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Theorem gsmsymgreqlem2 17674
Description: Lemma 2 for gsmsymgreq 17675. (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 13251 . . . . . . . . 9 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (#‘(𝑋 ++ ⟨“𝐶”⟩)) = ((#‘𝑋) + 1))
213ad2ant1 1075 . . . . . . . 8 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (#‘(𝑋 ++ ⟨“𝐶”⟩)) = ((#‘𝑋) + 1))
32oveq2d 6565 . . . . . . 7 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩))) = (0..^((#‘𝑋) + 1)))
4 lencl 13179 . . . . . . . . . . 11 (𝑋 ∈ Word 𝐵 → (#‘𝑋) ∈ ℕ0)
5 elnn0uz 11601 . . . . . . . . . . 11 ((#‘𝑋) ∈ ℕ0 ↔ (#‘𝑋) ∈ (ℤ‘0))
64, 5sylib 207 . . . . . . . . . 10 (𝑋 ∈ Word 𝐵 → (#‘𝑋) ∈ (ℤ‘0))
76adantr 480 . . . . . . . . 9 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (#‘𝑋) ∈ (ℤ‘0))
8 fzosplitsn 12442 . . . . . . . . 9 ((#‘𝑋) ∈ (ℤ‘0) → (0..^((#‘𝑋) + 1)) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
97, 8syl 17 . . . . . . . 8 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (0..^((#‘𝑋) + 1)) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
1093ad2ant1 1075 . . . . . . 7 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (0..^((#‘𝑋) + 1)) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
113, 10eqtrd 2644 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(#‘𝑋)) ∪ {(#‘𝑋)}))
1211raleqdv 3121 . . . . 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)))
134adantr 480 . . . . . . 7 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (#‘𝑋) ∈ ℕ0)
14133ad2ant1 1075 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (#‘𝑋) ∈ ℕ0)
15 fveq2 6103 . . . . . . . . . 10 (𝑖 = (#‘𝑋) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = ((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋)))
1615fveq1d 6105 . . . . . . . . 9 (𝑖 = (#‘𝑋) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛))
17 fveq2 6103 . . . . . . . . . 10 (𝑖 = (#‘𝑋) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = ((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋)))
1817fveq1d 6105 . . . . . . . . 9 (𝑖 = (#‘𝑋) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛))
1916, 18eqeq12d 2625 . . . . . . . 8 (𝑖 = (#‘𝑋) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛)))
2019ralbidv 2969 . . . . . . 7 (𝑖 = (#‘𝑋) → (∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛)))
2120ralunsn 4360 . . . . . 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))))
2214, 21syl 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))))
23 simpl 472 . . . . . . . . . . . . 13 ((𝑋 ∈ Word 𝐵𝐶𝐵) → 𝑋 ∈ Word 𝐵)
24233ad2ant1 1075 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → 𝑋 ∈ Word 𝐵)
2524adantr 480 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑋 ∈ Word 𝐵)
26 simpr 476 . . . . . . . . . . . . 13 ((𝑋 ∈ Word 𝐵𝐶𝐵) → 𝐶𝐵)
27263ad2ant1 1075 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → 𝐶𝐵)
2827adantr 480 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝐶𝐵)
29 simpr 476 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑖 ∈ (0..^(#‘𝑋)))
30 ccats1val1 13255 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝐵𝐶𝐵𝑖 ∈ (0..^(#‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋𝑖))
3125, 28, 29, 30syl3anc 1318 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋𝑖))
3231fveq1d 6105 . . . . . . . . 9 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = ((𝑋𝑖)‘𝑛))
33 simpl2l 1107 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑌 ∈ Word 𝑃)
34 simpl2r 1108 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑅𝑃)
35 oveq2 6557 . . . . . . . . . . . . . . 15 ((#‘𝑋) = (#‘𝑌) → (0..^(#‘𝑋)) = (0..^(#‘𝑌)))
3635eleq2d 2673 . . . . . . . . . . . . . 14 ((#‘𝑋) = (#‘𝑌) → (𝑖 ∈ (0..^(#‘𝑋)) ↔ 𝑖 ∈ (0..^(#‘𝑌))))
3736biimpd 218 . . . . . . . . . . . . 13 ((#‘𝑋) = (#‘𝑌) → (𝑖 ∈ (0..^(#‘𝑋)) → 𝑖 ∈ (0..^(#‘𝑌))))
38373ad2ant3 1077 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑖 ∈ (0..^(#‘𝑋)) → 𝑖 ∈ (0..^(#‘𝑌))))
3938imp 444 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → 𝑖 ∈ (0..^(#‘𝑌)))
40 ccats1val1 13255 . . . . . . . . . . 11 ((𝑌 ∈ Word 𝑃𝑅𝑃𝑖 ∈ (0..^(#‘𝑌))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌𝑖))
4133, 34, 39, 40syl3anc 1318 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌𝑖))
4241fveq1d 6105 . . . . . . . . 9 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛))
4332, 42eqeq12d 2625 . . . . . . . 8 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) ∧ 𝑖 ∈ (0..^(#‘𝑋))) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛)))
4443ralbidv 2969 . . . . . . 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)))
4544ralbidva 2968 . . . . . 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)))
46 eqidd 2611 . . . . . . . . . . . 12 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (#‘𝑋) = (#‘𝑋))
4723, 26, 463jca 1235 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (𝑋 ∈ Word 𝐵𝐶𝐵 ∧ (#‘𝑋) = (#‘𝑋)))
48473ad2ant1 1075 . . . . . . . . . 10 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑋 ∈ Word 𝐵𝐶𝐵 ∧ (#‘𝑋) = (#‘𝑋)))
49 ccats1val2 13256 . . . . . . . . . 10 ((𝑋 ∈ Word 𝐵𝐶𝐵 ∧ (#‘𝑋) = (#‘𝑋)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋)) = 𝐶)
5048, 49syl 17 . . . . . . . . 9 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋)) = 𝐶)
5150fveq1d 6105 . . . . . . . 8 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (𝐶𝑛))
52 df-3an 1033 . . . . . . . . . . 11 ((𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (#‘𝑋) = (#‘𝑌)) ↔ ((𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)))
5352biimpri 217 . . . . . . . . . 10 (((𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (#‘𝑋) = (#‘𝑌)))
54533adant1 1072 . . . . . . . . 9 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (#‘𝑋) = (#‘𝑌)))
55 ccats1val2 13256 . . . . . . . . . 10 ((𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (#‘𝑋) = (#‘𝑌)) → ((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋)) = 𝑅)
5655fveq1d 6105 . . . . . . . . 9 ((𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (#‘𝑋) = (#‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) = (𝑅𝑛))
5754, 56syl 17 . . . . . . . 8 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) = (𝑅𝑛))
5851, 57eqeq12d 2625 . . . . . . 7 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → ((((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) ↔ (𝐶𝑛) = (𝑅𝑛)))
5958ralbidv 2969 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌)) → (∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(#‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(#‘𝑋))‘𝑛) ↔ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛)))
6045, 59anbi12d 743 . . . . 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))))
6112, 22, 603bitrd 293 . . . 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))))
6261ad2antlr 759 . . 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))))
63 pm3.35 609 . . . . . . 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))
64 fveq2 6103 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝑆 Σg 𝑋)‘𝑛) = ((𝑆 Σg 𝑋)‘𝑗))
65 fveq2 6103 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝑍 Σg 𝑌)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑗))
6664, 65eqeq12d 2625 . . . . . . . . . 10 (𝑛 = 𝑗 → (((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)))
6766cbvralv 3147 . . . . . . . . 9 (∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
68 simp-4l 802 . . . . . . . . . . . . . . 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)
69 simp-4r 803 . . . . . . . . . . . . . . 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)
70 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)
7168, 69, 703jca 1235 . . . . . . . . . . . . . 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))
7271adantr 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))
73 simp-4r 803 . . . . . . . . . . . . 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)))
74 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))
7574adantr 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))
7675anim1i 590 . . . . . . . . . . . . 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)))
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 = (𝑁𝑀)
8277, 78, 79, 80, 81gsmsymgreqlem1 17673 . . . . . . . . . . . . . 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)))
8382imp 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))
8472, 73, 76, 83syl21anc 1317 . . . . . . . . . . . 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))
8584ex 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)))
8685ralimdva 2945 . . . . . . . . . 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)))
8786expcom 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))))
8867, 87sylbi 206 . . . . . . . 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))))
8988com23 84 . . . . . . 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))))
9063, 89syl 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))))
9190impancom 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))))
9291com13 86 . . . 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))))
9392imp 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)))
9462, 93sylbid 229 . 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)))
9594ex 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  cun 3538  cin 3539  {csn 4125  cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816   + caddc 9818  0cn0 11169  cuz 11563  ..^cfzo 12334  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149  Basecbs 15695   Σg cgsu 15924  SymGrpcsymg 17620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-tset 15787  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-symg 17621
This theorem is referenced by:  gsmsymgreq  17675
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