Theorem psgndif 21562

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 2735	. . . . . . . . . 10 ⊢ ran (pmTrsp‘(𝑁 ∖ {𝐾})) = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
3		eqid 2735	. . . . . . . . . 10 ⊢ (SymGrp‘(𝑁 ∖ {𝐾})) = (SymGrp‘(𝑁 ∖ {𝐾}))
4		eqid 2735	. . . . . . . . . 10 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
5		eqid 2735	. . . . . . . . . 10 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁)
6	1, 2, 3, 4, 5	psgnfix2 21559	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)𝑄 = ((SymGrp‘𝑁) Σg 𝑟)))
7	6	imp 406	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)𝑄 = ((SymGrp‘𝑁) Σg 𝑟))
8	7	ad2antrr 726	. . . . . . 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 21561	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 1361	. . . . . . . . . 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 721	. . . . . . . . 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 2739	. . . . . . . . . . 11 ⊢ (𝑠 = (-1↑(♯‘𝑤)) → (𝑠 = (-1↑(♯‘𝑟)) ↔ (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑟))))
14	13	ad2antll 729	. . . . . . . . . 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 480	. . . . . . . . 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 259	. . . . . . . 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 3132	. . . . . . 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 3105	. . . . . 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 3143	. . . . 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 21558	. . . . . . . . 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 406	. . . . . . . 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 726	. . . . . . 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 782	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 2741	. . . . . . . . . . 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 412	. . . . . . . . . 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 721	. . . . . . . . 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 2739	. . . . . . . . . . 11 ⊢ (𝑠 = (-1↑(♯‘𝑟)) → (𝑠 = (-1↑(♯‘𝑤)) ↔ (-1↑(♯‘𝑟)) = (-1↑(♯‘𝑤))))
36	35	ad2antll 729	. . . . . . . . . 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 480	. . . . . . . . 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 259	. . . . . . . 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 3132	. . . . . . 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 3105	. . . . . 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 3143	. . . . 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 212	. . . 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 6515	. . 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 9189	. . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐾}) ∈ Fin)
45	44	ad2antrr 726	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑁 ∖ {𝐾}) ∈ Fin)
46		eqid 2735	. . . . . 6 ⊢ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}
47		eqid 2735	. . . . . 6 ⊢ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
48		eqid 2735	. . . . . 6 ⊢ (𝑁 ∖ {𝐾}) = (𝑁 ∖ {𝐾})
49	1, 46, 47, 48	symgfixelsi 19416	. . . . 5 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))))
50	49	adantll 714	. . . 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 19495	. . . 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 482	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin)
55		elrabi 3666	. . . 4 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → 𝑄 ∈ 𝑃)
56		psgndif.s	. . . . 5 ⊢ 𝑆 = (pmSgn‘𝑁)
57	4, 1, 5, 56	psgnvalfi 19495	. . . 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 2780	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (𝑆‘𝑄))
60	59	ex 412	1 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (𝑆‘𝑄)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∃wrex 3060  {crab 3415   ∖ cdif 3923  {csn 4601  ran crn 5655   ↾ cres 5656  ℩cio 6482  ‘cfv 6531  (class class class)co 7405  Fincfn 8959  1c1 11130  -cneg 11467  ↑cexp 14079  ♯chash 14348  Word cword 14531  Basecbs 17228   Σ_g cgsu 17454  SymGrpcsymg 19350  pmTrspcpmtr 19422  pmSgncpsgn 19470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-xnn0 12575  df-z 12589  df-uz 12853  df-rp 13009  df-fz 13525  df-fzo 13672  df-seq 14020  df-exp 14080  df-hash 14349  df-word 14532  df-lsw 14581  df-concat 14589  df-s1 14614  df-substr 14659  df-pfx 14689  df-splice 14768  df-reverse 14777  df-s2 14867  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-tset 17290  df-0g 17455  df-gsum 17456  df-mre 17598  df-mrc 17599  df-acs 17601  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-efmnd 18847  df-grp 18919  df-minusg 18920  df-subg 19106  df-ghm 19196  df-gim 19242  df-oppg 19329  df-symg 19351  df-pmtr 19423  df-psgn 19472

This theorem is referenced by:  copsgndif  21563