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Theorem c0snmgmhm 20381

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 18649	. . . . 5 ⊢ (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
2	1	anim1i 615	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3	2	3adant3 1132	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
4	3	ancomd 461	. 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm))
5		zrrhm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑇)
6	5	fvexi 6836	. . . . 5 ⊢ 𝐵 ∈ V
7		hash1snb 14326	. . . . 5 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏}))
8	6, 7	ax-mp 5	. . . 4 ⊢ ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏})
9		eqid 2731	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		zrrhm.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑆)
11	9, 10	mndidcl 18657	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆))
12	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 0 ∈ (Base‘𝑆))
13	12	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 0 ∈ (Base‘𝑆))
14	13	adantr 480	. . . . . . . 8 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 ∈ 𝐵) → 0 ∈ (Base‘𝑆))
15		zrrhm.h	. . . . . . . 8 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )
16	14, 15	fmptd 7047	. . . . . . 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 2732	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 = 𝑏) → 0 = 0 )
19		vsnid 4616	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ {𝑏}
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐵 = {𝑏} → 𝑏 ∈ {𝑏})
21		eleq2 2820	. . . . . . . . . . . 12 ⊢ (𝐵 = {𝑏} → (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ {𝑏}))
22	20, 21	mpbird 257	. . . . . . . . . . 11 ⊢ (𝐵 = {𝑏} → 𝑏 ∈ 𝐵)
23	22	adantl 481	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑏 ∈ 𝐵)
24	17, 18, 23, 13	fvmptd 6936	. . . . . . . . 9 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘𝑏) = 0 )
25		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → (𝐻‘𝑏) = 0 )
26	25, 25	oveq12d 7364	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)) = ( 0 (+_g‘𝑆) 0 ))
27		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
28	9, 27, 10	mndlid 18662	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Mnd ∧ 0 ∈ (Base‘𝑆)) → ( 0 (+_g‘𝑆) 0 ) = 0 )
29	11, 28	mpdan 687	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Mnd → ( 0 (+_g‘𝑆) 0 ) = 0 )
30	29	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ( 0 (+_g‘𝑆) 0 ) = 0 )
31	30	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ( 0 (+_g‘𝑆) 0 ) = 0 )
32	31	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → ( 0 (+_g‘𝑆) 0 ) = 0 )
33		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 𝑇 ∈ Mgm)
34	33	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑇 ∈ Mgm)
35	34	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → 𝑇 ∈ Mgm)
36		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
37		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝑇) = (+_g‘𝑇)
38	5, 37	mgmcl 18551	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ Mgm ∧ 𝑏 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) ∈ 𝐵)
39	35, 36, 36, 38	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) ∈ 𝐵)
40		eleq2 2820	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = {𝑏} → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 ↔ (𝑏(+_g‘𝑇)𝑏) ∈ {𝑏}))
41		elsni 4593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏(+_g‘𝑇)𝑏) ∈ {𝑏} → (𝑏(+_g‘𝑇)𝑏) = 𝑏)
42	40, 41	biimtrdi 253	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = {𝑏} → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 → (𝑏(+_g‘𝑇)𝑏) = 𝑏))
43	42	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 → (𝑏(+_g‘𝑇)𝑏) = 𝑏))
44	43	adantr 480	. . . . . . . . . . . . . . 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 687	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝑏(+_g‘𝑇)𝑏) = 𝑏)
47	46	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = (𝐻‘𝑏))
48	47	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = (𝐻‘𝑏))
49	48, 25	eqtr2d 2767	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → 0 = (𝐻‘(𝑏(+_g‘𝑇)𝑏)))
50	26, 32, 49	3eqtrrd 2771	. . . . . . . . 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 687	. . . . . . . 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 3292	. . . . . . . . . . 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 3312	. . . . . . . . . 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 481	. . . . . . . . 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 7369	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = (𝐻‘(𝑏(+_g‘𝑇)𝑐)))
57		fveq2 6822	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (𝐻‘𝑎) = (𝐻‘𝑏))
58	57	oveq1d 7361	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐)))
59	56, 58	eqeq12d 2747	. . . . . . . . . . 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 7354	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑏(+_g‘𝑇)𝑐) = (𝑏(+_g‘𝑇)𝑏))
61	60	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → (𝐻‘(𝑏(+_g‘𝑇)𝑐)) = (𝐻‘(𝑏(+_g‘𝑇)𝑏)))
62		fveq2 6822	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝐻‘𝑐) = (𝐻‘𝑏))
63	62	oveq2d 7362	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)))
64	61, 63	eqeq12d 2747	. . . . . . . . . . 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 4631	. . . . . . . . . 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 3443	. . . . . . . . 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 287	. . . . . . . 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 257	. . . . . . 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 511	. . . . . 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 412	. . . . 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 1934	. . . 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 242	. . 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 1117	. 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 18604	. 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 583	1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∃wex 1780 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 {csn 4576 ↦ cmpt 5172 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 1c1 11007 ♯chash 14237 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 Mgmcmgm 18546 MgmHom cmgmhm 18598 Mndcmnd 18642

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-oadd 8389 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-hash 14238 df-0g 17345 df-mgm 18548 df-mgmhm 18600 df-sgrp 18627 df-mnd 18643

This theorem is referenced by: c0snmhm 20382

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