Theorem psgndiflemB 21619

Description: Lemma 1 for psgndif 21621. (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 3686	. . . . 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 19394	. . . . 5 ⊢ (𝑄 ∈ 𝑃 → 𝑄:𝑁⟶𝑁)
5		ffn 6735	. . . . 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 6908	. . . . . . 7 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘𝑍)
15	3, 14	eqtri 2764	. . . . . 6 ⊢ 𝑃 = (Base‘𝑍)
16		psgnfix.r	. . . . . 6 ⊢ 𝑅 = ran (pmTrsp‘𝑁)
17	12, 15, 16	gsmtrcl 19535	. . . . 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 19394	. . . 4 ⊢ ((𝑍 Σg 𝑈) ∈ 𝑃 → (𝑍 Σg 𝑈):𝑁⟶𝑁)
20		ffn 6735	. . . 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 19488	. . . . . . . . . . . . . . 15 ⊢ 𝑅 ⊆ (Base‘𝑍)
27		sswrd 14561	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ⊆ (Base‘𝑍) → Word 𝑅 ⊆ Word (Base‘𝑍))
28	27	sseld 3981	. . . . . . . . . . . . . . 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 3082	. . . . . . . . . . . . . 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 7440	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑈) = (♯‘𝑊) → (0..^(♯‘𝑈)) = (0..^(♯‘𝑊)))
38	37	eqcoms 2744	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) = (♯‘𝑈) → (0..^(♯‘𝑈)) = (0..^(♯‘𝑊)))
39	38	raleqdv 3325	. . . . . . . . . . . . . 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 19447	. . . . . . . . . . 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 6905	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑄‘𝑘) = (𝑄‘𝐾))
48		fveq1 6904	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑄 → (𝑞‘𝐾) = (𝑄‘𝐾))
49	48	eqeq1d 2738	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑄 → ((𝑞‘𝐾) = 𝐾 ↔ (𝑄‘𝐾) = 𝐾))
50	49	elrab 3691	. . . . . . . . . . 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 2798	. . . . . . . 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 6905	. . . . . . . . 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 2786	. . . . . . 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 6904	. . . . . . . . 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 2947	. . . . . . . . . . . . 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 4785	. . . . . . . . . . . 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 6925	. . . . . . . . . 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 6905	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((𝑆 Σg 𝑊)‘𝑛) = ((𝑆 Σg 𝑊)‘𝑘))
75		fveq2 6905	. . . . . . . 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 9216	. . . . . . . . . . . . 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 19488	. . . . . . . . . . . . . 14 ⊢ 𝑇 ⊆ (Base‘𝑆)
85		sswrd 14561	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ⊆ (Base‘𝑆) → Word 𝑇 ⊆ Word (Base‘𝑆))
86	85	sseld 3981	. . . . . . . . . . . . . 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 1195	. . . . . . . . . . 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 3082	. . . . . . . . . . 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 4208	. . . . . . . . . . 11 ⊢ ((𝑁 ∖ {𝐾}) ∩ 𝑁) = (𝑁 ∩ (𝑁 ∖ {𝐾}))
100		indif 4279	. . . . . . . . . . 11 ⊢ (𝑁 ∩ (𝑁 ∖ {𝐾})) = (𝑁 ∖ {𝐾})
101	99, 100	eqtri 2764	. . . . . . . . . 10 ⊢ ((𝑁 ∖ {𝐾}) ∩ 𝑁) = (𝑁 ∖ {𝐾})
102	101	eqcomi 2745	. . . . . . . . 9 ⊢ (𝑁 ∖ {𝐾}) = ((𝑁 ∖ {𝐾}) ∩ 𝑁)
103	82, 83, 12, 25, 102	gsmsymgreq 19451	. . . . . . . 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 3622	. . . . . 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 2784	. . . . 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 7053	. 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 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  {crab 3435   ∖ cdif 3947   ∩ cin 3949   ⊆ wss 3950  {csn 4625  ran crn 5685   ↾ cres 5686   Fn wfn 6555  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432  Fincfn 8986  0cc0 11156  ..^cfzo 13695  ♯chash 14370  Word cword 14553  Basecbs 17248   Σ_g cgsu 17486  SymGrpcsymg 19387  pmTrspcpmtr 19460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-xnn0 12602  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-word 14554  df-lsw 14602  df-concat 14610  df-s1 14635  df-substr 14680  df-pfx 14710  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-tset 17317  df-0g 17487  df-gsum 17488  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-submnd 18798  df-efmnd 18883  df-grp 18955  df-minusg 18956  df-subg 19142  df-symg 19388  df-pmtr 19461  df-psgn 19510

This theorem is referenced by:  psgndiflemA  21620