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Theorem psgnghm 20726

Description: The sign is a homomorphism from the finitary permutation group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)

**Hypotheses**
Ref	Expression
psgnghm.s	⊢ 𝑆 = (SymGrp‘𝐷)
psgnghm.n	⊢ 𝑁 = (pmSgn‘𝐷)
psgnghm.f	⊢ 𝐹 = (𝑆 ↾_s dom 𝑁)
psgnghm.u	⊢ 𝑈 = ((mulGrp‘ℂ_fld) ↾_s {1, -1})

**Assertion**
Ref	Expression
psgnghm	⊢ (𝐷 ∈ 𝑉 → 𝑁 ∈ (𝐹 GrpHom 𝑈))

Proof of Theorem psgnghm Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgnghm.s	. . . . . 6 ⊢ 𝑆 = (SymGrp‘𝐷)
2		eqid 2823	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2823	. . . . . 6 ⊢ {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
4		psgnghm.n	. . . . . 6 ⊢ 𝑁 = (pmSgn‘𝐷)
5	1, 2, 3, 4	psgnfn 18631	. . . . 5 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
6		fndm 6457	. . . . 5 ⊢ (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin})
7	5, 6	ax-mp 5	. . . 4 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
8	7	ssrab3 4059	. . 3 ⊢ dom 𝑁 ⊆ (Base‘𝑆)
9		psgnghm.f	. . . 4 ⊢ 𝐹 = (𝑆 ↾_s dom 𝑁)
10	9, 2	ressbas2 16557	. . 3 ⊢ (dom 𝑁 ⊆ (Base‘𝑆) → dom 𝑁 = (Base‘𝐹))
11	8, 10	ax-mp 5	. 2 ⊢ dom 𝑁 = (Base‘𝐹)
12		psgnghm.u	. . 3 ⊢ 𝑈 = ((mulGrp‘ℂ_fld) ↾_s {1, -1})
13	12	cnmsgnbas 20724	. 2 ⊢ {1, -1} = (Base‘𝑈)
14	11	fvexi 6686	. . 3 ⊢ dom 𝑁 ∈ V
15		eqid 2823	. . . 4 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
16	9, 15	ressplusg 16614	. . 3 ⊢ (dom 𝑁 ∈ V → (+_g‘𝑆) = (+_g‘𝐹))
17	14, 16	ax-mp 5	. 2 ⊢ (+_g‘𝑆) = (+_g‘𝐹)
18		prex 5335	. . 3 ⊢ {1, -1} ∈ V
19		eqid 2823	. . . . 5 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
20		cnfldmul 20553	. . . . 5 ⊢ · = (._r‘ℂ_fld)
21	19, 20	mgpplusg 19245	. . . 4 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
22	12, 21	ressplusg 16614	. . 3 ⊢ ({1, -1} ∈ V → · = (+_g‘𝑈))
23	18, 22	ax-mp 5	. 2 ⊢ · = (+_g‘𝑈)
24	1, 4	psgndmsubg 18632	. . 3 ⊢ (𝐷 ∈ 𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
25	9	subggrp 18284	. . 3 ⊢ (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
26	24, 25	syl 17	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝐹 ∈ Grp)
27	12	cnmsgngrp 20725	. . 3 ⊢ 𝑈 ∈ Grp
28	27	a1i 11	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝑈 ∈ Grp)
29		fnfun 6455	. . . . . 6 ⊢ (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
30	5, 29	ax-mp 5	. . . . 5 ⊢ Fun 𝑁
31		funfn 6387	. . . . 5 ⊢ (Fun 𝑁 ↔ 𝑁 Fn dom 𝑁)
32	30, 31	mpbi 232	. . . 4 ⊢ 𝑁 Fn dom 𝑁
33	32	a1i 11	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝑁 Fn dom 𝑁)
34		eqid 2823	. . . . . 6 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
35	1, 34, 4	psgnvali 18638	. . . . 5 ⊢ (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))))
36		lencl 13885	. . . . . . . . . 10 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℕ₀)
37	36	nn0zd 12088	. . . . . . . . 9 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℤ)
38		m1expcl2 13454	. . . . . . . . . 10 ⊢ ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {-1, 1})
39		prcom 4670	. . . . . . . . . 10 ⊢ {-1, 1} = {1, -1}
40	38, 39	eleqtrdi 2925	. . . . . . . . 9 ⊢ ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {1, -1})
41		eleq1a 2910	. . . . . . . . 9 ⊢ ((-1↑(♯‘𝑧)) ∈ {1, -1} → ((𝑁‘𝑥) = (-1↑(♯‘𝑧)) → (𝑁‘𝑥) ∈ {1, -1}))
42	37, 40, 41	3syl 18	. . . . . . . 8 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → ((𝑁‘𝑥) = (-1↑(♯‘𝑧)) → (𝑁‘𝑥) ∈ {1, -1}))
43	42	adantld 493	. . . . . . 7 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → ((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) → (𝑁‘𝑥) ∈ {1, -1}))
44	43	rexlimiv 3282	. . . . . 6 ⊢ (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) → (𝑁‘𝑥) ∈ {1, -1})
45	44	a1i 11	. . . . 5 ⊢ (𝐷 ∈ 𝑉 → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) → (𝑁‘𝑥) ∈ {1, -1}))
46	35, 45	syl5 34	. . . 4 ⊢ (𝐷 ∈ 𝑉 → (𝑥 ∈ dom 𝑁 → (𝑁‘𝑥) ∈ {1, -1}))
47	46	ralrimiv 3183	. . 3 ⊢ (𝐷 ∈ 𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁‘𝑥) ∈ {1, -1})
48		ffnfv 6884	. . 3 ⊢ (𝑁:dom 𝑁⟶{1, -1} ↔ (𝑁 Fn dom 𝑁 ∧ ∀𝑥 ∈ dom 𝑁(𝑁‘𝑥) ∈ {1, -1}))
49	33, 47, 48	sylanbrc 585	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝑁:dom 𝑁⟶{1, -1})
50		ccatcl 13928	. . . . . . 7 ⊢ ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
51	1, 34, 4	psgnvalii 18639	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷)) → (𝑁‘(𝑆 Σ_g (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
52	50, 51	sylan2 594	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σ_g (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
53	1	symggrp 18530	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Grp)
54		grpmnd 18112	. . . . . . . . . 10 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
55	53, 54	syl 17	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Mnd)
56	34, 1, 2	symgtrf 18599	. . . . . . . . . . 11 ⊢ ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
57		sswrd 13872	. . . . . . . . . . 11 ⊢ (ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
58	56, 57	ax-mp 5	. . . . . . . . . 10 ⊢ Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆)
59	58	sseli 3965	. . . . . . . . 9 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
60	58	sseli 3965	. . . . . . . . 9 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
61	2, 15	gsumccat 18008	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝑧 ∈ Word (Base‘𝑆) ∧ 𝑤 ∈ Word (Base‘𝑆)) → (𝑆 Σ_g (𝑧 ++ 𝑤)) = ((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤)))
62	55, 59, 60, 61	syl3an 1156	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑆 Σ_g (𝑧 ++ 𝑤)) = ((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤)))
63	62	3expb 1116	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑆 Σ_g (𝑧 ++ 𝑤)) = ((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤)))
64	63	fveq2d 6676	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σ_g (𝑧 ++ 𝑤))) = (𝑁‘((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤))))
65		ccatlen 13929	. . . . . . . . 9 ⊢ ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
66	65	adantl 484	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
67	66	oveq2d 7174	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = (-1↑((♯‘𝑧) + (♯‘𝑤))))
68		neg1cn 11754	. . . . . . . . 9 ⊢ -1 ∈ ℂ
69	68	a1i 11	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
70		lencl 13885	. . . . . . . . 9 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑤) ∈ ℕ₀)
71	70	ad2antll 727	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑤) ∈ ℕ₀)
72	36	ad2antrl 726	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑧) ∈ ℕ₀)
73	69, 71, 72	expaddd 13515	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((♯‘𝑧) + (♯‘𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
74	67, 73	eqtrd 2858	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
75	52, 64, 74	3eqtr3d 2866	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
76		oveq12 7167	. . . . . . . 8 ⊢ ((𝑥 = (𝑆 Σ_g 𝑧) ∧ 𝑦 = (𝑆 Σ_g 𝑤)) → (𝑥(+_g‘𝑆)𝑦) = ((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤)))
77	76	fveq2d 6676	. . . . . . 7 ⊢ ((𝑥 = (𝑆 Σ_g 𝑧) ∧ 𝑦 = (𝑆 Σ_g 𝑤)) → (𝑁‘(𝑥(+_g‘𝑆)𝑦)) = (𝑁‘((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤))))
78		oveq12 7167	. . . . . . 7 ⊢ (((𝑁‘𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤))) → ((𝑁‘𝑥) · (𝑁‘𝑦)) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
79	77, 78	eqeqan12d 2840	. . . . . 6 ⊢ (((𝑥 = (𝑆 Σ_g 𝑧) ∧ 𝑦 = (𝑆 Σ_g 𝑤)) ∧ ((𝑁‘𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))) → ((𝑁‘(𝑥(+_g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤)))))
80	79	an4s 658	. . . . 5 ⊢ (((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))) → ((𝑁‘(𝑥(+_g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤)))))
81	75, 80	syl5ibrcom 249	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))) → (𝑁‘(𝑥(+_g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦))))
82	81	rexlimdvva 3296	. . 3 ⊢ (𝐷 ∈ 𝑉 → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))) → (𝑁‘(𝑥(+_g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦))))
83	1, 34, 4	psgnvali 18638	. . . . 5 ⊢ (𝑦 ∈ dom 𝑁 → ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤))))
84	35, 83	anim12i 614	. . . 4 ⊢ ((𝑥 ∈ dom 𝑁 ∧ 𝑦 ∈ dom 𝑁) → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))))
85		reeanv 3369	. . . 4 ⊢ (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))) ↔ (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))))
86	84, 85	sylibr 236	. . 3 ⊢ ((𝑥 ∈ dom 𝑁 ∧ 𝑦 ∈ dom 𝑁) → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))))
87	82, 86	impel 508	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑥 ∈ dom 𝑁 ∧ 𝑦 ∈ dom 𝑁)) → (𝑁‘(𝑥(+_g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦)))
88	11, 13, 17, 23, 26, 28, 49, 87	isghmd 18369	1 ⊢ (𝐷 ∈ 𝑉 → 𝑁 ∈ (𝐹 GrpHom 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 {crab 3144 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 {cpr 4571 I cid 5461 dom cdm 5557 ran crn 5558 Fun wfun 6351 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℂcc 10537 1c1 10540 + caddc 10542 · cmul 10544 -cneg 10873 ℕ₀cn0 11900 ℤcz 11984 ↑cexp 13432 ♯chash 13693 Word cword 13864 ++ cconcat 13924 Basecbs 16485 ↾_s cress 16486 +_gcplusg 16567 Σ_g cgsu 16716 Mndcmnd 17913 Grpcgrp 18105 SubGrpcsubg 18275 GrpHom cghm 18357 SymGrpcsymg 18497 pmTrspcpmtr 18571 pmSgncpsgn 18619 mulGrpcmgp 19241 ℂ_fldccnfld 20547

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 ax-addf 10618 ax-mulf 10619

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-dec 12102 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-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 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 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-cnfld 20548

This theorem is referenced by: psgnghm2 20727 evpmss 20732

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