Theorem psgndif 20748

Description: Embedding of permutation signs restricted to a set without a single element into a ring. (Contributed by AV, 31-Jan-2019.)

Hypotheses

Ref	Expression
psgndif.p	⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
psgndif.s	⊢ 𝑆 = (pmSgn‘𝑁)
psgndif.z	⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))

Assertion

Ref	Expression
psgndif	⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (𝑆‘𝑄)))

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

Allowed substitution hints:   𝑆(𝑞)   𝑁(𝑞)   𝑍(𝑞)

Proof of Theorem psgndif

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

Step	Hyp	Ref	Expression
1		psgndif.p	. . . . . . . . . 10 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
2		eqid 2823	. . . . . . . . . 10 ⊢ ran (pmTrsp‘(𝑁 ∖ {𝐾})) = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
3		eqid 2823	. . . . . . . . . 10 ⊢ (SymGrp‘(𝑁 ∖ {𝐾})) = (SymGrp‘(𝑁 ∖ {𝐾}))
4		eqid 2823	. . . . . . . . . 10 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
5		eqid 2823	. . . . . . . . . 10 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁)
6	1, 2, 3, 4, 5	psgnfix2 20745	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)𝑄 = ((SymGrp‘𝑁) Σg 𝑟)))
7	6	imp 409	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)𝑄 = ((SymGrp‘𝑁) Σg 𝑟))
8	7	ad2antrr 724	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ ((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) → ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)𝑄 = ((SymGrp‘𝑁) Σg 𝑟))
9	1, 2, 3, 4, 5	psgndiflemA 20747	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ((𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾})) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑟)))))
10	9	imp 409	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ (𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾})) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁))) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑟))))
11	10	3anassrs 1356	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤)) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑟))))
12	11	adantlrr 719	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ ((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑟))))
13		eqeq1 2827	. . . . . . . . . . 11 ⊢ (𝑠 = (-1↑(♯‘𝑤)) → (𝑠 = (-1↑(♯‘𝑟)) ↔ (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑟))))
14	13	ad2antll 727	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ ((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) → (𝑠 = (-1↑(♯‘𝑟)) ↔ (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑟))))
15	14	adantr 483	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ ((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) → (𝑠 = (-1↑(♯‘𝑟)) ↔ (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑟))))
16	12, 15	sylibrd 261	. . . . . . . 8 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ ((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) → 𝑠 = (-1↑(♯‘𝑟))))
17	16	ralrimiva 3184	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ ((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) → ∀𝑟 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑟) → 𝑠 = (-1↑(♯‘𝑟))))
18	8, 17	r19.29imd 3259	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ ((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) → ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟))))
19	18	rexlimdva2 3289	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (∃𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) → ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))))
20	1, 2, 3	psgnfix1 20744	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))(𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤)))
21	20	imp 409	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∃𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))(𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤))
22	21	ad2antrr 724	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))) → ∃𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))(𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤))
23		simp-4l 781	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑟)) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤)) → ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}))
24		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑟)) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) → 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾})))
25	24	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑟)) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤)) → 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾})))
26		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑟)) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤)) → (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤))
27		simp-4r 782	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑟)) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤)) → 𝑟 ∈ Word ran (pmTrsp‘𝑁))
28	25, 26, 27	3jca 1124	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑟)) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤)) → (𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾})) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)))
29		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑟)) → 𝑄 = ((SymGrp‘𝑁) Σg 𝑟))
30	29	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑟)) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤)) → 𝑄 = ((SymGrp‘𝑁) Σg 𝑟))
31	23, 28, 30, 9	syl3c 66	. . . . . . . . . . . 12 ⊢ (((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑟)) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤)) → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑟)))
32	31	eqcomd 2829	. . . . . . . . . . 11 ⊢ (((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑟)) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤)) → (-1↑(♯‘𝑟)) = (-1↑(♯‘𝑤)))
33	32	ex 415	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑟)) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) → ((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) → (-1↑(♯‘𝑟)) = (-1↑(♯‘𝑤))))
34	33	adantlrr 719	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) → ((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) → (-1↑(♯‘𝑟)) = (-1↑(♯‘𝑤))))
35		eqeq1 2827	. . . . . . . . . . 11 ⊢ (𝑠 = (-1↑(♯‘𝑟)) → (𝑠 = (-1↑(♯‘𝑤)) ↔ (-1↑(♯‘𝑟)) = (-1↑(♯‘𝑤))))
36	35	ad2antll 727	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))) → (𝑠 = (-1↑(♯‘𝑤)) ↔ (-1↑(♯‘𝑟)) = (-1↑(♯‘𝑤))))
37	36	adantr 483	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) → (𝑠 = (-1↑(♯‘𝑤)) ↔ (-1↑(♯‘𝑟)) = (-1↑(♯‘𝑤))))
38	34, 37	sylibrd 261	. . . . . . . 8 ⊢ ((((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))) ∧ 𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))) → ((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) → 𝑠 = (-1↑(♯‘𝑤))))
39	38	ralrimiva 3184	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))) → ∀𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) → 𝑠 = (-1↑(♯‘𝑤))))
40	22, 39	r19.29imd 3259	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ∧ 𝑟 ∈ Word ran (pmTrsp‘𝑁)) ∧ (𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))) → ∃𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))
41	40	rexlimdva2 3289	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (∃𝑟 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟))) → ∃𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
42	19, 41	impbid 214	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (∃𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))))
43	42	iotabidv 6341	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (℩𝑠∃𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) = (℩𝑠∃𝑟 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))))
44		diffi 8752	. . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐾}) ∈ Fin)
45	44	ad2antrr 724	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑁 ∖ {𝐾}) ∈ Fin)
46		eqid 2823	. . . . . 6 ⊢ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}
47		eqid 2823	. . . . . 6 ⊢ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
48		eqid 2823	. . . . . 6 ⊢ (𝑁 ∖ {𝐾}) = (𝑁 ∖ {𝐾})
49	1, 46, 47, 48	symgfixelsi 18565	. . . . 5 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))))
50	49	adantll 712	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))))
51		psgndif.z	. . . . 5 ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))
52	3, 47, 2, 51	psgnvalfi 18644	. . . 4 ⊢ (((𝑁 ∖ {𝐾}) ∈ Fin ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))) → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (℩𝑠∃𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
53	45, 50, 52	syl2anc 586	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (℩𝑠∃𝑤 ∈ Word ran (pmTrsp‘(𝑁 ∖ {𝐾}))((𝑄 ↾ (𝑁 ∖ {𝐾})) = ((SymGrp‘(𝑁 ∖ {𝐾})) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
54		simpl 485	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin)
55		elrabi 3677	. . . 4 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → 𝑄 ∈ 𝑃)
56		psgndif.s	. . . . 5 ⊢ 𝑆 = (pmSgn‘𝑁)
57	4, 1, 5, 56	psgnvalfi 18644	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) = (℩𝑠∃𝑟 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))))
58	54, 55, 57	syl2an 597	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑆‘𝑄) = (℩𝑠∃𝑟 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑟) ∧ 𝑠 = (-1↑(♯‘𝑟)))))
59	43, 53, 58	3eqtr4d 2868	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (𝑆‘𝑄))
60	59	ex 415	1 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (𝑆‘𝑄)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  {crab 3144   ∖ cdif 3935  {csn 4569  ran crn 5558   ↾ cres 5559  ℩cio 6314  ‘cfv 6357  (class class class)co 7158  Fincfn 8511  1c1 10540  -cneg 10873  ↑cexp 13432  ♯chash 13693  Word cword 13864  Basecbs 16485   Σ_g cgsu 16716  SymGrpcsymg 18497  pmTrspcpmtr 18571  pmSgncpsgn 18619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-xor 1502  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-ot 4578  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-tpos 7894  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-rp 12393  df-fz 12896  df-fzo 13037  df-seq 13373  df-exp 13433  df-hash 13694  df-word 13865  df-lsw 13917  df-concat 13925  df-s1 13952  df-substr 14005  df-pfx 14035  df-splice 14114  df-reverse 14123  df-s2 14212  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-tset 16586  df-0g 16717  df-gsum 16718  df-mre 16859  df-mrc 16860  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-submnd 17959  df-efmnd 18036  df-grp 18108  df-minusg 18109  df-subg 18278  df-ghm 18358  df-gim 18401  df-oppg 18476  df-symg 18498  df-pmtr 18572  df-psgn 18621

This theorem is referenced by:  copsgndif  20749