Theorem psgndif 21006

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 2736	. . . . . . . . . 10 ⊢ ran (pmTrsp‘(𝑁 ∖ {𝐾})) = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
3		eqid 2736	. . . . . . . . . 10 ⊢ (SymGrp‘(𝑁 ∖ {𝐾})) = (SymGrp‘(𝑁 ∖ {𝐾}))
4		eqid 2736	. . . . . . . . . 10 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
5		eqid 2736	. . . . . . . . . 10 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁)
6	1, 2, 3, 4, 5	psgnfix2 21003	. . . . . . . . 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 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 21005	. . . . . . . . . . . 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 1360	. . . . . . . . . 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 2740	. . . . . . . . . . 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 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 3143	. . . . . . 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 3121	. . . . . 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 3154	. . . . 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 21002	. . . . . . . . 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 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 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 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 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 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 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 2742	. . . . . . . . . . 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 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 2740	. . . . . . . . . . 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 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 3143	. . . . . . 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 3121	. . . . . 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 3154	. . . . 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 6480	. . 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 9123	. . . . 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 2736	. . . . . 6 ⊢ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}
47		eqid 2736	. . . . . 6 ⊢ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
48		eqid 2736	. . . . . 6 ⊢ (𝑁 ∖ {𝐾}) = (𝑁 ∖ {𝐾})
49	1, 46, 47, 48	symgfixelsi 19217	. . . . 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 19296	. . . 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 3639	. . . 4 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → 𝑄 ∈ 𝑃)
56		psgndif.s	. . . . 5 ⊢ 𝑆 = (pmSgn‘𝑁)
57	4, 1, 5, 56	psgnvalfi 19296	. . . 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 2786	. 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 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3073  {crab 3407   ∖ cdif 3907  {csn 4586  ran crn 5634   ↾ cres 5635  ℩cio 6446  ‘cfv 6496  (class class class)co 7357  Fincfn 8883  1c1 11052  -cneg 11386  ↑cexp 13967  ♯chash 14230  Word cword 14402  Basecbs 17083   Σ_g cgsu 17322  SymGrpcsymg 19148  pmTrspcpmtr 19223  pmSgncpsgn 19271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-xor 1510  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-splice 14638  df-reverse 14647  df-s2 14737  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-tset 17152  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-efmnd 18679  df-grp 18751  df-minusg 18752  df-subg 18925  df-ghm 19006  df-gim 19049  df-oppg 19124  df-symg 19149  df-pmtr 19224  df-psgn 19273

This theorem is referenced by:  copsgndif  21007