Theorem psgnghm 21487

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 2729	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2729	. . . . . 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 19380	. . . . 5 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
6	5	fndmi 6586	. . . 4 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
7	6	ssrab3 4033	. . 3 ⊢ dom 𝑁 ⊆ (Base‘𝑆)
8		psgnghm.f	. . . 4 ⊢ 𝐹 = (𝑆 ↾s dom 𝑁)
9	8, 2	ressbas2 17149	. . 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 21485	. 2 ⊢ {1, -1} = (Base‘𝑈)
13	10	fvexi 6836	. . 3 ⊢ dom 𝑁 ∈ V
14		eqid 2729	. . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆)
15	8, 14	ressplusg 17195	. . 3 ⊢ (dom 𝑁 ∈ V → (+g‘𝑆) = (+g‘𝐹))
16	13, 15	ax-mp 5	. 2 ⊢ (+g‘𝑆) = (+g‘𝐹)
17		prex 5376	. . 3 ⊢ {1, -1} ∈ V
18		eqid 2729	. . . . 5 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
19		cnfldmul 21269	. . . . 5 ⊢ · = (.r‘ℂfld)
20	18, 19	mgpplusg 20029	. . . 4 ⊢ · = (+g‘(mulGrp‘ℂfld))
21	11, 20	ressplusg 17195	. . 3 ⊢ ({1, -1} ∈ V → · = (+g‘𝑈))
22	17, 21	ax-mp 5	. 2 ⊢ · = (+g‘𝑈)
23	1, 4	psgndmsubg 19381	. . 3 ⊢ (𝐷 ∈ 𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
24	8	subggrp 19008	. . 3 ⊢ (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
25	23, 24	syl 17	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝐹 ∈ Grp)
26	11	cnmsgngrp 21486	. . 3 ⊢ 𝑈 ∈ Grp
27	26	a1i 11	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝑈 ∈ Grp)
28		fnfun 6582	. . . . . 6 ⊢ (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
29	5, 28	ax-mp 5	. . . . 5 ⊢ Fun 𝑁
30		funfn 6512	. . . . 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 2729	. . . . . 6 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
34	1, 33, 4	psgnvali 19387	. . . . 5 ⊢ (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁‘𝑥) = (-1↑(♯‘𝑧))))
35		lencl 14440	. . . . . . . . . 10 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℕ0)
36	35	nn0zd 12497	. . . . . . . . 9 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℤ)
37		m1expcl2 13992	. . . . . . . . . 10 ⊢ ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {-1, 1})
38		prcom 4684	. . . . . . . . . 10 ⊢ {-1, 1} = {1, -1}
39	37, 38	eleqtrdi 2838	. . . . . . . . 9 ⊢ ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {1, -1})
40		eleq1a 2823	. . . . . . . . 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 3123	. . . . . 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 3120	. . 3 ⊢ (𝐷 ∈ 𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁‘𝑥) ∈ {1, -1})
47		ffnfv 7053	. . 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 14481	. . . . . . 7 ⊢ ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
50	1, 33, 4	psgnvalii 19388	. . . . . . 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 19279	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Grp)
53	52	grpmndd 18825	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Mnd)
54	33, 1, 2	symgtrf 19348	. . . . . . . . . . 11 ⊢ ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
55		sswrd 14429	. . . . . . . . . . 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 3931	. . . . . . . . 9 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
58	56	sseli 3931	. . . . . . . . 9 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
59	2, 14	gsumccat 18715	. . . . . . . . 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 6826	. . . . . 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 14482	. . . . . . . . 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 7365	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = (-1↑((♯‘𝑧) + (♯‘𝑤))))
66		neg1cn 12113	. . . . . . . . 9 ⊢ -1 ∈ ℂ
67	66	a1i 11	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
68		lencl 14440	. . . . . . . . 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 14055	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((♯‘𝑧) + (♯‘𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
72	65, 71	eqtrd 2764	. . . . . 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 2772	. . . . 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 7358	. . . . . . . 8 ⊢ ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑥(+g‘𝑆)𝑦) = ((𝑆 Σg 𝑧)(+g‘𝑆)(𝑆 Σg 𝑤)))
75	74	fveq2d 6826	. . . . . . 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 7358	. . . . . . 7 ⊢ (((𝑁‘𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁‘𝑦) = (-1↑(♯‘𝑤))) → ((𝑁‘𝑥) · (𝑁‘𝑦)) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
77	75, 76	eqeqan12d 2743	. . . . . 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 3186	. . 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 19387	. . . . 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 3201	. . . 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 19104	1 ⊢ (𝐷 ∈ 𝑉 → 𝑁 ∈ (𝐹 GrpHom 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3394  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  {cpr 4579   I cid 5513  dom cdm 5619  ran crn 5620  Fun wfun 6476   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  Fincfn 8872  ℂcc 11007  1c1 11010   + caddc 11012   · cmul 11014  -cneg 11348  ℕ₀cn0 12384  ℤcz 12471  ↑cexp 13968  ♯chash 14237  Word cword 14420   ++ cconcat 14477  Basecbs 17120   ↾_s cress 17141  +_gcplusg 17161   Σ_g cgsu 17344  Mndcmnd 18608  Grpcgrp 18812  SubGrpcsubg 18999   GrpHom cghm 19091  SymGrpcsymg 19248  pmTrspcpmtr 19320  pmSgncpsgn 19368  mulGrpcmgp 20025  ℂ_fldccnfld 21261

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-addf 11088  ax-mulf 11089

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-tpos 8159  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-xnn0 12458  df-z 12472  df-dec 12592  df-uz 12736  df-rp 12894  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969  df-hash 14238  df-word 14421  df-lsw 14470  df-concat 14478  df-s1 14503  df-substr 14548  df-pfx 14578  df-splice 14656  df-reverse 14665  df-s2 14755  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-0g 17345  df-gsum 17346  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-submnd 18658  df-efmnd 18743  df-grp 18815  df-minusg 18816  df-subg 19002  df-ghm 19092  df-gim 19138  df-oppg 19225  df-symg 19249  df-pmtr 19321  df-psgn 19370  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-cring 20121  df-oppr 20222  df-dvdsr 20242  df-unit 20243  df-invr 20273  df-dvr 20286  df-drng 20616  df-cnfld 21262

This theorem is referenced by:  psgnghm2  21488  evpmss  21493