Theorem zrrnghm 20512

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 4064	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring)
2		ringrng 20261	. . . . 5 ⊢ (𝑇 ∈ Ring → 𝑇 ∈ Rng)
3	1, 2	syl 17	. . . 4 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Rng)
4	3	anim1i 622	. . 3 ⊢ ((𝑇 ∈ (Ring ∖ NzRing) ∧ 𝑆 ∈ Rng) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
5	4	ancoms 460	. 2 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
6		rngabl 20131	. . . . . 6 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel)
7		ablgrp 19755	. . . . . 6 ⊢ (𝑆 ∈ Abel → 𝑆 ∈ Grp)
8	6, 7	syl 17	. . . . 5 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Grp)
9	8	adantr 482	. . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑆 ∈ Grp)
10		ringgrp 20214	. . . . . 6 ⊢ (𝑇 ∈ Ring → 𝑇 ∈ Grp)
11	1, 10	syl 17	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Grp)
12	11	adantl 483	. . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑇 ∈ Grp)
13		zrrnghm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑇)
14		eqid 2741	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	13, 14	0ringbas 20504	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝐵 = {(0g‘𝑇)})
16	15	adantl 483	. . . 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 20439	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {(0g‘𝑇)}) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
20	9, 12, 16, 19	syl3anc 1380	. . 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 2742	. . . . . . . 8 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ 𝑥 = (0g‘𝑇)) → 0 = 0 )
23	13, 14	ring0cl 20243	. . . . . . . . . 10 ⊢ (𝑇 ∈ Ring → (0g‘𝑇) ∈ 𝐵)
24	1, 23	syl 17	. . . . . . . . 9 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → (0g‘𝑇) ∈ 𝐵)
25	24	ad2antlr 734	. . . . . . . 8 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (0g‘𝑇) ∈ 𝐵)
26	17	fvexi 6845	. . . . . . . . 9 ⊢ 0 ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → 0 ∈ V)
28	21, 22, 25, 27	fvmptd 6947	. . . . . . 7 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (𝐻‘(0g‘𝑇)) = 0 )
29		eqid 2741	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑆) = (Base‘𝑆)
30	29, 17	grpidcl 18936	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Grp → 0 ∈ (Base‘𝑆))
31	8, 30	syl 17	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Rng → 0 ∈ (Base‘𝑆))
32		eqid 2741	. . . . . . . . . . . . 13 ⊢ (.r‘𝑆) = (.r‘𝑆)
33	29, 32, 17	rnglz 20141	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Rng ∧ 0 ∈ (Base‘𝑆)) → ( 0 (.r‘𝑆) 0 ) = 0 )
34	31, 33	mpdan 694	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Rng → ( 0 (.r‘𝑆) 0 ) = 0 )
35	34	adantr 482	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ( 0 (.r‘𝑆) 0 ) = 0 )
36	35	adantr 482	. . . . . . . . 9 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ( 0 (.r‘𝑆) 0 ) = 0 )
37	36	adantr 482	. . . . . . . 8 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → ( 0 (.r‘𝑆) 0 ) = 0 )
38		simpr 486	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → (𝐻‘(0g‘𝑇)) = 0 )
39	38, 38	oveq12d 7378	. . . . . . . 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 2741	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑇) = (.r‘𝑇)
41	13, 40, 14	ringlz 20269	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ Ring ∧ (0g‘𝑇) ∈ 𝐵) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
42	1, 23, 41	syl2anc2 592	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
43	42	ad2antlr 734	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
44	43	adantr 482	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
45	44	fveq2d 6835	. . . . . . . . 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 2776	. . . . . . . 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 2787	. . . . . . 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 694	. . . . . 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 517	. . . . . . . . 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 734	. . . . . . 7 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ((0g‘𝑇) ∈ 𝐵 ∧ (0g‘𝑇) ∈ 𝐵))
52		fvoveq1 7383	. . . . . . . . 9 ⊢ (𝑎 = (0g‘𝑇) → (𝐻‘(𝑎(.r‘𝑇)𝑐)) = (𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)))
53		fveq2 6831	. . . . . . . . . 10 ⊢ (𝑎 = (0g‘𝑇) → (𝐻‘𝑎) = (𝐻‘(0g‘𝑇)))
54	53	oveq1d 7375	. . . . . . . . 9 ⊢ (𝑎 = (0g‘𝑇) → ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐)))
55	52, 54	eqeq12d 2757	. . . . . . . 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 7368	. . . . . . . . . 10 ⊢ (𝑐 = (0g‘𝑇) → ((0g‘𝑇)(.r‘𝑇)𝑐) = ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)))
57	56	fveq2d 6835	. . . . . . . . 9 ⊢ (𝑐 = (0g‘𝑇) → (𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)) = (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))))
58		fveq2 6831	. . . . . . . . . 10 ⊢ (𝑐 = (0g‘𝑇) → (𝐻‘𝑐) = (𝐻‘(0g‘𝑇)))
59	58	oveq2d 7376	. . . . . . . . 9 ⊢ (𝑐 = (0g‘𝑇) → ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘(0g‘𝑇))))
60	57, 59	eqeq12d 2757	. . . . . . . 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 4613	. . . . . . 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 259	. . . . 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 3296	. . . . . . 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 3309	. . . . . 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 483	. . . . 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 259	. . . 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 694	. . 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 517	. 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 20416	. 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 590	1 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHom 𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  Vcvv 3433   ∖ cdif 3882  {csn 4558   ↦ cmpt 5156  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  ._rcmulr 17216  0_gc0g 17397  Grpcgrp 18904   GrpHom cghm 19182  Abelcabl 19751  Rngcrng 20128  Ringcrg 20209   RngHom crnghm 20409  NzRingcnzr 20488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9820  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-n0 12433  df-xnn0 12506  df-z 12520  df-uz 12784  df-fz 13457  df-hash 14288  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-plusg 17228  df-0g 17399  df-mgm 18603  df-mgmhm 18655  df-sgrp 18682  df-mnd 18698  df-mhm 18746  df-grp 18907  df-minusg 18908  df-ghm 19183  df-cmn 19752  df-abl 19753  df-mgp 20117  df-rng 20129  df-ur 20158  df-ring 20211  df-rnghm 20411  df-nzr 20489

This theorem is referenced by:  zrinitorngc  20618