Theorem zrrnghm 45363

Description: The constant mapping to zero is a nonunital ring homomorphism from the zero ring to any nonunital ring. (Contributed by AV, 17-Apr-2020.)

Hypotheses

Ref	Expression
zrrhm.b	⊢ 𝐵 = (Base‘𝑇)
zrrhm.0	⊢ 0 = (0g‘𝑆)
zrrhm.h	⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )

Assertion

Ref	Expression
zrrnghm	⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHomo 𝑆))

Distinct variable groups:   𝑥,𝐵   𝑥,𝑆   𝑥,𝑇   𝑥, 0

Allowed substitution hint:   𝐻(𝑥)

Proof of Theorem zrrnghm

Dummy variables 𝑎 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 4057	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring)
2		ringrng 45325	. . . . 5 ⊢ (𝑇 ∈ Ring → 𝑇 ∈ Rng)
3	1, 2	syl 17	. . . 4 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Rng)
4	3	anim1i 614	. . 3 ⊢ ((𝑇 ∈ (Ring ∖ NzRing) ∧ 𝑆 ∈ Rng) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
5	4	ancoms 458	. 2 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
6		rngabl 45323	. . . . . 6 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel)
7		ablgrp 19306	. . . . . 6 ⊢ (𝑆 ∈ Abel → 𝑆 ∈ Grp)
8	6, 7	syl 17	. . . . 5 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Grp)
9	8	adantr 480	. . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑆 ∈ Grp)
10		ringgrp 19703	. . . . . 6 ⊢ (𝑇 ∈ Ring → 𝑇 ∈ Grp)
11	1, 10	syl 17	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Grp)
12	11	adantl 481	. . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑇 ∈ Grp)
13		zrrhm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑇)
14		eqid 2738	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	13, 14	0ringbas 45317	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝐵 = {(0g‘𝑇)})
16	15	adantl 481	. . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐵 = {(0g‘𝑇)})
17		zrrhm.0	. . . . 5 ⊢ 0 = (0g‘𝑆)
18		zrrhm.h	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )
19	13, 17, 18, 14	c0snghm 45362	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {(0g‘𝑇)}) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
20	9, 12, 16, 19	syl3anc 1369	. . 3 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
21	18	a1i 11	. . . . . . . 8 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ))
22		eqidd 2739	. . . . . . . 8 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ 𝑥 = (0g‘𝑇)) → 0 = 0 )
23	13, 14	ring0cl 19723	. . . . . . . . . 10 ⊢ (𝑇 ∈ Ring → (0g‘𝑇) ∈ 𝐵)
24	1, 23	syl 17	. . . . . . . . 9 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → (0g‘𝑇) ∈ 𝐵)
25	24	ad2antlr 723	. . . . . . . 8 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (0g‘𝑇) ∈ 𝐵)
26	17	fvexi 6770	. . . . . . . . 9 ⊢ 0 ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → 0 ∈ V)
28	21, 22, 25, 27	fvmptd 6864	. . . . . . 7 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (𝐻‘(0g‘𝑇)) = 0 )
29		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑆) = (Base‘𝑆)
30	29, 17	grpidcl 18522	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Grp → 0 ∈ (Base‘𝑆))
31	8, 30	syl 17	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Rng → 0 ∈ (Base‘𝑆))
32		eqid 2738	. . . . . . . . . . . . 13 ⊢ (.r‘𝑆) = (.r‘𝑆)
33	29, 32, 17	rnglz 45330	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Rng ∧ 0 ∈ (Base‘𝑆)) → ( 0 (.r‘𝑆) 0 ) = 0 )
34	31, 33	mpdan 683	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Rng → ( 0 (.r‘𝑆) 0 ) = 0 )
35	34	adantr 480	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ( 0 (.r‘𝑆) 0 ) = 0 )
36	35	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ( 0 (.r‘𝑆) 0 ) = 0 )
37	36	adantr 480	. . . . . . . 8 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → ( 0 (.r‘𝑆) 0 ) = 0 )
38		simpr 484	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → (𝐻‘(0g‘𝑇)) = 0 )
39	38, 38	oveq12d 7273	. . . . . . . 8 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘(0g‘𝑇))) = ( 0 (.r‘𝑆) 0 ))
40		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑇) = (.r‘𝑇)
41	13, 40, 14	ringlz 19741	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ Ring ∧ (0g‘𝑇) ∈ 𝐵) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
42	1, 23, 41	syl2anc2 584	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
43	42	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
45	44	fveq2d 6760	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))) = (𝐻‘(0g‘𝑇)))
46	45, 38	eqtrd 2778	. . . . . . . 8 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))) = 0 )
47	37, 39, 46	3eqtr4rd 2789	. . . . . . 7 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘(0g‘𝑇))))
48	28, 47	mpdan 683	. . . . . 6 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘(0g‘𝑇))))
49	23, 23	jca 511	. . . . . . . . 9 ⊢ (𝑇 ∈ Ring → ((0g‘𝑇) ∈ 𝐵 ∧ (0g‘𝑇) ∈ 𝐵))
50	1, 49	syl 17	. . . . . . . 8 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → ((0g‘𝑇) ∈ 𝐵 ∧ (0g‘𝑇) ∈ 𝐵))
51	50	ad2antlr 723	. . . . . . 7 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ((0g‘𝑇) ∈ 𝐵 ∧ (0g‘𝑇) ∈ 𝐵))
52		fvoveq1 7278	. . . . . . . . 9 ⊢ (𝑎 = (0g‘𝑇) → (𝐻‘(𝑎(.r‘𝑇)𝑐)) = (𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)))
53		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑎 = (0g‘𝑇) → (𝐻‘𝑎) = (𝐻‘(0g‘𝑇)))
54	53	oveq1d 7270	. . . . . . . . 9 ⊢ (𝑎 = (0g‘𝑇) → ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐)))
55	52, 54	eqeq12d 2754	. . . . . . . 8 ⊢ (𝑎 = (0g‘𝑇) → ((𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)) ↔ (𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐))))
56		oveq2 7263	. . . . . . . . . 10 ⊢ (𝑐 = (0g‘𝑇) → ((0g‘𝑇)(.r‘𝑇)𝑐) = ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)))
57	56	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑐 = (0g‘𝑇) → (𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)) = (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))))
58		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑐 = (0g‘𝑇) → (𝐻‘𝑐) = (𝐻‘(0g‘𝑇)))
59	58	oveq2d 7271	. . . . . . . . 9 ⊢ (𝑐 = (0g‘𝑇) → ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘(0g‘𝑇))))
60	57, 59	eqeq12d 2754	. . . . . . . 8 ⊢ (𝑐 = (0g‘𝑇) → ((𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐)) ↔ (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘(0g‘𝑇)))))
61	55, 60	2ralsng 4609	. . . . . . 7 ⊢ (((0g‘𝑇) ∈ 𝐵 ∧ (0g‘𝑇) ∈ 𝐵) → (∀𝑎 ∈ {(0g‘𝑇)}∀𝑐 ∈ {(0g‘𝑇)} (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)) ↔ (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘(0g‘𝑇)))))
62	51, 61	syl 17	. . . . . 6 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (∀𝑎 ∈ {(0g‘𝑇)}∀𝑐 ∈ {(0g‘𝑇)} (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)) ↔ (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘(0g‘𝑇)))))
63	48, 62	mpbird 256	. . . . 5 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ∀𝑎 ∈ {(0g‘𝑇)}∀𝑐 ∈ {(0g‘𝑇)} (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)))
64		raleq 3333	. . . . . . 7 ⊢ (𝐵 = {(0g‘𝑇)} → (∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)) ↔ ∀𝑐 ∈ {(0g‘𝑇)} (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐))))
65	64	raleqbi1dv 3331	. . . . . 6 ⊢ (𝐵 = {(0g‘𝑇)} → (∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)) ↔ ∀𝑎 ∈ {(0g‘𝑇)}∀𝑐 ∈ {(0g‘𝑇)} (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐))))
66	65	adantl 481	. . . . 5 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)) ↔ ∀𝑎 ∈ {(0g‘𝑇)}∀𝑐 ∈ {(0g‘𝑇)} (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐))))
67	63, 66	mpbird 256	. . . 4 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)))
68	16, 67	mpdan 683	. . 3 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)))
69	20, 68	jca 511	. 2 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝐻 ∈ (𝑇 GrpHom 𝑆) ∧ ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐))))
70	13, 40, 32	isrnghm 45338	. 2 ⊢ (𝐻 ∈ (𝑇 RngHomo 𝑆) ↔ ((𝑇 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐻 ∈ (𝑇 GrpHom 𝑆) ∧ ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(.r‘𝑇)𝑐)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)))))
71	5, 69, 70	sylanbrc 582	1 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHomo 𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∖ cdif 3880  {csn 4558   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  ._rcmulr 16889  0_gc0g 17067  Grpcgrp 18492   GrpHom cghm 18746  Abelcabl 19302  Ringcrg 19698  NzRingcnzr 20441  Rngcrng 45320   RngHomo crngh 45331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-plusg 16901  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-grp 18495  df-minusg 18496  df-ghm 18747  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-nzr 20442  df-mgmhm 45221  df-rng0 45321  df-rnghomo 45333

This theorem is referenced by:  zrinitorngc  45446