Theorem psgndiflemA 21153

Description: Lemma 2 for psgndif 21154. (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 6891	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
2	1	eqeq1d 2734	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) = (♯‘𝑟) ↔ (♯‘𝑊) = (♯‘𝑟)))
3	1	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (0..^(♯‘𝑤)) = (0..^(♯‘𝑊)))
4		fveq1 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → (𝑤‘𝑖) = (𝑊‘𝑖))
5	4	fveq1d 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → ((𝑤‘𝑖)‘𝑛) = ((𝑊‘𝑖)‘𝑛))
6	5	eqeq1d 2734	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑊 → (((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛) ↔ ((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
7	6	ralbidv 3177	. . . . . . . . . . . . 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 3342	. . . . . . . . . . 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 3178	. . . . . . . . 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 3609	. . . . . . . 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 19353	. . . . . . . 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 725	. . . 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 407	. . 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 7416	. . . . . . . . 9 ⊢ ((♯‘𝑊) = (♯‘𝑟) → (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑟)))
22	21	adantr 481	. . . . . . . 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 729	. . . . . . 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 773	. . . . . . . . 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 725	. . . . . . . 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 773	. . . . . . . 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 481	. . . . . . . 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 775	. . . . . . . . . 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 482	. . . . . . . . . . 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 725	. . . . . . . . . 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 775	. . . . . . . . . . 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 481	. . . . . . . . . 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 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))) → ∀𝑖 ∈ (0..^(♯‘𝑊))(((𝑟‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
37	36	adantr 481	. . . . . . . . . 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 21152	. . . . . . . . . . . 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 418	. . . . . . . . . . 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 2738	. . . . . . . . . 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 2741	. . . . . . . . . . . 12 ⊢ (SymGrp‘𝑁) = 𝑍
46	45	oveq1i 7418	. . . . . . . . . . 11 ⊢ ((SymGrp‘𝑁) Σg 𝑈) = (𝑍 Σg 𝑈)
47	44, 46	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → 𝑄 = (𝑍 Σg 𝑈))
48	47	adantl 482	. . . . . . . . 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 2772	. . . . . . . 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 19366	. . . . . . 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 2772	. . . . . 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 413	. . . . 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 413	. . . 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 3147	. . 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 413	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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  {crab 3432   ∖ cdif 3945  {csn 4628  ran crn 5677   ↾ cres 5678  ‘cfv 6543  (class class class)co 7408  Fincfn 8938  0cc0 11109  1c1 11110  -cneg 11444  ..^cfzo 13626  ↑cexp 14026  ♯chash 14289  Word cword 14463  Basecbs 17143   Σ_g cgsu 17385  SymGrpcsymg 19233  pmTrspcpmtr 19308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-xor 1510  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-tpos 8210  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-2o 8466  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-xnn0 12544  df-z 12558  df-uz 12822  df-rp 12974  df-fz 13484  df-fzo 13627  df-seq 13966  df-exp 14027  df-hash 14290  df-word 14464  df-lsw 14512  df-concat 14520  df-s1 14545  df-substr 14590  df-pfx 14620  df-splice 14699  df-reverse 14708  df-s2 14798  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-tset 17215  df-0g 17386  df-gsum 17387  df-mre 17529  df-mrc 17530  df-acs 17532  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-mhm 18670  df-submnd 18671  df-efmnd 18749  df-grp 18821  df-minusg 18822  df-subg 19002  df-ghm 19089  df-gim 19132  df-oppg 19209  df-symg 19234  df-pmtr 19309  df-psgn 19358

This theorem is referenced by:  psgndif  21154