Theorem zrrnghm 20699

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 4081	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring)
2		ringrng 20427	. . . . 5 ⊢ (𝑇 ∈ Ring → 𝑇 ∈ Rng)
3	1, 2	syl 18	. . . 4 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Rng)
4	3	anim1i 627	. . 3 ⊢ ((𝑇 ∈ (Ring ∖ NzRing) ∧ 𝑆 ∈ Rng) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
5	4	ancoms 464	. 2 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
6		rngabl 20291	. . . . . 6 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel)
7		ablgrp 19913	. . . . . 6 ⊢ (𝑆 ∈ Abel → 𝑆 ∈ Grp)
8	6, 7	syl 18	. . . . 5 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Grp)
9	8	adantr 486	. . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑆 ∈ Grp)
10		ringgrp 20378	. . . . . 6 ⊢ (𝑇 ∈ Ring → 𝑇 ∈ Grp)
11	1, 10	syl 18	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Grp)
12	11	adantl 487	. . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑇 ∈ Grp)
13		zrrnghm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑇)
14		eqid 2762	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	13, 14	0ringbas 20690	. . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝐵 = {(0g‘𝑇)})
16	15	adantl 487	. . . 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 20606	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {(0g‘𝑇)}) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
20	9, 12, 16, 19	syl3anc 1398	. . 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 2763	. . . . . . . 8 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ 𝑥 = (0g‘𝑇)) → 0 = 0 )
23	13, 14	ring0cl 20409	. . . . . . . . . 10 ⊢ (𝑇 ∈ Ring → (0g‘𝑇) ∈ 𝐵)
24	1, 23	syl 18	. . . . . . . . 9 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → (0g‘𝑇) ∈ 𝐵)
25	24	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (0g‘𝑇) ∈ 𝐵)
26	17	fvexi 6896	. . . . . . . . 9 ⊢ 0 ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → 0 ∈ V)
28	21, 22, 25, 27	fvmptd 6998	. . . . . . 7 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → (𝐻‘(0g‘𝑇)) = 0 )
29		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑆) = (Base‘𝑆)
30	29, 17	grpidcl 19090	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Grp → 0 ∈ (Base‘𝑆))
31	8, 30	syl 18	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Rng → 0 ∈ (Base‘𝑆))
32		eqid 2762	. . . . . . . . . . . . 13 ⊢ (.r‘𝑆) = (.r‘𝑆)
33	29, 32, 17	rnglz 20301	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Rng ∧ 0 ∈ (Base‘𝑆)) → ( 0 (.r‘𝑆) 0 ) = 0 )
34	31, 33	mpdan 700	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Rng → ( 0 (.r‘𝑆) 0 ) = 0 )
35	34	adantr 486	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ( 0 (.r‘𝑆) 0 ) = 0 )
36	35	adantr 486	. . . . . . . . 9 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ( 0 (.r‘𝑆) 0 ) = 0 )
37	36	adantr 486	. . . . . . . 8 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → ( 0 (.r‘𝑆) 0 ) = 0 )
38		simpr 490	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → (𝐻‘(0g‘𝑇)) = 0 )
39	38, 38	oveq12d 7434	. . . . . . . 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 2762	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑇) = (.r‘𝑇)
41	13, 40, 14	ringlz 20436	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ Ring ∧ (0g‘𝑇) ∈ 𝐵) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
42	1, 23, 41	syl2anc2 597	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
43	42	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
44	43	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) ∧ (𝐻‘(0g‘𝑇)) = 0 ) → ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)) = (0g‘𝑇))
45	44	fveq2d 6886	. . . . . . . . 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 2797	. . . . . . . 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 2808	. . . . . . 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 700	. . . . . 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 521	. . . . . . . . 9 ⊢ (𝑇 ∈ Ring → ((0g‘𝑇) ∈ 𝐵 ∧ (0g‘𝑇) ∈ 𝐵))
50	1, 49	syl 18	. . . . . . . 8 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → ((0g‘𝑇) ∈ 𝐵 ∧ (0g‘𝑇) ∈ 𝐵))
51	50	ad2antlr 740	. . . . . . 7 ⊢ (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g‘𝑇)}) → ((0g‘𝑇) ∈ 𝐵 ∧ (0g‘𝑇) ∈ 𝐵))
52		fvoveq1 7439	. . . . . . . . 9 ⊢ (𝑎 = (0g‘𝑇) → (𝐻‘(𝑎(.r‘𝑇)𝑐)) = (𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)))
53		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑎 = (0g‘𝑇) → (𝐻‘𝑎) = (𝐻‘(0g‘𝑇)))
54	53	oveq1d 7431	. . . . . . . . 9 ⊢ (𝑎 = (0g‘𝑇) → ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐)))
55	52, 54	eqeq12d 2778	. . . . . . . 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 7424	. . . . . . . . . 10 ⊢ (𝑐 = (0g‘𝑇) → ((0g‘𝑇)(.r‘𝑇)𝑐) = ((0g‘𝑇)(.r‘𝑇)(0g‘𝑇)))
57	56	fveq2d 6886	. . . . . . . . 9 ⊢ (𝑐 = (0g‘𝑇) → (𝐻‘((0g‘𝑇)(.r‘𝑇)𝑐)) = (𝐻‘((0g‘𝑇)(.r‘𝑇)(0g‘𝑇))))
58		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑐 = (0g‘𝑇) → (𝐻‘𝑐) = (𝐻‘(0g‘𝑇)))
59	58	oveq2d 7432	. . . . . . . . 9 ⊢ (𝑐 = (0g‘𝑇) → ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘𝑐)) = ((𝐻‘(0g‘𝑇))(.r‘𝑆)(𝐻‘(0g‘𝑇))))
60	57, 59	eqeq12d 2778	. . . . . . . 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 4642	. . . . . . 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 18	. . . . . 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 260	. . . . 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 3318	. . . . . . 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 487	. . . . 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 260	. . . 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 700	. . 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 521	. 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 20583	. 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 595	1 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHom 𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ∖ cdif 3899  {csn 4587   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  ._rcmulr 17347  0_gc0g 17528  Grpcgrp 19058   GrpHom cghm 19341  Abelcabl 19909  Rngcrng 20288  Ringcrg 20373   RngHom crnghm 20576  NzRingcnzr 20673

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-oadd 8462  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9909  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-n0 12532  df-xnn0 12605  df-z 12619  df-uz 12891  df-fz 13564  df-hash 14397  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-plusg 17359  df-0g 17530  df-mgm 18734  df-mgmhm 18796  df-sgrp 18823  df-mnd 18839  df-mhm 18892  df-grp 19061  df-minusg 19062  df-ghm 19342  df-cmn 19910  df-abl 19911  df-mgp 20275  df-rng 20289  df-ur 20322  df-ring 20375  df-rnghm 20578  df-nzr 20674

This theorem is used by:  zrinitorngc  20805