Theorem psgnghm 21621

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 2740	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2740	. . . . . 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 19543	. . . . 5 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
6	5	fndmi 6683	. . . 4 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
7	6	ssrab3 4105	. . 3 ⊢ dom 𝑁 ⊆ (Base‘𝑆)
8		psgnghm.f	. . . 4 ⊢ 𝐹 = (𝑆 ↾s dom 𝑁)
9	8, 2	ressbas2 17296	. . 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 21619	. 2 ⊢ {1, -1} = (Base‘𝑈)
13	10	fvexi 6934	. . 3 ⊢ dom 𝑁 ∈ V
14		eqid 2740	. . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆)
15	8, 14	ressplusg 17349	. . 3 ⊢ (dom 𝑁 ∈ V → (+g‘𝑆) = (+g‘𝐹))
16	13, 15	ax-mp 5	. 2 ⊢ (+g‘𝑆) = (+g‘𝐹)
17		prex 5452	. . 3 ⊢ {1, -1} ∈ V
18		eqid 2740	. . . . 5 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
19		cnfldmul 21395	. . . . 5 ⊢ · = (.r‘ℂfld)
20	18, 19	mgpplusg 20165	. . . 4 ⊢ · = (+g‘(mulGrp‘ℂfld))
21	11, 20	ressplusg 17349	. . 3 ⊢ ({1, -1} ∈ V → · = (+g‘𝑈))
22	17, 21	ax-mp 5	. 2 ⊢ · = (+g‘𝑈)
23	1, 4	psgndmsubg 19544	. . 3 ⊢ (𝐷 ∈ 𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
24	8	subggrp 19169	. . 3 ⊢ (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
25	23, 24	syl 17	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝐹 ∈ Grp)
26	11	cnmsgngrp 21620	. . 3 ⊢ 𝑈 ∈ Grp
27	26	a1i 11	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝑈 ∈ Grp)
28		fnfun 6679	. . . . . 6 ⊢ (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
29	5, 28	ax-mp 5	. . . . 5 ⊢ Fun 𝑁
30		funfn 6608	. . . . 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 2740	. . . . . 6 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
34	1, 33, 4	psgnvali 19550	. . . . 5 ⊢ (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))))
35		lencl 14581	. . . . . . . . . 10 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℕ0)
36	35	nn0zd 12665	. . . . . . . . 9 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℤ)
37		m1expcl2 14136	. . . . . . . . . 10 ⊢ ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {-1, 1})
38		prcom 4757	. . . . . . . . . 10 ⊢ {-1, 1} = {1, -1}
39	37, 38	eleqtrdi 2854	. . . . . . . . 9 ⊢ ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {1, -1})
40		eleq1a 2839	. . . . . . . . 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 3154	. . . . . 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 3151	. . 3 ⊢ (𝐷 ∈ 𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁‘𝑥) ∈ {1, -1})
47		ffnfv 7153	. . 3 ⊢ (𝑁:dom 𝑁⟶{1, -1} ↔ (𝑁 Fn dom 𝑁 ∧ ∀𝑥 ∈ dom 𝑁(𝑁‘𝑥) ∈ {1, -1}))
48	32, 46, 47	sylanbrc 582	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝑁:dom 𝑁⟶{1, -1})
49		ccatcl 14622	. . . . . . 7 ⊢ ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
50	1, 33, 4	psgnvalii 19551	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷)) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
51	49, 50	sylan2 592	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
52	1	symggrp 19442	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Grp)
53	52	grpmndd 18986	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Mnd)
54	33, 1, 2	symgtrf 19511	. . . . . . . . . . 11 ⊢ ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
55		sswrd 14570	. . . . . . . . . . 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 4004	. . . . . . . . 9 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
58	56	sseli 4004	. . . . . . . . 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 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 6924	. . . . . 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 14623	. . . . . . . . 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 7464	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = (-1↑((♯‘𝑧) + (♯‘𝑤))))
66		neg1cn 12407	. . . . . . . . 9 ⊢ -1 ∈ ℂ
67	66	a1i 11	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
68		lencl 14581	. . . . . . . . 9 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑤) ∈ ℕ0)
69	68	ad2antll 728	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑤) ∈ ℕ0)
70	35	ad2antrl 727	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑧) ∈ ℕ0)
71	67, 69, 70	expaddd 14198	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((♯‘𝑧) + (♯‘𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
72	65, 71	eqtrd 2780	. . . . . 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 2788	. . . . 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 7457	. . . . . . . 8 ⊢ ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑥(+g‘𝑆)𝑦) = ((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤)))
75	74	fveq2d 6924	. . . . . . 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 7457	. . . . . . 7 ⊢ (((𝑁‘𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤))) → ((𝑁‘𝑥) · (𝑁‘𝑦)) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
77	75, 76	eqeqan12d 2754	. . . . . 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 659	. . . . 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 3219	. . 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 19550	. . . . 5 ⊢ (𝑦 ∈ dom 𝑁 → ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤))))
82	34, 81	anim12i 612	. . . 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 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 19265	1 ⊢ (𝐷 ∈ 𝑉 → 𝑁 ∈ (𝐹 GrpHom 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  {cpr 4650   I cid 5592  dom cdm 5700  ran crn 5701  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  ℂcc 11182  1c1 11185   + caddc 11187   · cmul 11189  -cneg 11521  ℕ₀cn0 12553  ℤcz 12639  ↑cexp 14112  ♯chash 14379  Word cword 14562   ++ cconcat 14618  Basecbs 17258   ↾_s cress 17287  +_gcplusg 17311   Σ_g cgsu 17500  Mndcmnd 18772  Grpcgrp 18973  SubGrpcsubg 19160   GrpHom cghm 19252  SymGrpcsymg 19410  pmTrspcpmtr 19483  pmSgncpsgn 19531  mulGrpcmgp 20161  ℂ_fldccnfld 21387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-addf 11263  ax-mulf 11264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-xor 1509  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-xnn0 12626  df-z 12640  df-dec 12759  df-uz 12904  df-rp 13058  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-word 14563  df-lsw 14611  df-concat 14619  df-s1 14644  df-substr 14689  df-pfx 14719  df-splice 14798  df-reverse 14807  df-s2 14897  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-starv 17326  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-0g 17501  df-gsum 17502  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-efmnd 18904  df-grp 18976  df-minusg 18977  df-subg 19163  df-ghm 19253  df-gim 19299  df-oppg 19386  df-symg 19411  df-pmtr 19484  df-psgn 19533  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-cring 20263  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-dvr 20427  df-drng 20753  df-cnfld 21388

This theorem is referenced by:  psgnghm2  21622  evpmss  21627