Theorem psgnghm 21496

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‘ℂfld) ↾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 2730	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2730	. . . . . 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 19438	. . . . 5 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
6	5	fndmi 6625	. . . 4 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
7	6	ssrab3 4048	. . 3 ⊢ dom 𝑁 ⊆ (Base‘𝑆)
8		psgnghm.f	. . . 4 ⊢ 𝐹 = (𝑆 ↾s dom 𝑁)
9	8, 2	ressbas2 17215	. . 3 ⊢ (dom 𝑁 ⊆ (Base‘𝑆) → dom 𝑁 = (Base‘𝐹))
10	7, 9	ax-mp 5	. 2 ⊢ dom 𝑁 = (Base‘𝐹)
11		psgnghm.u	. . 3 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})
12	11	cnmsgnbas 21494	. 2 ⊢ {1, -1} = (Base‘𝑈)
13	10	fvexi 6875	. . 3 ⊢ dom 𝑁 ∈ V
14		eqid 2730	. . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆)
15	8, 14	ressplusg 17261	. . 3 ⊢ (dom 𝑁 ∈ V → (+g‘𝑆) = (+g‘𝐹))
16	13, 15	ax-mp 5	. 2 ⊢ (+g‘𝑆) = (+g‘𝐹)
17		prex 5395	. . 3 ⊢ {1, -1} ∈ V
18		eqid 2730	. . . . 5 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
19		cnfldmul 21279	. . . . 5 ⊢ · = (.r‘ℂfld)
20	18, 19	mgpplusg 20060	. . . 4 ⊢ · = (+g‘(mulGrp‘ℂfld))
21	11, 20	ressplusg 17261	. . 3 ⊢ ({1, -1} ∈ V → · = (+g‘𝑈))
22	17, 21	ax-mp 5	. 2 ⊢ · = (+g‘𝑈)
23	1, 4	psgndmsubg 19439	. . 3 ⊢ (𝐷 ∈ 𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
24	8	subggrp 19068	. . 3 ⊢ (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
25	23, 24	syl 17	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝐹 ∈ Grp)
26	11	cnmsgngrp 21495	. . 3 ⊢ 𝑈 ∈ Grp
27	26	a1i 11	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝑈 ∈ Grp)
28		fnfun 6621	. . . . . 6 ⊢ (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
29	5, 28	ax-mp 5	. . . . 5 ⊢ Fun 𝑁
30		funfn 6549	. . . . 5 ⊢ (Fun 𝑁 ↔ 𝑁 Fn dom 𝑁)
31	29, 30	mpbi 230	. . . 4 ⊢ 𝑁 Fn dom 𝑁
32	31	a1i 11	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝑁 Fn dom 𝑁)
33		eqid 2730	. . . . . 6 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
34	1, 33, 4	psgnvali 19445	. . . . 5 ⊢ (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))))
35		lencl 14505	. . . . . . . . . 10 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℕ0)
36	35	nn0zd 12562	. . . . . . . . 9 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℤ)
37		m1expcl2 14057	. . . . . . . . . 10 ⊢ ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {-1, 1})
38		prcom 4699	. . . . . . . . . 10 ⊢ {-1, 1} = {1, -1}
39	37, 38	eleqtrdi 2839	. . . . . . . . 9 ⊢ ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {1, -1})
40		eleq1a 2824	. . . . . . . . 9 ⊢ ((-1↑(♯‘𝑧)) ∈ {1, -1} → ((𝑁‘𝑥) = (-1↑(♯‘𝑧)) → (𝑁‘𝑥) ∈ {1, -1}))
41	36, 39, 40	3syl 18	. . . . . . . 8 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → ((𝑁‘𝑥) = (-1↑(♯‘𝑧)) → (𝑁‘𝑥) ∈ {1, -1}))
42	41	adantld 490	. . . . . . 7 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → ((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) → (𝑁‘𝑥) ∈ {1, -1}))
43	42	rexlimiv 3128	. . . . . 6 ⊢ (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) → (𝑁‘𝑥) ∈ {1, -1})
44	43	a1i 11	. . . . 5 ⊢ (𝐷 ∈ 𝑉 → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) → (𝑁‘𝑥) ∈ {1, -1}))
45	34, 44	syl5 34	. . . 4 ⊢ (𝐷 ∈ 𝑉 → (𝑥 ∈ dom 𝑁 → (𝑁‘𝑥) ∈ {1, -1}))
46	45	ralrimiv 3125	. . 3 ⊢ (𝐷 ∈ 𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁‘𝑥) ∈ {1, -1})
47		ffnfv 7094	. . 3 ⊢ (𝑁:dom 𝑁⟶{1, -1} ↔ (𝑁 Fn dom 𝑁 ∧ ∀𝑥 ∈ dom 𝑁(𝑁‘𝑥) ∈ {1, -1}))
48	32, 46, 47	sylanbrc 583	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝑁:dom 𝑁⟶{1, -1})
49		ccatcl 14546	. . . . . . 7 ⊢ ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
50	1, 33, 4	psgnvalii 19446	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷)) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
51	49, 50	sylan2 593	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
52	1	symggrp 19337	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Grp)
53	52	grpmndd 18885	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Mnd)
54	33, 1, 2	symgtrf 19406	. . . . . . . . . . 11 ⊢ ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
55		sswrd 14494	. . . . . . . . . . 11 ⊢ (ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
56	54, 55	ax-mp 5	. . . . . . . . . 10 ⊢ Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆)
57	56	sseli 3945	. . . . . . . . 9 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
58	56	sseli 3945	. . . . . . . . 9 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
59	2, 14	gsumccat 18775	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝑧 ∈ Word (Base‘𝑆) ∧ 𝑤 ∈ Word (Base‘𝑆)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤)))
60	53, 57, 58, 59	syl3an 1160	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤)))
61	60	3expb 1120	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤)))
62	61	fveq2d 6865	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (𝑁‘((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤))))
63		ccatlen 14547	. . . . . . . . 9 ⊢ ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
64	63	adantl 481	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
65	64	oveq2d 7406	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = (-1↑((♯‘𝑧) + (♯‘𝑤))))
66		neg1cn 12178	. . . . . . . . 9 ⊢ -1 ∈ ℂ
67	66	a1i 11	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
68		lencl 14505	. . . . . . . . 9 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑤) ∈ ℕ0)
69	68	ad2antll 729	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑤) ∈ ℕ0)
70	35	ad2antrl 728	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑧) ∈ ℕ0)
71	67, 69, 70	expaddd 14120	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((♯‘𝑧) + (♯‘𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
72	65, 71	eqtrd 2765	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
73	51, 62, 72	3eqtr3d 2773	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
74		oveq12 7399	. . . . . . . 8 ⊢ ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑥(+g‘𝑆)𝑦) = ((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤)))
75	74	fveq2d 6865	. . . . . . 7 ⊢ ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑁‘(𝑥(+g‘𝑆)𝑦)) = (𝑁‘((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤))))
76		oveq12 7399	. . . . . . 7 ⊢ (((𝑁‘𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤))) → ((𝑁‘𝑥) · (𝑁‘𝑦)) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
77	75, 76	eqeqan12d 2744	. . . . . 6 ⊢ (((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) ∧ ((𝑁‘𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))) → ((𝑁‘(𝑥(+g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤)))))
78	77	an4s 660	. . . . 5 ⊢ (((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))) → ((𝑁‘(𝑥(+g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤)))))
79	73, 78	syl5ibrcom 247	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))) → (𝑁‘(𝑥(+g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦))))
80	79	rexlimdvva 3195	. . 3 ⊢ (𝐷 ∈ 𝑉 → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))) → (𝑁‘(𝑥(+g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦))))
81	1, 33, 4	psgnvali 19445	. . . . 5 ⊢ (𝑦 ∈ dom 𝑁 → ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤))))
82	34, 81	anim12i 613	. . . 4 ⊢ ((𝑥 ∈ dom 𝑁 ∧ 𝑦 ∈ dom 𝑁) → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))))
83		reeanv 3210	. . . 4 ⊢ (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))) ↔ (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))))
84	82, 83	sylibr 234	. . 3 ⊢ ((𝑥 ∈ dom 𝑁 ∧ 𝑦 ∈ dom 𝑁) → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤)))))
85	80, 84	impel 505	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑥 ∈ dom 𝑁 ∧ 𝑦 ∈ dom 𝑁)) → (𝑁‘(𝑥(+g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦)))
86	10, 12, 16, 22, 25, 27, 48, 85	isghmd 19164	1 ⊢ (𝐷 ∈ 𝑉 → 𝑁 ∈ (𝐹 GrpHom 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  {crab 3408  Vcvv 3450   ∖ cdif 3914   ⊆ wss 3917  {cpr 4594   I cid 5535  dom cdm 5641  ran crn 5642  Fun wfun 6508   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Fincfn 8921  ℂcc 11073  1c1 11076   + caddc 11078   · cmul 11080  -cneg 11413  ℕ₀cn0 12449  ℤcz 12536  ↑cexp 14033  ♯chash 14302  Word cword 14485   ++ cconcat 14542  Basecbs 17186   ↾_s cress 17207  +_gcplusg 17227   Σ_g cgsu 17410  Mndcmnd 18668  Grpcgrp 18872  SubGrpcsubg 19059   GrpHom cghm 19151  SymGrpcsymg 19306  pmTrspcpmtr 19378  pmSgncpsgn 19426  mulGrpcmgp 20056  ℂ_fldccnfld 21271

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-addf 11154  ax-mulf 11155

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-xnn0 12523  df-z 12537  df-dec 12657  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-seq 13974  df-exp 14034  df-hash 14303  df-word 14486  df-lsw 14535  df-concat 14543  df-s1 14568  df-substr 14613  df-pfx 14643  df-splice 14722  df-reverse 14731  df-s2 14821  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-starv 17242  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-0g 17411  df-gsum 17412  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-efmnd 18803  df-grp 18875  df-minusg 18876  df-subg 19062  df-ghm 19152  df-gim 19198  df-oppg 19285  df-symg 19307  df-pmtr 19379  df-psgn 19428  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-cring 20152  df-oppr 20253  df-dvdsr 20273  df-unit 20274  df-invr 20304  df-dvr 20317  df-drng 20647  df-cnfld 21272

This theorem is referenced by:  psgnghm2  21497  evpmss  21502