Theorem psgndiflemA 19766

Description: Lemma 2 for psgndif 19767. (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		psgnfix.t	. . . . . . . . . 10 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
2		psgnfix.r	. . . . . . . . . 10 ⊢ 𝑅 = ran (pmTrsp‘𝑁)
3	1, 2	pmtrdifwrdel2 17729	. . . . . . . . 9 ⊢ (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))))
4		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
5	4	eqeq1d 2612	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ((#‘𝑤) = (#‘𝑟) ↔ (#‘𝑊) = (#‘𝑟)))
6	4	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (0..^(#‘𝑤)) = (0..^(#‘𝑊)))
7		fveq1 6102	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑊 → (𝑤‘𝑖) = (𝑊‘𝑖))
8	7	fveq1d 6105	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → ((𝑤‘𝑖)‘𝑛) = ((𝑊‘𝑖)‘𝑛))
9	8	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → (((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛) ↔ ((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
10	9	ralbidv 2969	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑊 → (∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
11	10	anbi2d 736	. . . . . . . . . . . . 13 ⊢ (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))))
12	6, 11	raleqbidv 3129	. . . . . . . . . . . 12 ⊢ (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))))
13	5, 12	anbi12d 743	. . . . . . . . . . 11 ⊢ (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)))))
14	13	rexbidv 3034	. . . . . . . . . 10 ⊢ (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)))))
15	14	rspccv 3279	. . . . . . . . 9 ⊢ (∀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)))))
16	3, 15	syl 17	. . . . . . . 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)))))
17	16	com12 32	. . . . . . 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)))))
18	17	3ad2ant1 1075	. . . . . 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)))))
19	18	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)))))
20	19	ad2antlr 759	. . . 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)))))
21	20	imp 444	. . 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))))
22		oveq2 6557	. . . . . . . . 9 ⊢ ((#‘𝑊) = (#‘𝑟) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑟)))
23	22	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)))
24	23	ad3antlr 763	. . . . . . 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)))
25		psgnfix.z	. . . . . . . 8 ⊢ 𝑍 = (SymGrp‘𝑁)
26		simplll 794	. . . . . . . . 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)
27	26	ad2antlr 759	. . . . . . . 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)
28		simplll 794	. . . . . . . 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)
29		simprr3 1104	. . . . . . . . 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)
30	29	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)
31		simplrl 796	. . . . . . . . . 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}))
32		3simpa 1051	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)))
33	32	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)))
34	33	ad2antlr 759	. . . . . . . . . 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)))
35		simplrl 796	. . . . . . . . . . 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))
36	35	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))
37		simplrr 797	. . . . . . . . . . 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)))
38	37	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)))
39		psgnfix.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
40		psgnfix.s	. . . . . . . . . . . . 13 ⊢ 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾}))
41	39, 1, 40, 25, 2	psgndiflemB 19765	. . . . . . . . . . . 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))))
42	41	imp31 447	. . . . . . . . . . 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))
43	42	eqcomd 2616	. . . . . . . . . 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)
44	31, 34, 28, 36, 38, 43	syl23anc 1325	. . . . . . . . 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)
45		id 22	. . . . . . . . . . 11 ⊢ (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → 𝑄 = ((SymGrp‘𝑁) Σg 𝑈))
46	25	eqcomi 2619	. . . . . . . . . . . 12 ⊢ (SymGrp‘𝑁) = 𝑍
47	46	oveq1i 6559	. . . . . . . . . . 11 ⊢ ((SymGrp‘𝑁) Σg 𝑈) = (𝑍 Σg 𝑈)
48	45, 47	syl6eq 2660	. . . . . . . . . 10 ⊢ (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → 𝑄 = (𝑍 Σg 𝑈))
49	48	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))
50	44, 49	eqtrd 2644	. . . . . . . 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))
51	25, 2, 27, 28, 30, 50	psgnuni 17742	. . . . . . 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)))
52	24, 51	eqtrd 2644	. . . . . 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)))
53	52	ex 449	. . . . 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))))
54	53	ex 449	. . . 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)))))
55	54	rexlimiva 3010	. . 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)))))
56	21, 55	mpcom 37	. 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))))
57	56	ex 449	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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537  {csn 4125  ran crn 5039   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816  -cneg 10146  ..^cfzo 12334  ↑cexp 12722  #chash 12979  Word cword 13146  Basecbs 15695   Σ_g cgsu 15924  SymGrpcsymg 17620  pmTrspcpmtr 17684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-xor 1457  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-tset 15787  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-subg 17414  df-ghm 17481  df-gim 17524  df-oppg 17599  df-symg 17621  df-pmtr 17685  df-psgn 17734

This theorem is referenced by:  psgndif  19767