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Theorem psgndiflemB 20736
Description: Lemma 1 for psgndif 20738. (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 3673 . . . . 5 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} → 𝑄𝑃)
2 eqid 2819 . . . . . 6 (SymGrp‘𝑁) = (SymGrp‘𝑁)
3 psgnfix.p . . . . . 6 𝑃 = (Base‘(SymGrp‘𝑁))
42, 3symgbasf 18496 . . . . 5 (𝑄𝑃𝑄:𝑁𝑁)
5 ffn 6507 . . . . 5 (𝑄:𝑁𝑁𝑄 Fn 𝑁)
61, 4, 53syl 18 . . . 4 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} → 𝑄 Fn 𝑁)
76ad3antlr 729 . . 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 485 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝐾𝑁) → 𝑁 ∈ Fin)
98adantr 483 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) → 𝑁 ∈ Fin)
109adantr 483 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) → 𝑁 ∈ Fin)
11 simp1 1131 . . . . 5 ((𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))) → 𝑈 ∈ Word 𝑅)
12 psgnfix.z . . . . . 6 𝑍 = (SymGrp‘𝑁)
1312eqcomi 2828 . . . . . . . 8 (SymGrp‘𝑁) = 𝑍
1413fveq2i 6666 . . . . . . 7 (Base‘(SymGrp‘𝑁)) = (Base‘𝑍)
153, 14eqtri 2842 . . . . . 6 𝑃 = (Base‘𝑍)
16 psgnfix.r . . . . . 6 𝑅 = ran (pmTrsp‘𝑁)
1712, 15, 16gsmtrcl 18636 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑈 ∈ Word 𝑅) → (𝑍 Σg 𝑈) ∈ 𝑃)
1810, 11, 17syl2an 597 . . . 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)
192, 3symgbasf 18496 . . . 4 ((𝑍 Σg 𝑈) ∈ 𝑃 → (𝑍 Σg 𝑈):𝑁𝑁)
20 ffn 6507 . . . 4 ((𝑍 Σg 𝑈):𝑁𝑁 → (𝑍 Σg 𝑈) Fn 𝑁)
2118, 19, 203syl 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)
228ad3antrrr 728 . . . . . . . . . . . 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)
23 simpr 487 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝐾𝑁) → 𝐾𝑁)
2423ad3antrrr 728 . . . . . . . . . . . 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)
25 eqid 2819 . . . . . . . . . . . . . . . 16 (Base‘𝑍) = (Base‘𝑍)
2616, 12, 25symgtrf 18589 . . . . . . . . . . . . . . 15 𝑅 ⊆ (Base‘𝑍)
27 sswrd 13861 . . . . . . . . . . . . . . . 16 (𝑅 ⊆ (Base‘𝑍) → Word 𝑅 ⊆ Word (Base‘𝑍))
2827sseld 3964 . . . . . . . . . . . . . . 15 (𝑅 ⊆ (Base‘𝑍) → (𝑈 ∈ Word 𝑅𝑈 ∈ Word (Base‘𝑍)))
2926, 28ax-mp 5 . . . . . . . . . . . . . 14 (𝑈 ∈ Word 𝑅𝑈 ∈ Word (Base‘𝑍))
30293ad2ant1 1128 . . . . . . . . . . . . 13 ((𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))) → 𝑈 ∈ Word (Base‘𝑍))
3130adantl 484 . . . . . . . . . . . 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))
3222, 24, 313jca 1123 . . . . . . . . . . 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)))
33 simpl 485 . . . . . . . . . . . . . . 15 ((((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)) → ((𝑈𝑖)‘𝐾) = 𝐾)
3433ralimi 3158 . . . . . . . . . . . . . 14 (∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)) → ∀𝑖 ∈ (0..^(♯‘𝑊))((𝑈𝑖)‘𝐾) = 𝐾)
35343ad2ant3 1130 . . . . . . . . . . . . 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)
3635adantl 484 . . . . . . . . . . . 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)
37 oveq2 7156 . . . . . . . . . . . . . . . 16 ((♯‘𝑈) = (♯‘𝑊) → (0..^(♯‘𝑈)) = (0..^(♯‘𝑊)))
3837eqcoms 2827 . . . . . . . . . . . . . . 15 ((♯‘𝑊) = (♯‘𝑈) → (0..^(♯‘𝑈)) = (0..^(♯‘𝑊)))
3938raleqdv 3414 . . . . . . . . . . . . . 14 ((♯‘𝑊) = (♯‘𝑈) → (∀𝑖 ∈ (0..^(♯‘𝑈))((𝑈𝑖)‘𝐾) = 𝐾 ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))((𝑈𝑖)‘𝐾) = 𝐾))
40393ad2ant2 1129 . . . . . . . . . . . . 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))
4140adantl 484 . . . . . . . . . . . 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))
4236, 41mpbird 259 . . . . . . . . . . 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)
4312, 25gsmsymgrfix 18548 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝐾𝑁𝑈 ∈ Word (Base‘𝑍)) → (∀𝑖 ∈ (0..^(♯‘𝑈))((𝑈𝑖)‘𝐾) = 𝐾 → ((𝑍 Σg 𝑈)‘𝐾) = 𝐾))
4432, 42, 43sylc 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)
4544eqcomd 2825 . . . . . . . . 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))
4645adantr 483 . . . . . . . 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))
47 fveq2 6663 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑄𝑘) = (𝑄𝐾))
48 fveq1 6662 . . . . . . . . . . . . 13 (𝑞 = 𝑄 → (𝑞𝐾) = (𝑄𝐾))
4948eqeq1d 2821 . . . . . . . . . . . 12 (𝑞 = 𝑄 → ((𝑞𝐾) = 𝐾 ↔ (𝑄𝐾) = 𝐾))
5049elrab 3678 . . . . . . . . . . 11 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} ↔ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐾))
5150simprbi 499 . . . . . . . . . 10 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} → (𝑄𝐾) = 𝐾)
5251ad3antlr 729 . . . . . . . . 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)))) → (𝑄𝐾) = 𝐾)
5347, 52sylan9eqr 2876 . . . . . . . 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) = 𝐾)
54 fveq2 6663 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑍 Σg 𝑈)‘𝑘) = ((𝑍 Σg 𝑈)‘𝐾))
5554adantl 484 . . . . . . . 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) → ((𝑍 Σg 𝑈)‘𝑘) = ((𝑍 Σg 𝑈)‘𝐾))
5646, 53, 553eqtr4d 2864 . . . . . . 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))
5756ex 415 . . . . . 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)))
5857adantr 483 . . . . 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)))
5958com12 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)))
60 fveq1 6662 . . . . . . . . 9 ((𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
6160adantl 484 . . . . . . . 8 ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
6261ad3antlr 729 . . . . . . 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))
6362adantl 484 . . . . . 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))
64 simpr 487 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑘𝑁) → 𝑘𝑁)
65 neqne 3022 . . . . . . . . . . . . 13 𝑘 = 𝐾𝑘𝐾)
6664, 65anim12i 614 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑘𝑁) ∧ ¬ 𝑘 = 𝐾) → (𝑘𝑁𝑘𝐾))
67 eldifsn 4711 . . . . . . . . . . . 12 (𝑘 ∈ (𝑁 ∖ {𝐾}) ↔ (𝑘𝑁𝑘𝐾))
6866, 67sylibr 236 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑘𝑁) ∧ ¬ 𝑘 = 𝐾) → 𝑘 ∈ (𝑁 ∖ {𝐾}))
6968fvresd 6683 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑘𝑁) ∧ ¬ 𝑘 = 𝐾) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄𝑘))
7069exp31 422 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝐾𝑁) → (𝑘𝑁 → (¬ 𝑘 = 𝐾 → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄𝑘))))
7170ad3antrrr 728 . . . . . . . 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))))
7271imp 409 . . . . . . 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)))
7372impcom 410 . . . . . 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))
74 fveq2 6663 . . . . . . . 8 (𝑛 = 𝑘 → ((𝑆 Σg 𝑊)‘𝑛) = ((𝑆 Σg 𝑊)‘𝑘))
75 fveq2 6663 . . . . . . . 8 (𝑛 = 𝑘 → ((𝑍 Σg 𝑈)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑘))
7674, 75eqeq12d 2835 . . . . . . 7 (𝑛 = 𝑘 → (((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛) ↔ ((𝑆 Σg 𝑊)‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
77 diffi 8742 . . . . . . . . . . . . 13 (𝑁 ∈ Fin → (𝑁 ∖ {𝐾}) ∈ Fin)
7877ancri 552 . . . . . . . . . . . 12 (𝑁 ∈ Fin → ((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin))
7978adantr 483 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝐾𝑁) → ((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin))
8079ad3antrrr 728 . . . . . . . . . 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))
81 psgnfix.t . . . . . . . . . . . . . . 15 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
82 psgnfix.s . . . . . . . . . . . . . . 15 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾}))
83 eqid 2819 . . . . . . . . . . . . . . 15 (Base‘𝑆) = (Base‘𝑆)
8481, 82, 83symgtrf 18589 . . . . . . . . . . . . . 14 𝑇 ⊆ (Base‘𝑆)
85 sswrd 13861 . . . . . . . . . . . . . . 15 (𝑇 ⊆ (Base‘𝑆) → Word 𝑇 ⊆ Word (Base‘𝑆))
8685sseld 3964 . . . . . . . . . . . . . 14 (𝑇 ⊆ (Base‘𝑆) → (𝑊 ∈ Word 𝑇𝑊 ∈ Word (Base‘𝑆)))
8784, 86ax-mp 5 . . . . . . . . . . . . 13 (𝑊 ∈ Word 𝑇𝑊 ∈ Word (Base‘𝑆))
8887ad2antrl 726 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) → 𝑊 ∈ Word (Base‘𝑆))
8988adantr 483 . . . . . . . . . . 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))
90 simpr2 1190 . . . . . . . . . . 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))
9189, 31, 903jca 1123 . . . . . . . . . 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)))
9280, 91jca 514 . . . . . . . . 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))))
9392ad2antrl 726 . . . . . . . 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))))
94 simpr 487 . . . . . . . . . . . 12 ((((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)) → ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))
9594ralimi 3158 . . . . . . . . . . 11 (∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛)) → ∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))
96953ad2ant3 1130 . . . . . . . . . 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))
9796adantl 484 . . . . . . . . 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))
9897ad2antrl 726 . . . . . . . 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))
99 incom 4176 . . . . . . . . . . 11 ((𝑁 ∖ {𝐾}) ∩ 𝑁) = (𝑁 ∩ (𝑁 ∖ {𝐾}))
100 indif 4244 . . . . . . . . . . 11 (𝑁 ∩ (𝑁 ∖ {𝐾})) = (𝑁 ∖ {𝐾})
10199, 100eqtri 2842 . . . . . . . . . 10 ((𝑁 ∖ {𝐾}) ∩ 𝑁) = (𝑁 ∖ {𝐾})
102101eqcomi 2828 . . . . . . . . 9 (𝑁 ∖ {𝐾}) = ((𝑁 ∖ {𝐾}) ∩ 𝑁)
10382, 83, 12, 25, 102gsmsymgreq 18552 . . . . . . . 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)))
10493, 98, 103sylc 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))
10565anim2i 618 . . . . . . . . . . 11 ((𝑘𝑁 ∧ ¬ 𝑘 = 𝐾) → (𝑘𝑁𝑘𝐾))
106105, 67sylibr 236 . . . . . . . . . 10 ((𝑘𝑁 ∧ ¬ 𝑘 = 𝐾) → 𝑘 ∈ (𝑁 ∖ {𝐾}))
107106ex 415 . . . . . . . . 9 (𝑘𝑁 → (¬ 𝑘 = 𝐾𝑘 ∈ (𝑁 ∖ {𝐾})))
108107adantl 484 . . . . . . . 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})))
109108impcom 410 . . . . . . 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}))
11076, 104, 109rspcdva 3623 . . . . . 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))
11163, 73, 1103eqtr3d 2862 . . . . 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))
112111ex 415 . . . 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)))
11359, 112pm2.61i 184 . . 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))
1147, 21, 113eqfnfvd 6798 . 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))
115114exp31 422 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 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wne 3014  wral 3136  {crab 3140  cdif 3931  cin 3933  wss 3934  {csn 4559  ran crn 5549  cres 5550   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7148  Fincfn 8501  0cc0 10529  ..^cfzo 13025  chash 13682  Word cword 13853  Basecbs 16475   Σg cgsu 16706  SymGrpcsymg 18487  pmTrspcpmtr 18561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-seq 13362  df-hash 13683  df-word 13854  df-lsw 13907  df-concat 13915  df-s1 13942  df-substr 13995  df-pfx 14025  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-tset 16576  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-efmnd 18026  df-grp 18098  df-minusg 18099  df-subg 18268  df-symg 18488  df-pmtr 18562  df-psgn 18611
This theorem is referenced by:  psgndiflemA  20737
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