Theorem psgndiflemB 21565

Description: Lemma 1 for psgndif 21567. (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 ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘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.

Step	Hyp	Ref	Expression
1		elrabi 3671	. . . . 5 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → 𝑄 ∈ 𝑃)
2		eqid 2736	. . . . . 6 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
3		psgnfix.p	. . . . . 6 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
4	2, 3	symgbasf 19362	. . . . 5 ⊢ (𝑄 ∈ 𝑃 → 𝑄:𝑁⟶𝑁)
5		ffn 6711	. . . . 5 ⊢ (𝑄:𝑁⟶𝑁 → 𝑄 Fn 𝑁)
6	1, 4, 5	3syl 18	. . . 4 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → 𝑄 Fn 𝑁)
7	6	ad3antlr 731	. . 3 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → 𝑄 Fn 𝑁)
8		simpl 482	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin)
9	8	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → 𝑁 ∈ Fin)
10	9	adantr 480	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) → 𝑁 ∈ Fin)
11		simp1 1136	. . . . 5 ⊢ ((𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) → 𝑈 ∈ Word 𝑅)
12		psgnfix.z	. . . . . 6 ⊢ 𝑍 = (SymGrp‘𝑁)
13	12	eqcomi 2745	. . . . . . . 8 ⊢ (SymGrp‘𝑁) = 𝑍
14	13	fveq2i 6884	. . . . . . 7 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘𝑍)
15	3, 14	eqtri 2759	. . . . . 6 ⊢ 𝑃 = (Base‘𝑍)
16		psgnfix.r	. . . . . 6 ⊢ 𝑅 = ran (pmTrsp‘𝑁)
17	12, 15, 16	gsmtrcl 19502	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑈 ∈ Word 𝑅) → (𝑍 Σg 𝑈) ∈ 𝑃)
18	10, 11, 17	syl2an 596	. . . 4 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → (𝑍 Σg 𝑈) ∈ 𝑃)
19	2, 3	symgbasf 19362	. . . 4 ⊢ ((𝑍 Σg 𝑈) ∈ 𝑃 → (𝑍 Σg 𝑈):𝑁⟶𝑁)
20		ffn 6711	. . . 4 ⊢ ((𝑍 Σg 𝑈):𝑁⟶𝑁 → (𝑍 Σg 𝑈) Fn 𝑁)
21	18, 19, 20	3syl 18	. . 3 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → (𝑍 Σg 𝑈) Fn 𝑁)
22	8	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → 𝑁 ∈ Fin)
23		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → 𝐾 ∈ 𝑁)
24	23	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → 𝐾 ∈ 𝑁)
25		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑍) = (Base‘𝑍)
26	16, 12, 25	symgtrf 19455	. . . . . . . . . . . . . . 15 ⊢ 𝑅 ⊆ (Base‘𝑍)
27		sswrd 14545	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ⊆ (Base‘𝑍) → Word 𝑅 ⊆ Word (Base‘𝑍))
28	27	sseld 3962	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ⊆ (Base‘𝑍) → (𝑈 ∈ Word 𝑅 → 𝑈 ∈ Word (Base‘𝑍)))
29	26, 28	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ Word 𝑅 → 𝑈 ∈ Word (Base‘𝑍))
30	29	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) → 𝑈 ∈ Word (Base‘𝑍))
31	30	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → 𝑈 ∈ Word (Base‘𝑍))
32	22, 24, 31	3jca 1128	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → (𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁 ∧ 𝑈 ∈ Word (Base‘𝑍)))
33		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)) → ((𝑈‘𝑖)‘𝐾) = 𝐾)
34	33	ralimi 3074	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)) → ∀𝑖 ∈ (0..^(♯‘𝑊))((𝑈‘𝑖)‘𝐾) = 𝐾)
35	34	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) → ∀𝑖 ∈ (0..^(♯‘𝑊))((𝑈‘𝑖)‘𝐾) = 𝐾)
36	35	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → ∀𝑖 ∈ (0..^(♯‘𝑊))((𝑈‘𝑖)‘𝐾) = 𝐾)
37		oveq2 7418	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑈) = (♯‘𝑊) → (0..^(♯‘𝑈)) = (0..^(♯‘𝑊)))
38	37	eqcoms 2744	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) = (♯‘𝑈) → (0..^(♯‘𝑈)) = (0..^(♯‘𝑊)))
39	38	raleqdv 3309	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) = (♯‘𝑈) → (∀𝑖 ∈ (0..^(♯‘𝑈))((𝑈‘𝑖)‘𝐾) = 𝐾 ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))((𝑈‘𝑖)‘𝐾) = 𝐾))
40	39	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) → (∀𝑖 ∈ (0..^(♯‘𝑈))((𝑈‘𝑖)‘𝐾) = 𝐾 ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))((𝑈‘𝑖)‘𝐾) = 𝐾))
41	40	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → (∀𝑖 ∈ (0..^(♯‘𝑈))((𝑈‘𝑖)‘𝐾) = 𝐾 ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))((𝑈‘𝑖)‘𝐾) = 𝐾))
42	36, 41	mpbird 257	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → ∀𝑖 ∈ (0..^(♯‘𝑈))((𝑈‘𝑖)‘𝐾) = 𝐾)
43	12, 25	gsmsymgrfix 19414	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁 ∧ 𝑈 ∈ Word (Base‘𝑍)) → (∀𝑖 ∈ (0..^(♯‘𝑈))((𝑈‘𝑖)‘𝐾) = 𝐾 → ((𝑍 Σg 𝑈)‘𝐾) = 𝐾))
44	32, 42, 43	sylc 65	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → ((𝑍 Σg 𝑈)‘𝐾) = 𝐾)
45	44	eqcomd 2742	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → 𝐾 = ((𝑍 Σg 𝑈)‘𝐾))
46	45	adantr 480	. . . . . . . 8 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 = 𝐾) → 𝐾 = ((𝑍 Σg 𝑈)‘𝐾))
47		fveq2 6881	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑄‘𝑘) = (𝑄‘𝐾))
48		fveq1 6880	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑄 → (𝑞‘𝐾) = (𝑄‘𝐾))
49	48	eqeq1d 2738	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑄 → ((𝑞‘𝐾) = 𝐾 ↔ (𝑄‘𝐾) = 𝐾))
50	49	elrab 3676	. . . . . . . . . . 11 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↔ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐾))
51	50	simprbi 496	. . . . . . . . . 10 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → (𝑄‘𝐾) = 𝐾)
52	51	ad3antlr 731	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → (𝑄‘𝐾) = 𝐾)
53	47, 52	sylan9eqr 2793	. . . . . . . 8 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 = 𝐾) → (𝑄‘𝑘) = 𝐾)
54		fveq2 6881	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑍 Σg 𝑈)‘𝑘) = ((𝑍 Σg 𝑈)‘𝐾))
55	54	adantl 481	. . . . . . . 8 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 = 𝐾) → ((𝑍 Σg 𝑈)‘𝑘) = ((𝑍 Σg 𝑈)‘𝐾))
56	46, 53, 55	3eqtr4d 2781	. . . . . . 7 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 = 𝐾) → (𝑄‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘))
57	56	ex 412	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → (𝑘 = 𝐾 → (𝑄‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
58	57	adantr 480	. . . . 5 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁) → (𝑘 = 𝐾 → (𝑄‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
59	58	com12 32	. . . 4 ⊢ (𝑘 = 𝐾 → ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁) → (𝑄‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
60		fveq1 6880	. . . . . . . . 9 ⊢ ((𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
61	60	adantl 481	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
62	61	ad3antlr 731	. . . . . . 7 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
63	62	adantl 481	. . . . . 6 ⊢ ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁)) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
64		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
65		neqne 2941	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 𝐾 → 𝑘 ≠ 𝐾)
66	64, 65	anim12i 613	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐾) → (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐾))
67		eldifsn 4767	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐾}) ↔ (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐾))
68	66, 67	sylibr 234	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐾) → 𝑘 ∈ (𝑁 ∖ {𝐾}))
69	68	fvresd 6901	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐾) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄‘𝑘))
70	69	exp31 419	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑘 ∈ 𝑁 → (¬ 𝑘 = 𝐾 → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄‘𝑘))))
71	70	ad3antrrr 730	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → (𝑘 ∈ 𝑁 → (¬ 𝑘 = 𝐾 → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄‘𝑘))))
72	71	imp 406	. . . . . . 7 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁) → (¬ 𝑘 = 𝐾 → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄‘𝑘)))
73	72	impcom 407	. . . . . 6 ⊢ ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁)) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄‘𝑘))
74		fveq2 6881	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((𝑆 Σg 𝑊)‘𝑛) = ((𝑆 Σg 𝑊)‘𝑘))
75		fveq2 6881	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((𝑍 Σg 𝑈)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑘))
76	74, 75	eqeq12d 2752	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛) ↔ ((𝑆 Σg 𝑊)‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
77		diffi 9194	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐾}) ∈ Fin)
78	77	ancri 549	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ Fin → ((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin))
79	78	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → ((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin))
80	79	ad3antrrr 730	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘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 2736	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑆) = (Base‘𝑆)
84	81, 82, 83	symgtrf 19455	. . . . . . . . . . . . . 14 ⊢ 𝑇 ⊆ (Base‘𝑆)
85		sswrd 14545	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ⊆ (Base‘𝑆) → Word 𝑇 ⊆ Word (Base‘𝑆))
86	85	sseld 3962	. . . . . . . . . . . . . 14 ⊢ (𝑇 ⊆ (Base‘𝑆) → (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Word (Base‘𝑆)))
87	84, 86	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Word (Base‘𝑆))
88	87	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) → 𝑊 ∈ Word (Base‘𝑆))
89	88	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → 𝑊 ∈ Word (Base‘𝑆))
90		simpr2 1196	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → (♯‘𝑊) = (♯‘𝑈))
91	89, 31, 90	3jca 1128	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → (𝑊 ∈ Word (Base‘𝑆) ∧ 𝑈 ∈ Word (Base‘𝑍) ∧ (♯‘𝑊) = (♯‘𝑈)))
92	80, 91	jca 511	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → (((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin) ∧ (𝑊 ∈ Word (Base‘𝑆) ∧ 𝑈 ∈ Word (Base‘𝑍) ∧ (♯‘𝑊) = (♯‘𝑈))))
93	92	ad2antrl 728	. . . . . . . 8 ⊢ ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁)) → (((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin) ∧ (𝑊 ∈ Word (Base‘𝑆) ∧ 𝑈 ∈ Word (Base‘𝑍) ∧ (♯‘𝑊) = (♯‘𝑈))))
94		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)) → ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))
95	94	ralimi 3074	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)) → ∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))
96	95	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) → ∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))
97	96	adantl 481	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → ∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))
98	97	ad2antrl 728	. . . . . . . 8 ⊢ ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁)) → ∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))
99		incom 4189	. . . . . . . . . . 11 ⊢ ((𝑁 ∖ {𝐾}) ∩ 𝑁) = (𝑁 ∩ (𝑁 ∖ {𝐾}))
100		indif 4260	. . . . . . . . . . 11 ⊢ (𝑁 ∩ (𝑁 ∖ {𝐾})) = (𝑁 ∖ {𝐾})
101	99, 100	eqtri 2759	. . . . . . . . . 10 ⊢ ((𝑁 ∖ {𝐾}) ∩ 𝑁) = (𝑁 ∖ {𝐾})
102	101	eqcomi 2745	. . . . . . . . 9 ⊢ (𝑁 ∖ {𝐾}) = ((𝑁 ∖ {𝐾}) ∩ 𝑁)
103	82, 83, 12, 25, 102	gsmsymgreq 19418	. . . . . . . 8 ⊢ ((((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin) ∧ (𝑊 ∈ Word (Base‘𝑆) ∧ 𝑈 ∈ Word (Base‘𝑍) ∧ (♯‘𝑊) = (♯‘𝑈))) → (∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) → ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛)))
104	93, 98, 103	sylc 65	. . . . . . 7 ⊢ ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁)) → ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛))
105	65	anim2i 617	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ 𝑁 ∧ ¬ 𝑘 = 𝐾) → (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐾))
106	105, 67	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝑁 ∧ ¬ 𝑘 = 𝐾) → 𝑘 ∈ (𝑁 ∖ {𝐾}))
107	106	ex 412	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑁 → (¬ 𝑘 = 𝐾 → 𝑘 ∈ (𝑁 ∖ {𝐾})))
108	107	adantl 481	. . . . . . . 8 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁) → (¬ 𝑘 = 𝐾 → 𝑘 ∈ (𝑁 ∖ {𝐾})))
109	108	impcom 407	. . . . . . 7 ⊢ ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁)) → 𝑘 ∈ (𝑁 ∖ {𝐾}))
110	76, 104, 109	rspcdva 3607	. . . . . 6 ⊢ ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁)) → ((𝑆 Σg 𝑊)‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘))
111	63, 73, 110	3eqtr3d 2779	. . . . 5 ⊢ ((¬ 𝑘 = 𝐾 ∧ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁)) → (𝑄‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘))
112	111	ex 412	. . . 4 ⊢ (¬ 𝑘 = 𝐾 → ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁) → (𝑄‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
113	59, 112	pm2.61i 182	. . 3 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) ∧ 𝑘 ∈ 𝑁) → (𝑄‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘))
114	7, 21, 113	eqfnfvd 7029	. 2 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)))) → 𝑄 = (𝑍 Σg 𝑈))
115	114	exp31 419	1 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑈 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑈) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) → 𝑄 = (𝑍 Σg 𝑈))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  {crab 3420   ∖ cdif 3928   ∩ cin 3930   ⊆ wss 3931  {csn 4606  ran crn 5660   ↾ cres 5661   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Fincfn 8964  0cc0 11134  ..^cfzo 13676  ♯chash 14353  Word cword 14536  Basecbs 17233   Σ_g cgsu 17459  SymGrpcsymg 19355  pmTrspcpmtr 19427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-xnn0 12580  df-z 12594  df-uz 12858  df-fz 13530  df-fzo 13677  df-seq 14025  df-hash 14354  df-word 14537  df-lsw 14586  df-concat 14594  df-s1 14619  df-substr 14664  df-pfx 14694  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-tset 17295  df-0g 17460  df-gsum 17461  df-mre 17603  df-mrc 17604  df-acs 17606  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-submnd 18767  df-efmnd 18852  df-grp 18924  df-minusg 18925  df-subg 19111  df-symg 19356  df-pmtr 19428  df-psgn 19477

This theorem is referenced by:  psgndiflemA  21566