c0snmgmhm - Mathbox for Alexander van der Vekens

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Theorem c0snmgmhm 42442

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 17508	. . . . 5 ⊢ (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
2	1	anim1i 602	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3	2	3adant3 1126	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
4	3	ancomd 453	. 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm))
5		zrrhm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑇)
6		fvex 6342	. . . . . 6 ⊢ (Base‘𝑇) ∈ V
7	5, 6	eqeltri 2846	. . . . 5 ⊢ 𝐵 ∈ V
8		hash1snb 13409	. . . . 5 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏}))
9	7, 8	ax-mp 5	. . . 4 ⊢ ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏})
10		eqid 2771	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
11		zrrhm.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑆)
12	10, 11	mndidcl 17516	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆))
13	12	adantr 466	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 0 ∈ (Base‘𝑆))
14	13	adantr 466	. . . . . . . . 9 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 0 ∈ (Base‘𝑆))
15	14	adantr 466	. . . . . . . 8 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 ∈ 𝐵) → 0 ∈ (Base‘𝑆))
16		zrrhm.h	. . . . . . . 8 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )
17	15, 16	fmptd 6527	. . . . . . 7 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻:𝐵⟶(Base‘𝑆))
18	16	a1i 11	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ))
19		eqidd 2772	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 = 𝑏) → 0 = 0 )
20		vsnid 4348	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ {𝑏}
21	20	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐵 = {𝑏} → 𝑏 ∈ {𝑏})
22		eleq2 2839	. . . . . . . . . . . 12 ⊢ (𝐵 = {𝑏} → (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ {𝑏}))
23	21, 22	mpbird 247	. . . . . . . . . . 11 ⊢ (𝐵 = {𝑏} → 𝑏 ∈ 𝐵)
24	23	adantl 467	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑏 ∈ 𝐵)
25	18, 19, 24, 14	fvmptd 6430	. . . . . . . . 9 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘𝑏) = 0 )
26		simpr 471	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → (𝐻‘𝑏) = 0 )
27	26, 26	oveq12d 6811	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)) = ( 0 (+_g‘𝑆) 0 ))
28		eqid 2771	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
29	10, 28, 11	mndlid 17519	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Mnd ∧ 0 ∈ (Base‘𝑆)) → ( 0 (+_g‘𝑆) 0 ) = 0 )
30	12, 29	mpdan 667	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Mnd → ( 0 (+_g‘𝑆) 0 ) = 0 )
31	30	adantr 466	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ( 0 (+_g‘𝑆) 0 ) = 0 )
32	31	adantr 466	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ( 0 (+_g‘𝑆) 0 ) = 0 )
33	32	adantr 466	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → ( 0 (+_g‘𝑆) 0 ) = 0 )
34		simpr 471	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 𝑇 ∈ Mgm)
35	34	adantr 466	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑇 ∈ Mgm)
36	35	adantr 466	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → 𝑇 ∈ Mgm)
37		simpr 471	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
38		eqid 2771	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝑇) = (+_g‘𝑇)
39	5, 38	mgmcl 17453	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ Mgm ∧ 𝑏 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) ∈ 𝐵)
40	36, 37, 37, 39	syl3anc 1476	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) ∈ 𝐵)
41		eleq2 2839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = {𝑏} → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 ↔ (𝑏(+_g‘𝑇)𝑏) ∈ {𝑏}))
42		elsni 4333	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏(+_g‘𝑇)𝑏) ∈ {𝑏} → (𝑏(+_g‘𝑇)𝑏) = 𝑏)
43	41, 42	syl6bi 243	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = {𝑏} → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 → (𝑏(+_g‘𝑇)𝑏) = 𝑏))
44	43	adantl 467	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 → (𝑏(+_g‘𝑇)𝑏) = 𝑏))
45	44	adantr 466	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 → (𝑏(+_g‘𝑇)𝑏) = 𝑏))
46	40, 45	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) = 𝑏)
47	24, 46	mpdan 667	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝑏(+_g‘𝑇)𝑏) = 𝑏)
48	47	fveq2d 6336	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = (𝐻‘𝑏))
49	48	adantr 466	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = (𝐻‘𝑏))
50	49, 26	eqtr2d 2806	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → 0 = (𝐻‘(𝑏(+_g‘𝑇)𝑏)))
51	27, 33, 50	3eqtrrd 2810	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)))
52	25, 51	mpdan 667	. . . . . . . 8 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)))
53		id 22	. . . . . . . . . . 11 ⊢ (𝐵 = {𝑏} → 𝐵 = {𝑏})
54	53	raleqdv 3293	. . . . . . . . . . 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))))
55	53, 54	raleqbidv 3301	. . . . . . . . . 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))))
56	55	adantl 467	. . . . . . . . 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))))
57		vex 3354	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
58		fvoveq1 6816	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = (𝐻‘(𝑏(+_g‘𝑇)𝑐)))
59		fveq2 6332	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (𝐻‘𝑎) = (𝐻‘𝑏))
60	59	oveq1d 6808	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐)))
61	58, 60	eqeq12d 2786	. . . . . . . . . . 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))))
62		oveq2 6801	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑏(+_g‘𝑇)𝑐) = (𝑏(+_g‘𝑇)𝑏))
63	62	fveq2d 6336	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → (𝐻‘(𝑏(+_g‘𝑇)𝑐)) = (𝐻‘(𝑏(+_g‘𝑇)𝑏)))
64		fveq2 6332	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝐻‘𝑐) = (𝐻‘𝑏))
65	64	oveq2d 6809	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)))
66	63, 65	eqeq12d 2786	. . . . . . . . . . 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))))
67	61, 66	2ralsng 4358	. . . . . . . . . 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))))
68	57, 57, 67	mp2an 672	. . . . . . . . 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)))
69	56, 68	syl6bb 276	. . . . . . . 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))))
70	52, 69	mpbird 247	. . . . . . 7 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)))
71	17, 70	jca 501	. . . . . 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))))
72	71	ex 397	. . . . 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)))))
73	72	exlimdv 2013	. . . 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)))))
74	9, 73	syl5bi 232	. . 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)))))
75	74	3impia 1109	. 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))))
76	5, 10, 38, 28	ismgmhm 42311	. 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)))))
77	4, 75, 76	sylanbrc 572	1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∃wex 1852 ∈ wcel 2145 ∀wral 3061 Vcvv 3351 {csn 4316 ↦ cmpt 4863 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 1c1 10139 ♯chash 13321 Basecbs 16064 +_gcplusg 16149 0_gc0g 16308 Mgmcmgm 17448 Mndcmnd 17502 MgmHom cmgmhm 42305

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-card 8965 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-n0 11495 df-z 11580 df-uz 11889 df-fz 12534 df-hash 13322 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-mgmhm 42307

This theorem is referenced by: c0snmhm 42443

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