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Theorem psgndiflemB 19865
Description: Lemma 1 for psgndif 19867. (Contributed by AV, 27-Jan-2019.)
Hypotheses
Ref Expression
psgnfix.p 𝑃 = (Base‘(SymGrp‘𝑁))
psgnfix.t 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
psgnfix.s 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾}))
psgnfix.z 𝑍 = (SymGrp‘𝑁)
psgnfix.r 𝑅 = ran (pmTrsp‘𝑁)
Assertion
Ref Expression
psgndiflemB (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))) → 𝑄 = (𝑍 Σg 𝑈))))
Distinct variable groups:   𝐾,𝑞   𝑃,𝑞   𝑄,𝑞   𝑖,𝐾,𝑛   𝑖,𝑁,𝑛   𝑆,𝑖,𝑛   𝑈,𝑖,𝑛   𝑖,𝑊,𝑛   𝑖,𝑍,𝑛
Allowed substitution hints:   𝑃(𝑖,𝑛)   𝑄(𝑖,𝑛)   𝑅(𝑖,𝑛,𝑞)   𝑆(𝑞)   𝑇(𝑖,𝑛,𝑞)   𝑈(𝑞)   𝑁(𝑞)   𝑊(𝑞)   𝑍(𝑞)

Proof of Theorem psgndiflemB
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elrabi 3342 . . . . 5 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} → 𝑄𝑃)
2 eqid 2621 . . . . . 6 (SymGrp‘𝑁) = (SymGrp‘𝑁)
3 psgnfix.p . . . . . 6 𝑃 = (Base‘(SymGrp‘𝑁))
42, 3symgbasf 17725 . . . . 5 (𝑄𝑃𝑄:𝑁𝑁)
5 ffn 6002 . . . . 5 (𝑄:𝑁𝑁𝑄 Fn 𝑁)
61, 4, 53syl 18 . . . 4 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} → 𝑄 Fn 𝑁)
76ad3antlr 766 . . 3 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → 𝑄 Fn 𝑁)
8 simpl 473 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝐾𝑁) → 𝑁 ∈ Fin)
98adantr 481 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) → 𝑁 ∈ Fin)
109adantr 481 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) → 𝑁 ∈ Fin)
11 simp1 1059 . . . . . 6 ((𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))) → 𝑈 ∈ Word 𝑅)
1210, 11anim12i 589 . . . . 5 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → (𝑁 ∈ Fin ∧ 𝑈 ∈ Word 𝑅))
13 psgnfix.z . . . . . 6 𝑍 = (SymGrp‘𝑁)
1413eqcomi 2630 . . . . . . . 8 (SymGrp‘𝑁) = 𝑍
1514fveq2i 6151 . . . . . . 7 (Base‘(SymGrp‘𝑁)) = (Base‘𝑍)
163, 15eqtri 2643 . . . . . 6 𝑃 = (Base‘𝑍)
17 psgnfix.r . . . . . 6 𝑅 = ran (pmTrsp‘𝑁)
1813, 16, 17gsmtrcl 17857 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑈 ∈ Word 𝑅) → (𝑍 Σg 𝑈) ∈ 𝑃)
1912, 18syl 17 . . . 4 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → (𝑍 Σg 𝑈) ∈ 𝑃)
202, 3symgbasf 17725 . . . 4 ((𝑍 Σg 𝑈) ∈ 𝑃 → (𝑍 Σg 𝑈):𝑁𝑁)
21 ffn 6002 . . . 4 ((𝑍 Σg 𝑈):𝑁𝑁 → (𝑍 Σg 𝑈) Fn 𝑁)
2219, 20, 213syl 18 . . 3 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → (𝑍 Σg 𝑈) Fn 𝑁)
238ad3antrrr 765 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → 𝑁 ∈ Fin)
24 simpr 477 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝐾𝑁) → 𝐾𝑁)
2524ad3antrrr 765 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → 𝐾𝑁)
26 eqid 2621 . . . . . . . . . . . . . . . 16 (Base‘𝑍) = (Base‘𝑍)
2717, 13, 26symgtrf 17810 . . . . . . . . . . . . . . 15 𝑅 ⊆ (Base‘𝑍)
28 sswrd 13252 . . . . . . . . . . . . . . . 16 (𝑅 ⊆ (Base‘𝑍) → Word 𝑅 ⊆ Word (Base‘𝑍))
2928sseld 3582 . . . . . . . . . . . . . . 15 (𝑅 ⊆ (Base‘𝑍) → (𝑈 ∈ Word 𝑅𝑈 ∈ Word (Base‘𝑍)))
3027, 29ax-mp 5 . . . . . . . . . . . . . 14 (𝑈 ∈ Word 𝑅𝑈 ∈ Word (Base‘𝑍))
31303ad2ant1 1080 . . . . . . . . . . . . 13 ((𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))) → 𝑈 ∈ Word (Base‘𝑍))
3231adantl 482 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → 𝑈 ∈ Word (Base‘𝑍))
3323, 25, 323jca 1240 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → (𝑁 ∈ Fin ∧ 𝐾𝑁𝑈 ∈ Word (Base‘𝑍)))
34 simpl 473 . . . . . . . . . . . . . . 15 ((((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)) → ((𝑈𝑖)‘𝐾) = 𝐾)
3534ralimi 2947 . . . . . . . . . . . . . 14 (∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)) → ∀𝑖 ∈ (0..^(#‘𝑊))((𝑈𝑖)‘𝐾) = 𝐾)
36353ad2ant3 1082 . . . . . . . . . . . . 13 ((𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))) → ∀𝑖 ∈ (0..^(#‘𝑊))((𝑈𝑖)‘𝐾) = 𝐾)
3736adantl 482 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → ∀𝑖 ∈ (0..^(#‘𝑊))((𝑈𝑖)‘𝐾) = 𝐾)
38 oveq2 6612 . . . . . . . . . . . . . . . 16 ((#‘𝑈) = (#‘𝑊) → (0..^(#‘𝑈)) = (0..^(#‘𝑊)))
3938eqcoms 2629 . . . . . . . . . . . . . . 15 ((#‘𝑊) = (#‘𝑈) → (0..^(#‘𝑈)) = (0..^(#‘𝑊)))
4039raleqdv 3133 . . . . . . . . . . . . . 14 ((#‘𝑊) = (#‘𝑈) → (∀𝑖 ∈ (0..^(#‘𝑈))((𝑈𝑖)‘𝐾) = 𝐾 ↔ ∀𝑖 ∈ (0..^(#‘𝑊))((𝑈𝑖)‘𝐾) = 𝐾))
41403ad2ant2 1081 . . . . . . . . . . . . 13 ((𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))) → (∀𝑖 ∈ (0..^(#‘𝑈))((𝑈𝑖)‘𝐾) = 𝐾 ↔ ∀𝑖 ∈ (0..^(#‘𝑊))((𝑈𝑖)‘𝐾) = 𝐾))
4241adantl 482 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → (∀𝑖 ∈ (0..^(#‘𝑈))((𝑈𝑖)‘𝐾) = 𝐾 ↔ ∀𝑖 ∈ (0..^(#‘𝑊))((𝑈𝑖)‘𝐾) = 𝐾))
4337, 42mpbird 247 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → ∀𝑖 ∈ (0..^(#‘𝑈))((𝑈𝑖)‘𝐾) = 𝐾)
4413, 26gsmsymgrfix 17769 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝐾𝑁𝑈 ∈ Word (Base‘𝑍)) → (∀𝑖 ∈ (0..^(#‘𝑈))((𝑈𝑖)‘𝐾) = 𝐾 → ((𝑍 Σg 𝑈)‘𝐾) = 𝐾))
4533, 43, 44sylc 65 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → ((𝑍 Σg 𝑈)‘𝐾) = 𝐾)
4645eqcomd 2627 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → 𝐾 = ((𝑍 Σg 𝑈)‘𝐾))
4746adantr 481 . . . . . . . 8 ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘 = 𝐾) → 𝐾 = ((𝑍 Σg 𝑈)‘𝐾))
48 fveq2 6148 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑄𝑘) = (𝑄𝐾))
49 fveq1 6147 . . . . . . . . . . . . . 14 (𝑞 = 𝑄 → (𝑞𝐾) = (𝑄𝐾))
5049eqeq1d 2623 . . . . . . . . . . . . 13 (𝑞 = 𝑄 → ((𝑞𝐾) = 𝐾 ↔ (𝑄𝐾) = 𝐾))
5150elrab 3346 . . . . . . . . . . . 12 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} ↔ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐾))
5251simprbi 480 . . . . . . . . . . 11 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} → (𝑄𝐾) = 𝐾)
5352ad3antlr 766 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → (𝑄𝐾) = 𝐾)
5448, 53sylan9eqr 2677 . . . . . . . . 9 ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘 = 𝐾) → (𝑄𝑘) = 𝐾)
55 fveq2 6148 . . . . . . . . . 10 (𝑘 = 𝐾 → ((𝑍 Σg 𝑈)‘𝑘) = ((𝑍 Σg 𝑈)‘𝐾))
5655adantl 482 . . . . . . . . 9 ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘 = 𝐾) → ((𝑍 Σg 𝑈)‘𝑘) = ((𝑍 Σg 𝑈)‘𝐾))
5754, 56eqeq12d 2636 . . . . . . . 8 ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘 = 𝐾) → ((𝑄𝑘) = ((𝑍 Σg 𝑈)‘𝑘) ↔ 𝐾 = ((𝑍 Σg 𝑈)‘𝐾)))
5847, 57mpbird 247 . . . . . . 7 ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘 = 𝐾) → (𝑄𝑘) = ((𝑍 Σg 𝑈)‘𝑘))
5958ex 450 . . . . . 6 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → (𝑘 = 𝐾 → (𝑄𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
6059adantr 481 . . . . 5 ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁) → (𝑘 = 𝐾 → (𝑄𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
6160com12 32 . . . 4 (𝑘 = 𝐾 → ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁) → (𝑄𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
62 fveq1 6147 . . . . . . . . 9 ((𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
6362adantl 482 . . . . . . . 8 ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
6463ad3antlr 766 . . . . . . 7 ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
6564adantl 482 . . . . . 6 ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁)) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
66 simpr 477 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑘𝑁) → 𝑘𝑁)
67 df-ne 2791 . . . . . . . . . . . . . 14 (𝑘𝐾 ↔ ¬ 𝑘 = 𝐾)
6867biimpri 218 . . . . . . . . . . . . 13 𝑘 = 𝐾𝑘𝐾)
6966, 68anim12i 589 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑘𝑁) ∧ ¬ 𝑘 = 𝐾) → (𝑘𝑁𝑘𝐾))
70 eldifsn 4287 . . . . . . . . . . . 12 (𝑘 ∈ (𝑁 ∖ {𝐾}) ↔ (𝑘𝑁𝑘𝐾))
7169, 70sylibr 224 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑘𝑁) ∧ ¬ 𝑘 = 𝐾) → 𝑘 ∈ (𝑁 ∖ {𝐾}))
72 fvres 6164 . . . . . . . . . . 11 (𝑘 ∈ (𝑁 ∖ {𝐾}) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄𝑘))
7371, 72syl 17 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑘𝑁) ∧ ¬ 𝑘 = 𝐾) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄𝑘))
7473exp31 629 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝐾𝑁) → (𝑘𝑁 → (¬ 𝑘 = 𝐾 → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄𝑘))))
7574ad3antrrr 765 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → (𝑘𝑁 → (¬ 𝑘 = 𝐾 → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄𝑘))))
7675imp 445 . . . . . . 7 ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁) → (¬ 𝑘 = 𝐾 → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄𝑘)))
7776impcom 446 . . . . . 6 ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁)) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄𝑘))
7868anim2i 592 . . . . . . . . . . 11 ((𝑘𝑁 ∧ ¬ 𝑘 = 𝐾) → (𝑘𝑁𝑘𝐾))
7978, 70sylibr 224 . . . . . . . . . 10 ((𝑘𝑁 ∧ ¬ 𝑘 = 𝐾) → 𝑘 ∈ (𝑁 ∖ {𝐾}))
8079ex 450 . . . . . . . . 9 (𝑘𝑁 → (¬ 𝑘 = 𝐾𝑘 ∈ (𝑁 ∖ {𝐾})))
8180adantl 482 . . . . . . . 8 ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁) → (¬ 𝑘 = 𝐾𝑘 ∈ (𝑁 ∖ {𝐾})))
8281impcom 446 . . . . . . 7 ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁)) → 𝑘 ∈ (𝑁 ∖ {𝐾}))
83 diffi 8136 . . . . . . . . . . . . 13 (𝑁 ∈ Fin → (𝑁 ∖ {𝐾}) ∈ Fin)
8483ancri 574 . . . . . . . . . . . 12 (𝑁 ∈ Fin → ((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin))
8584adantr 481 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝐾𝑁) → ((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin))
8685ad3antrrr 765 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → ((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin))
87 psgnfix.t . . . . . . . . . . . . . . 15 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
88 psgnfix.s . . . . . . . . . . . . . . 15 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾}))
89 eqid 2621 . . . . . . . . . . . . . . 15 (Base‘𝑆) = (Base‘𝑆)
9087, 88, 89symgtrf 17810 . . . . . . . . . . . . . 14 𝑇 ⊆ (Base‘𝑆)
91 sswrd 13252 . . . . . . . . . . . . . . 15 (𝑇 ⊆ (Base‘𝑆) → Word 𝑇 ⊆ Word (Base‘𝑆))
9291sseld 3582 . . . . . . . . . . . . . 14 (𝑇 ⊆ (Base‘𝑆) → (𝑊 ∈ Word 𝑇𝑊 ∈ Word (Base‘𝑆)))
9390, 92ax-mp 5 . . . . . . . . . . . . 13 (𝑊 ∈ Word 𝑇𝑊 ∈ Word (Base‘𝑆))
9493ad2antrl 763 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) → 𝑊 ∈ Word (Base‘𝑆))
9594adantr 481 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → 𝑊 ∈ Word (Base‘𝑆))
96 simpr2 1066 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → (#‘𝑊) = (#‘𝑈))
9795, 32, 963jca 1240 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → (𝑊 ∈ Word (Base‘𝑆) ∧ 𝑈 ∈ Word (Base‘𝑍) ∧ (#‘𝑊) = (#‘𝑈)))
9886, 97jca 554 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → (((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin) ∧ (𝑊 ∈ Word (Base‘𝑆) ∧ 𝑈 ∈ Word (Base‘𝑍) ∧ (#‘𝑊) = (#‘𝑈))))
9998ad2antrl 763 . . . . . . . 8 ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁)) → (((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin) ∧ (𝑊 ∈ Word (Base‘𝑆) ∧ 𝑈 ∈ Word (Base‘𝑍) ∧ (#‘𝑊) = (#‘𝑈))))
100 simpr 477 . . . . . . . . . . . 12 ((((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)) → ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))
101100ralimi 2947 . . . . . . . . . . 11 (∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)) → ∀𝑖 ∈ (0..^(#‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))
1021013ad2ant3 1082 . . . . . . . . . 10 ((𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))) → ∀𝑖 ∈ (0..^(#‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))
103102adantl 482 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → ∀𝑖 ∈ (0..^(#‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))
104103ad2antrl 763 . . . . . . . 8 ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁)) → ∀𝑖 ∈ (0..^(#‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))
105 incom 3783 . . . . . . . . . . 11 ((𝑁 ∖ {𝐾}) ∩ 𝑁) = (𝑁 ∩ (𝑁 ∖ {𝐾}))
106 indif 3845 . . . . . . . . . . 11 (𝑁 ∩ (𝑁 ∖ {𝐾})) = (𝑁 ∖ {𝐾})
107105, 106eqtri 2643 . . . . . . . . . 10 ((𝑁 ∖ {𝐾}) ∩ 𝑁) = (𝑁 ∖ {𝐾})
108107eqcomi 2630 . . . . . . . . 9 (𝑁 ∖ {𝐾}) = ((𝑁 ∖ {𝐾}) ∩ 𝑁)
10988, 89, 13, 26, 108gsmsymgreq 17773 . . . . . . . 8 ((((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin) ∧ (𝑊 ∈ Word (Base‘𝑆) ∧ 𝑈 ∈ Word (Base‘𝑍) ∧ (#‘𝑊) = (#‘𝑈))) → (∀𝑖 ∈ (0..^(#‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛) → ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛)))
11099, 104, 109sylc 65 . . . . . . 7 ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁)) → ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛))
111 fveq2 6148 . . . . . . . . 9 (𝑛 = 𝑘 → ((𝑆 Σg 𝑊)‘𝑛) = ((𝑆 Σg 𝑊)‘𝑘))
112 fveq2 6148 . . . . . . . . 9 (𝑛 = 𝑘 → ((𝑍 Σg 𝑈)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑘))
113111, 112eqeq12d 2636 . . . . . . . 8 (𝑛 = 𝑘 → (((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛) ↔ ((𝑆 Σg 𝑊)‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
114113rspcva 3293 . . . . . . 7 ((𝑘 ∈ (𝑁 ∖ {𝐾}) ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛)) → ((𝑆 Σg 𝑊)‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘))
11582, 110, 114syl2anc 692 . . . . . 6 ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁)) → ((𝑆 Σg 𝑊)‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘))
11665, 77, 1153eqtr3d 2663 . . . . 5 ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁)) → (𝑄𝑘) = ((𝑍 Σg 𝑈)‘𝑘))
117116ex 450 . . . 4 𝑘 = 𝐾 → ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁) → (𝑄𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
11861, 117pm2.61i 176 . . 3 ((((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) ∧ 𝑘𝑁) → (𝑄𝑘) = ((𝑍 Σg 𝑈)‘𝑘))
1197, 22, 118eqfnfvd 6270 . 2 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)))) → 𝑄 = (𝑍 Σg 𝑈))
120119exp31 629 1 (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))) → 𝑄 = (𝑍 Σg 𝑈))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  {crab 2911  cdif 3552  cin 3554  wss 3555  {csn 4148  ran crn 5075  cres 5076   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Fincfn 7899  0cc0 9880  ..^cfzo 12406  #chash 13057  Word cword 13230  Basecbs 15781   Σg cgsu 16022  SymGrpcsymg 17718  pmTrspcpmtr 17782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-s1 13241  df-substr 13242  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-tset 15881  df-0g 16023  df-gsum 16024  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-grp 17346  df-minusg 17347  df-subg 17512  df-symg 17719  df-pmtr 17783  df-psgn 17832
This theorem is referenced by:  psgndiflemA  19866
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