Theorem psgndiflemA 20718

Description: Lemma 2 for psgndif 20719. (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 6756	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
2	1	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) = (♯‘𝑟) ↔ (♯‘𝑊) = (♯‘𝑟)))
3	1	oveq2d 7271	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (0..^(♯‘𝑤)) = (0..^(♯‘𝑊)))
4		fveq1 6755	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → (𝑤‘𝑖) = (𝑊‘𝑖))
5	4	fveq1d 6758	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → ((𝑤‘𝑖)‘𝑛) = ((𝑊‘𝑖)‘𝑛))
6	5	eqeq1d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑊 → (((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛) ↔ ((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
7	6	ralbidv 3120	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑟‘𝑖)‘𝑛)))
8	7	anbi2d 628	. . . . . . . . . . . 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 3327	. . . . . . . . . . 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 630	. . . . . . . . . 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 3225	. . . . . . . . 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 3549	. . . . . . . 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 19009	. . . . . . . 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 1131	. . . . . 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 723	. . . 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 7263	. . . . . . . . 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 727	. . . . . . 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 771	. . . . . . . . 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 723	. . . . . . . 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 771	. . . . . . . 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 1221	. . . . . . . . 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 773	. . . . . . . . . 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 1146	. . . . . . . . . . . 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 723	. . . . . . . . . 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 773	. . . . . . . . . . 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 774	. . . . . . . . . . 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 20717	. . . . . . . . . . . 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 2744	. . . . . . . . . 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 1375	. . . . . . . . 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 2747	. . . . . . . . . . . 12 ⊢ (SymGrp‘𝑁) = 𝑍
46	45	oveq1i 7265	. . . . . . . . . . 11 ⊢ ((SymGrp‘𝑁) Σg 𝑈) = (𝑍 Σg 𝑈)
47	44, 46	eqtrdi 2795	. . . . . . . . . 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 2778	. . . . . . . 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 19022	. . . . . . 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 2778	. . . . . 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 3209	. . 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 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  {crab 3067   ∖ cdif 3880  {csn 4558  ran crn 5581   ↾ cres 5582  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  0cc0 10802  1c1 10803  -cneg 11136  ..^cfzo 13311  ↑cexp 13710  ♯chash 13972  Word cword 14145  Basecbs 16840   Σ_g cgsu 17068  SymGrpcsymg 18889  pmTrspcpmtr 18964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-xor 1504  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-rp 12660  df-fz 13169  df-fzo 13312  df-seq 13650  df-exp 13711  df-hash 13973  df-word 14146  df-lsw 14194  df-concat 14202  df-s1 14229  df-substr 14282  df-pfx 14312  df-splice 14391  df-reverse 14400  df-s2 14489  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-tset 16907  df-0g 17069  df-gsum 17070  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-submnd 18346  df-efmnd 18423  df-grp 18495  df-minusg 18496  df-subg 18667  df-ghm 18747  df-gim 18790  df-oppg 18865  df-symg 18890  df-pmtr 18965  df-psgn 19014

This theorem is referenced by:  psgndif  20719