Theorem psgndiflemA 21642

Description: Lemma 2 for psgndif 21643. (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 6920	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
2	1	eqeq1d 2742	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) = (♯‘𝑟) ↔ (♯‘𝑊) = (♯‘𝑟)))
3	1	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (0..^(♯‘𝑤)) = (0..^(♯‘𝑊)))
4		fveq1 6919	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → (𝑤‘𝑖) = (𝑊‘𝑖))
5	4	fveq1d 6922	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → ((𝑤‘𝑖)‘𝑛) = ((𝑊‘𝑖)‘𝑛))
6	5	eqeq1d 2742	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑊 → (((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛) ↔ ((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
7	6	ralbidv 3184	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
8	7	anbi2d 629	. . . . . . . . . . . 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 3354	. . . . . . . . . . 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 631	. . . . . . . . . 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 3185	. . . . . . . . 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 3632	. . . . . . . 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 19528	. . . . . . . 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 33	. . . . . . 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 1133	. . . . . 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 32	. . . . 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 726	. . . 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 406	. . 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 7456	. . . . . . . . 9 ⊢ ((♯‘𝑊) = (♯‘𝑟) → (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑟)))
22	21	adantr 480	. . . . . . . 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 730	. . . . . . 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 774	. . . . . . . . 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 726	. . . . . . . 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 774	. . . . . . . 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 1223	. . . . . . . . 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 480	. . . . . . . 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 776	. . . . . . . . . 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 1148	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 726	. . . . . . . . . 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 776	. . . . . . . . . . 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 480	. . . . . . . . . 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 777	. . . . . . . . . . 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 480	. . . . . . . . . 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 21641	. . . . . . . . . . . 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 417	. . . . . . . . . . 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 2746	. . . . . . . . . 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 1377	. . . . . . . . 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 22	. . . . . . . . . . 11 ⊢ (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → 𝑄 = ((SymGrp‘𝑁) Σg 𝑈))
45	24	eqcomi 2749	. . . . . . . . . . . 12 ⊢ (SymGrp‘𝑁) = 𝑍
46	45	oveq1i 7458	. . . . . . . . . . 11 ⊢ ((SymGrp‘𝑁) Σg 𝑈) = (𝑍 Σg 𝑈)
47	44, 46	eqtrdi 2796	. . . . . . . . . 10 ⊢ (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → 𝑄 = (𝑍 Σg 𝑈))
48	47	adantl 481	. . . . . . . . 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 2780	. . . . . . . 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 19541	. . . . . . 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 2780	. . . . . 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 412	. . . . 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 412	. . . 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 3153	. . 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 38	. 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 412	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)))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443   ∖ cdif 3973  {csn 4648  ran crn 5701   ↾ cres 5702  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  0cc0 11184  1c1 11185  -cneg 11521  ..^cfzo 13711  ↑cexp 14112  ♯chash 14379  Word cword 14562  Basecbs 17258   Σ_g cgsu 17500  SymGrpcsymg 19410  pmTrspcpmtr 19483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-xor 1509  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-rp 13058  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-word 14563  df-lsw 14611  df-concat 14619  df-s1 14644  df-substr 14689  df-pfx 14719  df-splice 14798  df-reverse 14807  df-s2 14897  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-tset 17330  df-0g 17501  df-gsum 17502  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-efmnd 18904  df-grp 18976  df-minusg 18977  df-subg 19163  df-ghm 19253  df-gim 19299  df-oppg 19386  df-symg 19411  df-pmtr 19484  df-psgn 19533

This theorem is referenced by:  psgndif  21643