 2666 . . . . . . . 8 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛)))
4544ralbidv 3015 . . . . . . 7 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → (∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛)))
4645ralbidva 3014 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛)))
47 eqidd 2652 . . . . . . . . . . . 12 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (#‘𝑋) = (#‘𝑋))
4824, 27, 473jca 1261 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (𝑋 ∈ Word 𝐵𝐶𝐵 ∧ (#‘𝑋) = (#‘𝑋)))
49483ad2ant1 1102 . . . . . . . . . 10 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑋 ∈ Word 𝐵𝐶𝐵 ∧ (#‘𝑋) = (#‘𝑋)))
50 ccats1val2 13447 . . . . . . . . . 10 ((𝑋 ∈ Word 𝐵𝐶𝐵 ∧ (#‘𝑋) = (#‘𝑋)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋)) = 𝐶)
5149, 50syl 17 . . . . . . . . 9 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋)) = 𝐶)
5251fveq1d 6231 . . . . . . . 8 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (𝐶𝑛))
53 df-3an 1056 . . . . . . . . . . 11 ((𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (#‘𝑋) = (#‘𝑌)) ↔ ((𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)))
5453biimpri 218 . . . . . . . . . 10 (((𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (#‘𝑋) = (#‘𝑌)))
55543adant1 1099 . . . . . . . . 9 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (#‘𝑋) = (#‘𝑌)))
56 ccats1val2 13447 . . . . . . . . . 10 ((𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (#‘𝑋) = (#‘𝑌)) → ((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋)) = 𝑅)
5756fveq1d 6231 . . . . . . . . 9 ((𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (#‘𝑋) = (#‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) = (𝑅𝑛))
5855, 57syl 17 . . . . . . . 8 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) = (𝑅𝑛))
5952, 58eqeq12d 2666 . . . . . . 7 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → ((((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) ↔ (𝐶𝑛) = (𝑅𝑛)))
6059ralbidv 3015 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) ↔ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛)))
6146, 60anbi12d 747 . . . . 5 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛)) ↔ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛))))
6213, 23, 613bitrd 294 . . . 4 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛))))
6362ad2antlr 763 . . 3 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛))))
64 pm3.35 610 . . . . . . 7 ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))
65 fveq2 6229 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝑆 Σg 𝑋)‘𝑛) = ((𝑆 Σg 𝑋)‘𝑗))
66 fveq2 6229 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝑍 Σg 𝑌)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑗))
6765, 66eqeq12d 2666 . . . . . . . . . 10 (𝑛 = 𝑗 → (((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)))
6867cbvralv 3201 . . . . . . . . 9 (∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
69 simp-4l 823 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → 𝑁 ∈ Fin)
70 simp-4r 824 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → 𝑀 ∈ Fin)
71 simpr 476 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → 𝑛𝐼)
7269, 70, 713jca 1261 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛𝐼))
7372adantr 480 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) ∧ (𝐶𝑛) = (𝑅𝑛)) → (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛𝐼))
74 simp-4r 824 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) ∧ (𝐶𝑛) = (𝑅𝑛)) → ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)))
75 simpr 476 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) → ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
7675adantr 480 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
7776anim1i 591 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) ∧ (𝐶𝑛) = (𝑅𝑛)) → (∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶𝑛) = (𝑅𝑛)))
78 gsmsymgrfix.s . . . . . . . . . . . . . . 15 𝑆 = (SymGrp‘𝑁)
79 gsmsymgrfix.b . . . . . . . . . . . . . . 15 𝐵 = (Base‘𝑆)
80 gsmsymgreq.z . . . . . . . . . . . . . . 15 𝑍 = (SymGrp‘𝑀)
81 gsmsymgreq.p . . . . . . . . . . . . . . 15 𝑃 = (Base‘𝑍)
82 gsmsymgreq.i . . . . . . . . . . . . . . 15 𝐼 = (𝑁𝑀)
8378, 79, 80, 81, 82gsmsymgreqlem1 17896 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛𝐼) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ((∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶𝑛) = (𝑅𝑛)) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
8483imp 444 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛𝐼) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ (∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶𝑛) = (𝑅𝑛))) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))
8573, 74, 77, 84syl21anc 1365 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) ∧ (𝐶𝑛) = (𝑅𝑛)) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))
8685ex 449 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → ((𝐶𝑛) = (𝑅𝑛) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
8786ralimdva 2991 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) → (∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
8887expcom 450 . . . . . . . . 9 (∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → (∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
8968, 88sylbi 207 . . . . . . . 8 (∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → (∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
9089com23 86 . . . . . . 7 (∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) → (∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
9164, 90syl 17 . . . . . 6 ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
9291impancom 455 . . . . 5 ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛)) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
9392com13 88 . . . 4 (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛)) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
9493imp 444 . . 3 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛)) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
9563, 94sylbid 230 . 2 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) ∧ (∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
9695ex 449 1 (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ∪ cun 3605   ∩ cin 3606  {csn 4210  ‘cfv 5926  (class class class)co 6690  Fincfn 7997  0cc0 9974  1c1 9975   + caddc 9977  ℕ0cn0 11330  ℤ≥cuz 11725  ..^cfzo 12504  #chash 13157  Word cword 13323   ++ cconcat 13325  ⟨“cs1 13326  Basecbs 15904   Σg cgsu 16148  SymGrpcsymg 17843 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-tset 16007  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-symg 17844 This theorem is referenced by:  gsmsymgreq  17898
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