Theorem zrrnghm 20486

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

Hypotheses

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

Assertion

Ref	Expression
zrrnghm	⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHom 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 4085	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring)
2		ringrng 20237	. . . . 5 ⊢ (𝑇 ∈ Ring → 𝑇 ∈ Rng)
3	1, 2	syl 17	. . . 4 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Rng)
4	3	anim1i 616	. . 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 20107	. . . . . 6 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel)
7		ablgrp 19731	. . . . . 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 20190	. . . . . 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		zrrnghm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑇)
14		eqid 2737	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	13, 14	0ringbas 20478	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝐵 = {(0g‘𝑇)})
16	15	adantl 481	. . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐵 = {(0g‘𝑇)})
17		zrrnghm.0	. . . . 5 ⊢ 0 = (0g‘𝑆)
18		zrrnghm.h	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )
19	13, 17, 18, 14	c0snghm 20417	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {(0g‘𝑇)}) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
20	9, 12, 16, 19	syl3anc 1374	. . 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 2738	. . . . . . . 8 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ 𝑥 = (0g‘𝑇)) → 0 = 0 )
23	13, 14	ring0cl 20219	. . . . . . . . . 10 ⊢ (𝑇 ∈ Ring → (0g‘𝑇) ∈ 𝐵)
24	1, 23	syl 17	. . . . . . . . 9 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → (0g‘𝑇) ∈ 𝐵)
25	24	ad2antlr 728	. . . . . . . 8 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (0g‘𝑇) ∈ 𝐵)
26	17	fvexi 6858	. . . . . . . . 9 ⊢ 0 ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → 0 ∈ V)
28	21, 22, 25, 27	fvmptd 6959	. . . . . . 7 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (𝐻‘(0g‘𝑇)) = 0 )
29		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑆) = (Base‘𝑆)
30	29, 17	grpidcl 18912	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Grp → 0 ∈ (Base‘𝑆))
31	8, 30	syl 17	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Rng → 0 ∈ (Base‘𝑆))
32		eqid 2737	. . . . . . . . . . . . 13 ⊢ (.r‘𝑆) = (.r‘𝑆)
33	29, 32, 17	rnglz 20117	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Rng ∧ 0 ∈ (Base‘𝑆)) → ( 0 (.r‘𝑆) 0 ) = 0 )
34	31, 33	mpdan 688	. . . . . . . . . . 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 7388	. . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑇) = (.r‘𝑇)
41	13, 40, 14	ringlz 20245	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ Ring ∧ (0g‘𝑇) ∈ 𝐵) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
42	1, 23, 41	syl2anc2 586	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
43	42	ad2antlr 728	. . . . . . . . . . 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 6848	. . . . . . . . 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 2772	. . . . . . . 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 2783	. . . . . . 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 688	. . . . . 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 728	. . . . . . 7 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ((0g‘𝑇) ∈ 𝐵 ∧ (0g‘𝑇) ∈ 𝐵))
52		fvoveq1 7393	. . . . . . . . 9 ⊢ (𝑎 = (0g‘𝑇) → (𝐻‘(𝑎(.r‘𝑇)𝑐)) = (𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)))
53		fveq2 6844	. . . . . . . . . 10 ⊢ (𝑎 = (0g‘𝑇) → (𝐻‘𝑎) = (𝐻‘(0g‘𝑇)))
54	53	oveq1d 7385	. . . . . . . . 9 ⊢ (𝑎 = (0g‘𝑇) → ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐)))
55	52, 54	eqeq12d 2753	. . . . . . . 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 7378	. . . . . . . . . 10 ⊢ (𝑐 = (0g‘𝑇) → ((0g‘𝑇)(.r‘𝑇)𝑐) = ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)))
57	56	fveq2d 6848	. . . . . . . . 9 ⊢ (𝑐 = (0g‘𝑇) → (𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)) = (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))))
58		fveq2 6844	. . . . . . . . . 10 ⊢ (𝑐 = (0g‘𝑇) → (𝐻‘𝑐) = (𝐻‘(0g‘𝑇)))
59	58	oveq2d 7386	. . . . . . . . 9 ⊢ (𝑐 = (0g‘𝑇) → ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘(0g‘𝑇))))
60	57, 59	eqeq12d 2753	. . . . . . . 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 4637	. . . . . . 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 257	. . . . 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 3295	. . . . . . 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 3310	. . . . . 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 257	. . . 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 688	. . 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 20394	. 2 ⊢ (𝐻 ∈ (𝑇 RngHom 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 584	1 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHom 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ∖ cdif 3900  {csn 4582   ↦ cmpt 5181  ‘cfv 6502  (class class class)co 7370  Basecbs 17150  ._rcmulr 17192  0_gc0g 17373  Grpcgrp 18880   GrpHom cghm 19158  Abelcabl 19727  Rngcrng 20104  Ringcrg 20185   RngHom crnghm 20387  NzRingcnzr 20462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-oadd 8413  df-er 8647  df-map 8779  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-dju 9827  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-n0 12416  df-xnn0 12489  df-z 12503  df-uz 12766  df-fz 13438  df-hash 14268  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-plusg 17204  df-0g 17375  df-mgm 18579  df-mgmhm 18631  df-sgrp 18658  df-mnd 18674  df-mhm 18722  df-grp 18883  df-minusg 18884  df-ghm 19159  df-cmn 19728  df-abl 19729  df-mgp 20093  df-rng 20105  df-ur 20134  df-ring 20187  df-rnghm 20389  df-nzr 20463

This theorem is referenced by:  zrinitorngc  20592