Theorem psgndiflemA 21907

Description: Lemma 2 for psgndif 21908. (Contributed by AV, 31-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
psgndiflemA	⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → ( -1↑(♯‘𝑊)) = ( -1↑(♯‘𝑈)))))

Distinct variable groups:   𝐾,𝑞   𝑃,𝑞   𝑄,𝑞

Allowed substitution hints:   𝑅(𝑞)   𝑆(𝑞)   𝑇(𝑞)   𝑈(𝑞)   𝑁(𝑞)   𝑊(𝑞)   𝑍(𝑞)

Proof of Theorem psgndiflemA

Dummy variables 𝑤 𝑖 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
2	1	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) = (♯‘𝑟) ↔ (♯‘𝑊) = (♯‘𝑟)))
3	1	oveq2d 7436	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (0..^(♯‘𝑤)) = (0..^(♯‘𝑊)))
4		fveq1 6884	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → (𝑤‘𝑖) = (𝑊‘𝑖))
5	4	fveq1d 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → ((𝑤‘𝑖)‘𝑛) = ((𝑊‘𝑖)‘𝑛))
6	5	eqeq1d 2763	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑊 → (((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛) ↔ ((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
7	6	ralbidv 3186	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
8	7	anbi2d 642	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ((((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)) ↔ (((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛))))
9	3, 8	raleqbidv 3335	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (∀𝑖 ∈ (0..^(♯‘𝑤))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)) ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛))))
10	2, 9	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (((♯‘𝑤) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑤))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛))) ↔ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))))
11	10	rexbidv 3187	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (∃𝑟 ∈ Word 𝑅((♯‘𝑤) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑤))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛))) ↔ ∃𝑟 ∈ Word 𝑅((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))))
12	11	rspccv 3574	. . . . . . . 8 ⊢ (∀𝑤 ∈ Word 𝑇∃𝑟 ∈ Word 𝑅((♯‘𝑤) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑤))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛))) → (𝑊 ∈ Word 𝑇 → ∃𝑟 ∈ Word 𝑅((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))))
13		psgnfix.t	. . . . . . . . 9 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
14		psgnfix.r	. . . . . . . . 9 ⊢ 𝑅 = ran (pmTrsp‘𝑁)
15	13, 14	pmtrdifwrdel2 19700	. . . . . . . 8 ⊢ (𝐾 ∈ 𝑁 → ∀𝑤 ∈ Word 𝑇∃𝑟 ∈ Word 𝑅((♯‘𝑤) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑤))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛))))
16	12, 15	syl11 34	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑇 → (𝐾 ∈ 𝑁 → ∃𝑟 ∈ Word 𝑅((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))))
17	16	3ad2ant1 1151	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝐾 ∈ 𝑁 → ∃𝑟 ∈ Word 𝑅((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))))
18	17	com12 33	. . . . 5 ⊢ (𝐾 ∈ 𝑁 → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → ∃𝑟 ∈ Word 𝑅((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))))
19	18	ad2antlr 740	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → ∃𝑟 ∈ Word 𝑅((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))))
20	19	imp 412	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → ∃𝑟 ∈ Word 𝑅((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛))))
21		oveq2 7428	. . . . . . . . 9 ⊢ ((♯‘𝑊) = (♯‘𝑟) → ( -1↑(♯‘𝑊)) = ( -1↑(♯‘𝑟)))
22	21	adantr 486	. . . . . . . 8 ⊢ (((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛))) → ( -1↑(♯‘𝑊)) = ( -1↑(♯‘𝑟)))
23	22	ad3antlr 744	. . . . . . 7 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → ( -1↑(♯‘𝑊)) = ( -1↑(♯‘𝑟)))
24		psgnfix.z	. . . . . . . 8 ⊢ 𝑍 = (SymGrp‘𝑁)
25		simplll 787	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → 𝑁 ∈ Fin)
26	25	ad2antlr 740	. . . . . . . 8 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → 𝑁 ∈ Fin)
27		simplll 787	. . . . . . . 8 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → 𝑟 ∈ Word 𝑅)
28		simprr3 1242	. . . . . . . . 9 ⊢ (((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) → 𝑈 ∈ Word 𝑅)
29	28	adantr 486	. . . . . . . 8 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → 𝑈 ∈ Word 𝑅)
30		simplrl 789	. . . . . . . . . 10 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}))
31		3simpa 1166	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)))
32	31	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)))
33	32	ad2antlr 740	. . . . . . . . . 10 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)))
34		simplrl 789	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) → (♯‘𝑊) = (♯‘𝑟))
35	34	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → (♯‘𝑊) = (♯‘𝑟))
36		simplrr 790	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) → ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
37	36	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
38		psgnfix.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
39		psgnfix.s	. . . . . . . . . . . . 13 ⊢ 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾}))
40	38, 13, 39, 24, 14	psgndiflemB 21906	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑟 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛))) → 𝑄 = (𝑍 Σg 𝑟))))
41	40	imp31 423	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑟 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) → 𝑄 = (𝑍 Σg 𝑟))
42	41	eqcomd 2767	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑟 ∈ Word 𝑅 ∧ (♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) → (𝑍 Σg 𝑟) = 𝑄)
43	30, 33, 27, 35, 37, 42	syl23anc 1404	. . . . . . . . 9 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → (𝑍 Σg 𝑟) = 𝑄)
44		id 23	. . . . . . . . . . 11 ⊢ (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → 𝑄 = ((SymGrp‘𝑁) Σg 𝑈))
45	24	eqcomi 2770	. . . . . . . . . . . 12 ⊢ (SymGrp‘𝑁) = 𝑍
46	45	oveq1i 7430	. . . . . . . . . . 11 ⊢ ((SymGrp‘𝑁) Σg 𝑈) = (𝑍 Σg 𝑈)
47	44, 46	eqtrdi 2812	. . . . . . . . . 10 ⊢ (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → 𝑄 = (𝑍 Σg 𝑈))
48	47	adantl 487	. . . . . . . . 9 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → 𝑄 = (𝑍 Σg 𝑈))
49	43, 48	eqtrd 2796	. . . . . . . 8 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → (𝑍 Σg 𝑟) = (𝑍 Σg 𝑈))
50	24, 14, 26, 27, 29, 49	psgnuni 19713	. . . . . . 7 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → ( -1↑(♯‘𝑟)) = ( -1↑(♯‘𝑈)))
51	23, 50	eqtrd 2796	. . . . . 6 ⊢ ((((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → ( -1↑(♯‘𝑊)) = ( -1↑(♯‘𝑈)))
52	51	ex 418	. . . . 5 ⊢ (((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → ( -1↑(♯‘𝑊)) = ( -1↑(♯‘𝑈))))
53	52	ex 418	. . . 4 ⊢ ((𝑟 ∈ Word 𝑅 ∧ ((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))) → ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → ( -1↑(♯‘𝑊)) = ( -1↑(♯‘𝑈)))))
54	53	rexlimiva 3156	. . 3 ⊢ (∃𝑟 ∈ Word 𝑅((♯‘𝑊) = (♯‘𝑟) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛))) → ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → ( -1↑(♯‘𝑊)) = ( -1↑(♯‘𝑈)))))
55	20, 54	mpcom 39	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → ( -1↑(♯‘𝑊)) = ( -1↑(♯‘𝑈))))
56	55	ex 418	1 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → ( -1↑(♯‘𝑊)) = ( -1↑(♯‘𝑈)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413   ∖ cdif 3896  {csn 4584  ran crn 5652   ↾ cres 5653  ‘cfv 6538  (class class class)co 7420  Fincfn 8973  0cc0 11200  1c1 11201   -cneg 11542  ..^cfzo 13788  ↑cexp 14204  ♯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 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  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-xor 1542  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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-tpos 8243  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-div 11974  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-rp 13121  df-fz 13640  df-fzo 13789  df-seq 14145  df-exp 14205  df-hash 14475  df-word 14659  df-lsw 14708  df-concat 14716  df-s1 14743  df-substr 14789  df-pfx 14821  df-splice 14899  df-reverse 14908  df-s2 14999  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-mhm 18978  df-submnd 18979  df-efmnd 19065  df-grp 19147  df-minusg 19148  df-subg 19333  df-ghm 19428  df-gim 19473  df-oppg 19560  df-symg 19584  df-pmtr 19656  df-psgn 19705

This theorem is used by:  psgndif  21908