Theorem psgndif 20807

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 2738	. . . . . . . . . 10 ⊢ ran (pmTrsp‘(𝑁 ∖ {𝐾})) = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
3		eqid 2738	. . . . . . . . . 10 ⊢ (SymGrp‘(𝑁 ∖ {𝐾})) = (SymGrp‘(𝑁 ∖ {𝐾}))
4		eqid 2738	. . . . . . . . . 10 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
5		eqid 2738	. . . . . . . . . 10 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁)
6	1, 2, 3, 4, 5	psgnfix2 20804	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)𝑄 = ((SymGrp‘𝑁) Σg 𝑟)))
7	6	imp 407	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)𝑄 = ((SymGrp‘𝑁) Σg 𝑟))
8	7	ad2antrr 723	. . . . . . 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 20806	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 1359	. . . . . . . . . 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 718	. . . . . . . . 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 2742	. . . . . . . . . . 11 ⊢ (𝑠 = (-1↑(♯‘𝑤)) → (𝑠 = (-1↑(♯‘𝑟)) ↔ (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑟))))
14	13	ad2antll 726	. . . . . . . . . 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 481	. . . . . . . . 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 258	. . . . . . . 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 3103	. . . . . . 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 3186	. . . . . 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 3216	. . . . 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 20803	. . . . . . . . 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 407	. . . . . . . 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 723	. . . . . . 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 780	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 723	. . . . . . . . . . . . 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 2744	. . . . . . . . . . 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 413	. . . . . . . . . 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 718	. . . . . . . . 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 2742	. . . . . . . . . . 11 ⊢ (𝑠 = (-1↑(♯‘𝑟)) → (𝑠 = (-1↑(♯‘𝑤)) ↔ (-1↑(♯‘𝑟)) = (-1↑(♯‘𝑤))))
36	35	ad2antll 726	. . . . . . . . . 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 481	. . . . . . . . 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 258	. . . . . . . 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 3103	. . . . . . 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 3186	. . . . . 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 3216	. . . . 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 211	. . . 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 6417	. . 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 8962	. . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐾}) ∈ Fin)
45	44	ad2antrr 723	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑁 ∖ {𝐾}) ∈ Fin)
46		eqid 2738	. . . . . 6 ⊢ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}
47		eqid 2738	. . . . . 6 ⊢ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
48		eqid 2738	. . . . . 6 ⊢ (𝑁 ∖ {𝐾}) = (𝑁 ∖ {𝐾})
49	1, 46, 47, 48	symgfixelsi 19043	. . . . 5 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))))
50	49	adantll 711	. . . 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 19122	. . . 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 584	. . 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 483	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin)
55		elrabi 3618	. . . 4 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → 𝑄 ∈ 𝑃)
56		psgndif.s	. . . . 5 ⊢ 𝑆 = (pmSgn‘𝑁)
57	4, 1, 5, 56	psgnvalfi 19122	. . . 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 596	. . 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 2788	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (𝑆‘𝑄))
60	59	ex 413	1 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (𝑆‘𝑄)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  {crab 3068   ∖ cdif 3884  {csn 4561  ran crn 5590   ↾ cres 5591  ℩cio 6389  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  1c1 10872  -cneg 11206  ↑cexp 13782  ♯chash 14044  Word cword 14217  Basecbs 16912   Σ_g cgsu 17151  SymGrpcsymg 18974  pmTrspcpmtr 19049  pmSgncpsgn 19097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-xor 1507  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-word 14218  df-lsw 14266  df-concat 14274  df-s1 14301  df-substr 14354  df-pfx 14384  df-splice 14463  df-reverse 14472  df-s2 14561  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-tset 16981  df-0g 17152  df-gsum 17153  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-efmnd 18508  df-grp 18580  df-minusg 18581  df-subg 18752  df-ghm 18832  df-gim 18875  df-oppg 18950  df-symg 18975  df-pmtr 19050  df-psgn 19099

This theorem is referenced by:  copsgndif  20808