Theorem psgnghm 21633

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 2763	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2763	. . . . . 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 19542	. . . . 5 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
6	5	fndmi 6626	. . . 4 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
7	6	ssrab3 4036	. . 3 ⊢ dom 𝑁 ⊆ (Base‘𝑆)
8		psgnghm.f	. . . 4 ⊢ 𝐹 = (𝑆 ↾s dom 𝑁)
9	8, 2	ressbas2 17275	. . 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 21631	. 2 ⊢ {1, -1} = (Base‘𝑈)
13	10	fvexi 6882	. . 3 ⊢ dom 𝑁 ∈ V
14		eqid 2763	. . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆)
15	8, 14	ressplusg 17321	. . 3 ⊢ (dom 𝑁 ∈ V → (+g‘𝑆) = (+g‘𝐹))
16	13, 15	ax-mp 5	. 2 ⊢ (+g‘𝑆) = (+g‘𝐹)
17		prex 5396	. . 3 ⊢ {1, -1} ∈ V
18		eqid 2763	. . . . 5 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
19		cnfldmul 21433	. . . . 5 ⊢ · = (.r‘ℂfld)
20	18, 19	mgpplusg 20191	. . . 4 ⊢ · = (+g‘(mulGrp‘ℂfld))
21	11, 20	ressplusg 17321	. . 3 ⊢ ({1, -1} ∈ V → · = (+g‘𝑈))
22	17, 21	ax-mp 5	. 2 ⊢ · = (+g‘𝑈)
23	1, 4	psgndmsubg 19543	. . 3 ⊢ (𝐷 ∈ 𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
24	8	subggrp 19172	. . 3 ⊢ (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
25	23, 24	syl 17	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝐹 ∈ Grp)
26	11	cnmsgngrp 21632	. . 3 ⊢ 𝑈 ∈ Grp
27	26	a1i 11	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝑈 ∈ Grp)
28		fnfun 6622	. . . . . 6 ⊢ (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
29	5, 28	ax-mp 5	. . . . 5 ⊢ Fun 𝑁
30		funfn 6552	. . . . 5 ⊢ (Fun 𝑁 ↔ 𝑁 Fn dom 𝑁)
31	29, 30	mpbi 232	. . . 4 ⊢ 𝑁 Fn dom 𝑁
32	31	a1i 11	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝑁 Fn dom 𝑁)
33		eqid 2763	. . . . . 6 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
34	1, 33, 4	psgnvali 19549	. . . . 5 ⊢ (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))))
35		lencl 14547	. . . . . . . . . 10 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℕ0)
36	35	nn0zd 12594	. . . . . . . . 9 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℤ)
37		m1expcl2 14099	. . . . . . . . . 10 ⊢ ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {-1, 1})
38		prcom 4692	. . . . . . . . . 10 ⊢ {-1, 1} = {1, -1}
39	37, 38	eleqtrdi 2873	. . . . . . . . 9 ⊢ ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {1, -1})
40		eleq1a 2858	. . . . . . . . 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 494	. . . . . . 7 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → ((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))) → (𝑁‘𝑥) ∈ {1, -1}))
43	42	rexlimiv 3157	. . . . . 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 3154	. . 3 ⊢ (𝐷 ∈ 𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁‘𝑥) ∈ {1, -1})
47		ffnfv 7101	. . 3 ⊢ (𝑁:dom 𝑁⟶{1, -1} ↔ (𝑁 Fn dom 𝑁 ∧ ∀𝑥 ∈ dom 𝑁(𝑁‘𝑥) ∈ {1, -1}))
48	32, 46, 47	sylanbrc 592	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝑁:dom 𝑁⟶{1, -1})
49		ccatcl 14588	. . . . . . 7 ⊢ ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
50	1, 33, 4	psgnvalii 19550	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷)) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
51	49, 50	sylan2 602	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
52	1	symggrp 19441	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Grp)
53	52	grpmndd 18989	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Mnd)
54	33, 1, 2	symgtrf 19510	. . . . . . . . . . 11 ⊢ ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
55		sswrd 14536	. . . . . . . . . . 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 3933	. . . . . . . . 9 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
58	56	sseli 3933	. . . . . . . . 9 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
59	2, 14	gsumccat 18876	. . . . . . . . 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 1174	. . . . . . . 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 1134	. . . . . . 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 6872	. . . . . 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 14589	. . . . . . . . 9 ⊢ ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
64	63	adantl 485	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
65	64	oveq2d 7413	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = (-1↑((♯‘𝑧) + (♯‘𝑤))))
66		neg1cn 12181	. . . . . . . . 9 ⊢ -1 ∈ ℂ
67	66	a1i 11	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
68		lencl 14547	. . . . . . . . 9 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑤) ∈ ℕ0)
69	68	ad2antll 739	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑤) ∈ ℕ0)
70	35	ad2antrl 738	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑧) ∈ ℕ0)
71	67, 69, 70	expaddd 14162	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((♯‘𝑧) + (♯‘𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
72	65, 71	eqtrd 2798	. . . . . 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 2806	. . . . 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 7406	. . . . . . . 8 ⊢ ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑥(+g‘𝑆)𝑦) = ((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤)))
75	74	fveq2d 6872	. . . . . . 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 7406	. . . . . . 7 ⊢ (((𝑁‘𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤))) → ((𝑁‘𝑥) · (𝑁‘𝑦)) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
77	75, 76	eqeqan12d 2777	. . . . . 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 670	. . . . 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 249	. . . 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 3220	. . 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 19549	. . . . 5 ⊢ (𝑦 ∈ dom 𝑁 → ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤))))
82	34, 81	anim12i 622	. . . 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 3235	. . . 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 236	. . 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 513	. 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 19266	1 ⊢ (𝐷 ∈ 𝑉 → 𝑁 ∈ (𝐹 GrpHom 𝑈))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087  {crab 3415  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  {cpr 4585   I cid 5542  dom cdm 5648  ran crn 5649  Fun wfun 6516   Fn wfn 6517  ⟶wf 6518  ‘cfv 6522  (class class class)co 7397  Fincfn 8928  ℂcc 11072  1c1 11075   + caddc 11077   · cmul 11079  -cneg 11416  ℕ₀cn0 12482  ℤcz 12569  ↑cexp 14075  ♯chash 14344  Word cword 14527   ++ cconcat 14584  Basecbs 17246   ↾_s cress 17267  +_gcplusg 17287   Σ_g cgsu 17470  Mndcmnd 18769  Grpcgrp 18976  SubGrpcsubg 19163   GrpHom cghm 19254  SymGrpcsymg 19410  pmTrspcpmtr 19482  pmSgncpsgn 19530  mulGrpcmgp 20187  ℂ_fldccnfld 21425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151  ax-addf 11153  ax-mulf 11154

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-xor 1533  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-isom 6531  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-1st 7971  df-2nd 7972  df-tpos 8207  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-2o 8439  df-er 8679  df-map 8811  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932  df-card 9898  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-div 11846  df-nn 12212  df-2 12281  df-3 12282  df-4 12283  df-5 12284  df-6 12285  df-7 12286  df-8 12287  df-9 12288  df-n0 12483  df-xnn0 12556  df-z 12570  df-dec 12690  df-uz 12841  df-rp 12995  df-fz 13514  df-fzo 13661  df-seq 14016  df-exp 14076  df-hash 14345  df-word 14528  df-lsw 14577  df-concat 14585  df-s1 14611  df-substr 14656  df-pfx 14686  df-splice 14764  df-reverse 14773  df-s2 14862  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17247  df-ress 17268  df-plusg 17300  df-mulr 17301  df-starv 17302  df-tset 17306  df-ple 17307  df-ds 17309  df-unif 17310  df-0g 17471  df-gsum 17472  df-mre 17615  df-mrc 17616  df-acs 17618  df-mgm 18675  df-sgrp 18754  df-mnd 18770  df-mhm 18818  df-submnd 18819  df-efmnd 18904  df-grp 18979  df-minusg 18980  df-subg 19166  df-ghm 19255  df-gim 19300  df-oppg 19387  df-symg 19411  df-pmtr 19483  df-psgn 19532  df-cmn 19823  df-abl 19824  df-mgp 20188  df-rng 20200  df-ur 20233  df-ring 20286  df-cring 20287  df-oppr 20387  df-dvdsr 20407  df-unit 20408  df-invr 20438  df-dvr 20451  df-drng 20782  df-cnfld 21426

This theorem is referenced by:  psgnghm2  21634  evpmss  21639