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Theorem c0snmgmhm 20498

Description: The constant mapping to zero is a magma homomorphism from a magma with one element to any monoid. (Contributed by AV, 17-Apr-2020.)

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

**Assertion**
Ref	Expression
c0snmgmhm	⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))

Distinct variable groups: 𝑥,𝐵 𝑥,𝑆 𝑥,𝑇 𝑥, 0 Allowed substitution hint: 𝐻(𝑥)

Proof of Theorem c0snmgmhm Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mndmgm 18766	. . . . 5 ⊢ (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
2	1	anim1i 624	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3	2	3adant3 1144	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
4	3	ancomd 465	. 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm))
5		zrrhm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑇)
6	5	fvexi 6876	. . . . 5 ⊢ 𝐵 ∈ V
7		hash1snb 14426	. . . . 5 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏}))
8	6, 7	ax-mp 5	. . . 4 ⊢ ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏})
9		eqid 2761	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		zrrhm.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑆)
11	9, 10	mndidcl 18774	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆))
12	11	adantr 484	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 0 ∈ (Base‘𝑆))
13	12	adantr 484	. . . . . . . . 9 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 0 ∈ (Base‘𝑆))
14	13	adantr 484	. . . . . . . 8 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 ∈ 𝐵) → 0 ∈ (Base‘𝑆))
15		zrrhm.h	. . . . . . . 8 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )
16	14, 15	fmptd 7090	. . . . . . 7 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻:𝐵⟶(Base‘𝑆))
17	15	a1i 11	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ))
18		eqidd 2762	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 = 𝑏) → 0 = 0 )
19		vsnid 4619	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ {𝑏}
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐵 = {𝑏} → 𝑏 ∈ {𝑏})
21		eleq2 2850	. . . . . . . . . . . 12 ⊢ (𝐵 = {𝑏} → (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ {𝑏}))
22	20, 21	mpbird 259	. . . . . . . . . . 11 ⊢ (𝐵 = {𝑏} → 𝑏 ∈ 𝐵)
23	22	adantl 485	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑏 ∈ 𝐵)
24	17, 18, 23, 13	fvmptd 6978	. . . . . . . . 9 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘𝑏) = 0 )
25		simpr 488	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → (𝐻‘𝑏) = 0 )
26	25, 25	oveq12d 7409	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)) = ( 0 (+_g‘𝑆) 0 ))
27		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
28	9, 27, 10	mndlid 18779	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Mnd ∧ 0 ∈ (Base‘𝑆)) → ( 0 (+_g‘𝑆) 0 ) = 0 )
29	11, 28	mpdan 697	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Mnd → ( 0 (+_g‘𝑆) 0 ) = 0 )
30	29	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ( 0 (+_g‘𝑆) 0 ) = 0 )
31	30	adantr 484	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ( 0 (+_g‘𝑆) 0 ) = 0 )
32	31	adantr 484	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → ( 0 (+_g‘𝑆) 0 ) = 0 )
33		simpr 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 𝑇 ∈ Mgm)
34	33	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑇 ∈ Mgm)
35	34	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → 𝑇 ∈ Mgm)
36		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
37		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝑇) = (+_g‘𝑇)
38	5, 37	mgmcl 18668	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ Mgm ∧ 𝑏 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) ∈ 𝐵)
39	35, 36, 36, 38	syl3anc 1389	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) ∈ 𝐵)
40		eleq2 2850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = {𝑏} → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 ↔ (𝑏(+_g‘𝑇)𝑏) ∈ {𝑏}))
41		elsni 4596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏(+_g‘𝑇)𝑏) ∈ {𝑏} → (𝑏(+_g‘𝑇)𝑏) = 𝑏)
42	40, 41	biimtrdi 255	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = {𝑏} → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 → (𝑏(+_g‘𝑇)𝑏) = 𝑏))
43	42	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 → (𝑏(+_g‘𝑇)𝑏) = 𝑏))
44	43	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 → (𝑏(+_g‘𝑇)𝑏) = 𝑏))
45	39, 44	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) = 𝑏)
46	23, 45	mpdan 697	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝑏(+_g‘𝑇)𝑏) = 𝑏)
47	46	fveq2d 6866	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = (𝐻‘𝑏))
48	47	adantr 484	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = (𝐻‘𝑏))
49	48, 25	eqtr2d 2797	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → 0 = (𝐻‘(𝑏(+_g‘𝑇)𝑏)))
50	26, 32, 49	3eqtrrd 2801	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)))
51	24, 50	mpdan 697	. . . . . . . 8 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)))
52		id 22	. . . . . . . . . . 11 ⊢ (𝐵 = {𝑏} → 𝐵 = {𝑏})
53	52	raleqdv 3319	. . . . . . . . . . 11 ⊢ (𝐵 = {𝑏} → (∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) ↔ ∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐))))
54	52, 53	raleqbidv 3335	. . . . . . . . . 10 ⊢ (𝐵 = {𝑏} → (∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) ↔ ∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐))))
55	54	adantl 485	. . . . . . . . 9 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) ↔ ∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐))))
56		fvoveq1 7414	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = (𝐻‘(𝑏(+_g‘𝑇)𝑐)))
57		fveq2 6862	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (𝐻‘𝑎) = (𝐻‘𝑏))
58	57	oveq1d 7406	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐)))
59	56, 58	eqeq12d 2777	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ((𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) ↔ (𝐻‘(𝑏(+_g‘𝑇)𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐))))
60		oveq2 7399	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑏(+_g‘𝑇)𝑐) = (𝑏(+_g‘𝑇)𝑏))
61	60	fveq2d 6866	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → (𝐻‘(𝑏(+_g‘𝑇)𝑐)) = (𝐻‘(𝑏(+_g‘𝑇)𝑏)))
62		fveq2 6862	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝐻‘𝑐) = (𝐻‘𝑏))
63	62	oveq2d 7407	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)))
64	61, 63	eqeq12d 2777	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → ((𝐻‘(𝑏(+_g‘𝑇)𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐)) ↔ (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏))))
65	59, 64	2ralsng 4634	. . . . . . . . . 10 ⊢ ((𝑏 ∈ V ∧ 𝑏 ∈ V) → (∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) ↔ (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏))))
66	65	el2v 3460	. . . . . . . . 9 ⊢ (∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) ↔ (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)))
67	55, 66	bitrdi 289	. . . . . . . 8 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) ↔ (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏))))
68	51, 67	mpbird 259	. . . . . . 7 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)))
69	16, 68	jca 519	. . . . . 6 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐))))
70	69	ex 416	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝐵 = {𝑏} → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)))))
71	70	exlimdv 1952	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (∃𝑏 𝐵 = {𝑏} → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)))))
72	8, 71	biimtrid 244	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ((♯‘𝐵) = 1 → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)))))
73	72	3impia 1129	. 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐))))
74	5, 9, 37, 27	ismgmhm 18721	. 2 ⊢ (𝐻 ∈ (𝑇 MgmHom 𝑆) ↔ ((𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm) ∧ (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)))))
75	4, 73, 74	sylanbrc 592	1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∃wex 1798 ∈ wcel 2141 ∀wral 3075 Vcvv 3453 {csn 4579 ↦ cmpt 5178 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 1c1 11068 ♯chash 14337 Basecbs 17236 +_gcplusg 17277 0_gc0g 17459 Mgmcmgm 18663 MgmHom cmgmhm 18715 Mndcmnd 18759

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 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 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-oadd 8435 df-er 8672 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-dju 9853 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-n0 12476 df-z 12563 df-uz 12834 df-fz 13507 df-hash 14338 df-0g 17461 df-mgm 18665 df-mgmhm 18717 df-sgrp 18744 df-mnd 18760

This theorem is referenced by: c0snmhm 20499

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