Theorem psgndiflemB 21906

Description: Lemma 1 for psgndif 21908. (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 3640	. . . . 5 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → 𝑄 ∈ 𝑃)
2		eqid 2760	. . . . . 6 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
3		psgnfix.p	. . . . . 6 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
4	2, 3	symgbasf 19590	. . . . 5 ⊢ (𝑄 ∈ 𝑃 → 𝑄:𝑁⟶𝑁)
5		ffn 6709	. . . . 5 ⊢ (𝑄:𝑁⟶𝑁 → 𝑄 Fn 𝑁)
6	1, 4, 5	3syl 19	. . . 4 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → 𝑄 Fn 𝑁)
7	6	ad3antlr 744	. . 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 488	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin)
9	8	adantr 486	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → 𝑁 ∈ Fin)
10	9	adantr 486	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) → 𝑁 ∈ Fin)
11		simp1 1154	. . . . 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 2769	. . . . . . . 8 ⊢ (SymGrp‘𝑁) = 𝑍
14	13	fveq2i 6888	. . . . . . 7 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘𝑍)
15	3, 14	eqtri 2783	. . . . . 6 ⊢ 𝑃 = (Base‘𝑍)
16		psgnfix.r	. . . . . 6 ⊢ 𝑅 = ran (pmTrsp‘𝑁)
17	12, 15, 16	gsmtrcl 19730	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑈 ∈ Word 𝑅) → (𝑍 Σg 𝑈) ∈ 𝑃)
18	10, 11, 17	syl2an 608	. . . 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 19590	. . . 4 ⊢ ((𝑍 Σg 𝑈) ∈ 𝑃 → (𝑍 Σg 𝑈):𝑁⟶𝑁)
20		ffn 6709	. . . 4 ⊢ ((𝑍 Σg 𝑈):𝑁⟶𝑁 → (𝑍 Σg 𝑈) Fn 𝑁)
21	18, 19, 20	3syl 19	. . 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 743	. . . . . . . . . . . 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 490	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → 𝐾 ∈ 𝑁)
24	23	ad3antrrr 743	. . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑍) = (Base‘𝑍)
26	16, 12, 25	symgtrf 19683	. . . . . . . . . . . . . . 15 ⊢ 𝑅 ⊆ (Base‘𝑍)
27		sswrd 14667	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ⊆ (Base‘𝑍) → Word 𝑅 ⊆ Word (Base‘𝑍))
28	27	sseld 3907	. . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 1146	. . . . . . . . . . 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 488	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)) → ((𝑈‘𝑖)‘𝐾) = 𝐾)
34	33	ralimi 3099	. . . . . . . . . . . . . 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 1153	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑈) = (♯‘𝑊) → (0..^(♯‘𝑈)) = (0..^(♯‘𝑊)))
38	37	eqcoms 2768	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) = (♯‘𝑈) → (0..^(♯‘𝑈)) = (0..^(♯‘𝑊)))
39	38	raleqdv 3319	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) = (♯‘𝑈) → (∀𝑖 ∈ (0..^(♯‘𝑈))((𝑈‘𝑖)‘𝐾) = 𝐾 ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))((𝑈‘𝑖)‘𝐾) = 𝐾))
40	39	3ad2ant2 1152	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 260	. . . . . . . . . . 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 19642	. . . . . . . . . . 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 66	. . . . . . . . . 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 2766	. . . . . . . . 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 486	. . . . . . . 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 6885	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑄‘𝑘) = (𝑄‘𝐾))
48		fveq1 6884	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑄 → (𝑞‘𝐾) = (𝑄‘𝐾))
49	48	eqeq1d 2762	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑄 → ((𝑞‘𝐾) = 𝐾 ↔ (𝑄‘𝐾) = 𝐾))
50	49	elrab 3644	. . . . . . . . . . 11 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↔ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐾))
51	50	simprbi 503	. . . . . . . . . 10 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → (𝑄‘𝐾) = 𝐾)
52	51	ad3antlr 744	. . . . . . . . 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 2817	. . . . . . . 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 6885	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑍 Σg 𝑈)‘𝑘) = ((𝑍 Σg 𝑈)‘𝐾))
55	54	adantl 487	. . . . . . . 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 2805	. . . . . . 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 418	. . . . . 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 486	. . . . 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 33	. . . 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 6884	. . . . . . . . 9 ⊢ ((𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
61	60	adantl 487	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = ((𝑆 Σg 𝑊)‘𝑘))
62	61	ad3antlr 744	. . . . . . 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 487	. . . . . 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 490	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
65		neqne 2963	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 𝐾 → 𝑘 ≠ 𝐾)
66	64, 65	anim12i 625	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐾) → (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐾))
67		eldifsn 4747	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐾}) ↔ (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐾))
68	66, 67	sylibr 237	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐾) → 𝑘 ∈ (𝑁 ∖ {𝐾}))
69	68	fvresd 6905	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐾) → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄‘𝑘))
70	69	exp31 425	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑘 ∈ 𝑁 → (¬ 𝑘 = 𝐾 → ((𝑄 ↾ (𝑁 ∖ {𝐾}))‘𝑘) = (𝑄‘𝑘))))
71	70	ad3antrrr 743	. . . . . . . 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 412	. . . . . . 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 413	. . . . . 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 6885	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((𝑆 Σg 𝑊)‘𝑛) = ((𝑆 Σg 𝑊)‘𝑘))
75		fveq2 6885	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((𝑍 Σg 𝑈)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑘))
76	74, 75	eqeq12d 2776	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛) ↔ ((𝑆 Σg 𝑊)‘𝑘) = ((𝑍 Σg 𝑈)‘𝑘)))
77		diffi 9190	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐾}) ∈ Fin)
78	77	ancri 559	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ Fin → ((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin))
79	78	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → ((𝑁 ∖ {𝐾}) ∈ Fin ∧ 𝑁 ∈ Fin))
80	79	ad3antrrr 743	. . . . . . . . . 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 2760	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑆) = (Base‘𝑆)
84	81, 82, 83	symgtrf 19683	. . . . . . . . . . . . . 14 ⊢ 𝑇 ⊆ (Base‘𝑆)
85		sswrd 14667	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ⊆ (Base‘𝑆) → Word 𝑇 ⊆ Word (Base‘𝑆))
86	85	sseld 3907	. . . . . . . . . . . . . 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 741	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 1214	. . . . . . . . . . 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 1146	. . . . . . . . . 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 521	. . . . . . . . 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 741	. . . . . . . 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 490	. . . . . . . . . . . 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 3099	. . . . . . . . . . 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 1153	. . . . . . . . . 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 487	. . . . . . . . 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 741	. . . . . . . 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 4106	. . . . . . . . . . 11 ⊢ ((𝑁 ∖ {𝐾}) ∩ 𝑁) = (𝑁 ∩ (𝑁 ∖ {𝐾}))
100		indif 4225	. . . . . . . . . . 11 ⊢ (𝑁 ∩ (𝑁 ∖ {𝐾})) = (𝑁 ∖ {𝐾})
101	99, 100	eqtri 2783	. . . . . . . . . 10 ⊢ ((𝑁 ∖ {𝐾}) ∩ 𝑁) = (𝑁 ∖ {𝐾})
102	101	eqcomi 2769	. . . . . . . . 9 ⊢ (𝑁 ∖ {𝐾}) = ((𝑁 ∖ {𝐾}) ∩ 𝑁)
103	82, 83, 12, 25, 102	gsmsymgreq 19646	. . . . . . . 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 66	. . . . . . 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 629	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ 𝑁 ∧ ¬ 𝑘 = 𝐾) → (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐾))
106	105, 67	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝑁 ∧ ¬ 𝑘 = 𝐾) → 𝑘 ∈ (𝑁 ∖ {𝐾}))
107	106	ex 418	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑁 → (¬ 𝑘 = 𝐾 → 𝑘 ∈ (𝑁 ∖ {𝐾})))
108	107	adantl 487	. . . . . . . 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 413	. . . . . . 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 3577	. . . . . 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 2803	. . . . 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 418	. . . 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 184	. . 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 7032	. 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 425	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))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  {crab 3412   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  {csn 4583  ran crn 5651   ↾ cres 5652   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Fincfn 8973  0cc0 11200  ..^cfzo 13788  ♯chash 14474  Word cword 14658  Basecbs 17387   Σ_g cgsu 17611  SymGrpcsymg 19583  pmTrspcpmtr 19655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-xnn0 12680  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789  df-seq 14145  df-hash 14475  df-word 14659  df-lsw 14708  df-concat 14716  df-s1 14743  df-substr 14789  df-pfx 14821  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-tset 17447  df-0g 17612  df-gsum 17613  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-submnd 18979  df-efmnd 19065  df-grp 19147  df-minusg 19148  df-subg 19333  df-symg 19584  df-pmtr 19656  df-psgn 19705

This theorem is used by:  psgndiflemA  21907