Theorem psgndif 21920

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 2763	. . . . . . . . . 10 ⊢ ran (pmTrsp‘(𝑁 ∖ {𝐾})) = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
3		eqid 2763	. . . . . . . . . 10 ⊢ (SymGrp‘(𝑁 ∖ {𝐾})) = (SymGrp‘(𝑁 ∖ {𝐾}))
4		eqid 2763	. . . . . . . . . 10 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
5		eqid 2763	. . . . . . . . . 10 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁)
6	1, 2, 3, 4, 5	psgnfix2 21917	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)𝑄 = ((SymGrp‘𝑁) Σg 𝑟)))
7	6	imp 412	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∃𝑟 ∈ Word ran (pmTrsp‘𝑁)𝑄 = ((SymGrp‘𝑁) Σg 𝑟))
8	7	ad2antrr 739	. . . . . . 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 21919	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 1381	. . . . . . . . . 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 734	. . . . . . . . 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 2767	. . . . . . . . . . 11 ⊢ (𝑠 = ( -1↑(♯‘𝑤)) → (𝑠 = ( -1↑(♯‘𝑟)) ↔ ( -1↑(♯‘𝑤)) = ( -1↑(♯‘𝑟))))
14	13	ad2antll 742	. . . . . . . . . 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 486	. . . . . . . . 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 262	. . . . . . . 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 3157	. . . . . . 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 3130	. . . . . 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 3168	. . . . 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 21916	. . . . . . . . 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 412	. . . . . . . 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 739	. . . . . . 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 795	. . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . 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 490	. . . . . . . . . . . . . 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 739	. . . . . . . . . . . . 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 67	. . . . . . . . . . . 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 2769	. . . . . . . . . . 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 418	. . . . . . . . . 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 734	. . . . . . . . 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 2767	. . . . . . . . . . 11 ⊢ (𝑠 = ( -1↑(♯‘𝑟)) → (𝑠 = ( -1↑(♯‘𝑤)) ↔ ( -1↑(♯‘𝑟)) = ( -1↑(♯‘𝑤))))
36	35	ad2antll 742	. . . . . . . . . 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 486	. . . . . . . . 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 262	. . . . . . . 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 3157	. . . . . . 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 3130	. . . . . 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 3168	. . . . 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 215	. . . 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 6529	. . 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 9201	. . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐾}) ∈ Fin)
45	44	ad2antrr 739	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑁 ∖ {𝐾}) ∈ Fin)
46		eqid 2763	. . . . . 6 ⊢ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}
47		eqid 2763	. . . . . 6 ⊢ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
48		eqid 2763	. . . . . 6 ⊢ (𝑁 ∖ {𝐾}) = (𝑁 ∖ {𝐾})
49	1, 46, 47, 48	symgfixelsi 19661	. . . . 5 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))))
50	49	adantll 727	. . . 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 19740	. . . 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 596	. . 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 488	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin)
55		elrabi 3643	. . . 4 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → 𝑄 ∈ 𝑃)
56		psgndif.s	. . . . 5 ⊢ 𝑆 = (pmSgn‘𝑁)
57	4, 1, 5, 56	psgnvalfi 19740	. . . 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 608	. . 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 2808	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (𝑆‘𝑄))
60	59	ex 418	1 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (𝑆‘𝑄)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∃wrex 3089  {crab 3415   ∖ cdif 4047  {csn 4587  ran crn 5655   ↾ cres 5656  ℩cio 6498  ‘cfv 6546  (class class class)co 7427  Fincfn 8984  1c1 11211   -cneg 11552  ↑cexp 14214  ♯chash 14484  Word cword 14668  Basecbs 17399   Σ_g cgsu 17623  SymGrpcsymg 19595  pmTrspcpmtr 19667  pmSgncpsgn 19715

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1541  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4838  df-ot 4861  df-uni 4870  df-int 4910  df-iun 4955  df-iin 4956  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-se 5608  df-we 5609  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 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-tpos 8254  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-2o 8488  df-er 8728  df-map 8860  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 10030  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-div 11984  df-nn 12346  df-2 12415  df-3 12416  df-4 12417  df-5 12418  df-6 12419  df-7 12420  df-8 12421  df-9 12422  df-n0 12617  df-xnn0 12690  df-z 12704  df-uz 12976  df-rp 13131  df-fz 13650  df-fzo 13799  df-seq 14155  df-exp 14215  df-hash 14485  df-word 14669  df-lsw 14718  df-concat 14726  df-s1 14753  df-substr 14800  df-pfx 14832  df-splice 14910  df-reverse 14919  df-s2 15010  df-struct 17337  df-sets 17354  df-slot 17372  df-ndx 17384  df-base 17400  df-ress 17421  df-plusg 17453  df-tset 17459  df-0g 17624  df-gsum 17625  df-mre 17768  df-mrc 17769  df-acs 17771  df-mgm 18828  df-sgrp 18920  df-mnd 18936  df-mhm 18990  df-submnd 18991  df-efmnd 19077  df-grp 19159  df-minusg 19160  df-subg 19345  df-ghm 19440  df-gim 19485  df-oppg 19572  df-symg 19596  df-pmtr 19668  df-psgn 19717

This theorem is used by:  copsgndif  21921