Theorem zrrnghm 20471

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 4082	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring)
2		ringrng 20222	. . . . 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 20092	. . . . . 6 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel)
7		ablgrp 19716	. . . . . 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 20175	. . . . . 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 2735	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	13, 14	0ringbas 20463	. . . . 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 20402	. . . 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 2736	. . . . . . . 8 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ 𝑥 = (0g‘𝑇)) → 0 = 0 )
23	13, 14	ring0cl 20204	. . . . . . . . . 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 6847	. . . . . . . . 9 ⊢ 0 ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → 0 ∈ V)
28	21, 22, 25, 27	fvmptd 6948	. . . . . . 7 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (𝐻‘(0g‘𝑇)) = 0 )
29		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑆) = (Base‘𝑆)
30	29, 17	grpidcl 18897	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Grp → 0 ∈ (Base‘𝑆))
31	8, 30	syl 17	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Rng → 0 ∈ (Base‘𝑆))
32		eqid 2735	. . . . . . . . . . . . 13 ⊢ (.r‘𝑆) = (.r‘𝑆)
33	29, 32, 17	rnglz 20102	. . . . . . . . . . . 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 7376	. . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑇) = (.r‘𝑇)
41	13, 40, 14	ringlz 20230	. . . . . . . . . . . . 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 6837	. . . . . . . . 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 2770	. . . . . . . 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 2781	. . . . . . 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 7381	. . . . . . . . 9 ⊢ (𝑎 = (0g‘𝑇) → (𝐻‘(𝑎(.r‘𝑇)𝑐)) = (𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)))
53		fveq2 6833	. . . . . . . . . 10 ⊢ (𝑎 = (0g‘𝑇) → (𝐻‘𝑎) = (𝐻‘(0g‘𝑇)))
54	53	oveq1d 7373	. . . . . . . . 9 ⊢ (𝑎 = (0g‘𝑇) → ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐)))
55	52, 54	eqeq12d 2751	. . . . . . . 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 7366	. . . . . . . . . 10 ⊢ (𝑐 = (0g‘𝑇) → ((0g‘𝑇)(.r‘𝑇)𝑐) = ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)))
57	56	fveq2d 6837	. . . . . . . . 9 ⊢ (𝑐 = (0g‘𝑇) → (𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)) = (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))))
58		fveq2 6833	. . . . . . . . . 10 ⊢ (𝑐 = (0g‘𝑇) → (𝐻‘𝑐) = (𝐻‘(0g‘𝑇)))
59	58	oveq2d 7374	. . . . . . . . 9 ⊢ (𝑐 = (0g‘𝑇) → ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘(0g‘𝑇))))
60	57, 59	eqeq12d 2751	. . . . . . . 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 4634	. . . . . . 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 3292	. . . . . . 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 3307	. . . . . 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 20379	. 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 3050  Vcvv 3439   ∖ cdif 3897  {csn 4579   ↦ cmpt 5178  ‘cfv 6491  (class class class)co 7358  Basecbs 17138  ._rcmulr 17180  0_gc0g 17361  Grpcgrp 18865   GrpHom cghm 19143  Abelcabl 19712  Rngcrng 20089  Ringcrg 20170   RngHom crnghm 20372  NzRingcnzr 20447

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 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-map 8767  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-n0 12404  df-xnn0 12477  df-z 12491  df-uz 12754  df-fz 13426  df-hash 14256  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-plusg 17192  df-0g 17363  df-mgm 18567  df-mgmhm 18619  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-grp 18868  df-minusg 18869  df-ghm 19144  df-cmn 19713  df-abl 19714  df-mgp 20078  df-rng 20090  df-ur 20119  df-ring 20172  df-rnghm 20374  df-nzr 20448

This theorem is referenced by:  zrinitorngc  20577