c0snmgmhm - Mathbox for Alexander van der Vekens

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Theorem c0snmgmhm 41914

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 17300	. . . . 5 ⊢ (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
2	1	anim1i 592	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3	2	3adant3 1081	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
4	3	ancomd 467	. 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → (𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm))
5		zrrhm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑇)
6		fvex 6201	. . . . . 6 ⊢ (Base‘𝑇) ∈ V
7	5, 6	eqeltri 2697	. . . . 5 ⊢ 𝐵 ∈ V
8		hash1snb 13207	. . . . 5 ⊢ (𝐵 ∈ V → ((#‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏}))
9	7, 8	ax-mp 5	. . . 4 ⊢ ((#‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏})
10		eqid 2622	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
11		zrrhm.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑆)
12	10, 11	mndidcl 17308	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆))
13	12	adantr 481	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 0 ∈ (Base‘𝑆))
14	13	adantr 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 0 ∈ (Base‘𝑆))
15	14	adantr 481	. . . . . . . 8 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 ∈ 𝐵) → 0 ∈ (Base‘𝑆))
16		zrrhm.h	. . . . . . . 8 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )
17	15, 16	fmptd 6385	. . . . . . 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 2623	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 = 𝑏) → 0 = 0 )
20		vsnid 4209	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ {𝑏}
21	20	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐵 = {𝑏} → 𝑏 ∈ {𝑏})
22		eleq2 2690	. . . . . . . . . . . 12 ⊢ (𝐵 = {𝑏} → (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ {𝑏}))
23	21, 22	mpbird 247	. . . . . . . . . . 11 ⊢ (𝐵 = {𝑏} → 𝑏 ∈ 𝐵)
24	23	adantl 482	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑏 ∈ 𝐵)
25	18, 19, 24, 14	fvmptd 6288	. . . . . . . . 9 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘𝑏) = 0 )
26		simpr 477	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → (𝐻‘𝑏) = 0 )
27	26, 26	oveq12d 6668	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)) = ( 0 (+_g‘𝑆) 0 ))
28		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
29	10, 28, 11	mndlid 17311	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Mnd ∧ 0 ∈ (Base‘𝑆)) → ( 0 (+_g‘𝑆) 0 ) = 0 )
30	12, 29	mpdan 702	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Mnd → ( 0 (+_g‘𝑆) 0 ) = 0 )
31	30	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ( 0 (+_g‘𝑆) 0 ) = 0 )
32	31	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ( 0 (+_g‘𝑆) 0 ) = 0 )
33	32	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → ( 0 (+_g‘𝑆) 0 ) = 0 )
34		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 𝑇 ∈ Mgm)
35	34	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑇 ∈ Mgm)
36	35	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → 𝑇 ∈ Mgm)
37		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
38		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝑇) = (+_g‘𝑇)
39	5, 38	mgmcl 17245	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ Mgm ∧ 𝑏 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) ∈ 𝐵)
40	36, 37, 37, 39	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) ∈ 𝐵)
41		eleq2 2690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = {𝑏} → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 ↔ (𝑏(+_g‘𝑇)𝑏) ∈ {𝑏}))
42		elsni 4194	. . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 → (𝑏(+_g‘𝑇)𝑏) = 𝑏))
45	44	adantr 481	. . . . . . . . . . . . . . 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 702	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝑏(+_g‘𝑇)𝑏) = 𝑏)
48	47	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = (𝐻‘𝑏))
49	48	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → (𝐻‘(𝑏(+_g‘𝑇)𝑏)) = (𝐻‘𝑏))
50	49, 26	eqtr2d 2657	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → 0 = (𝐻‘(𝑏(+_g‘𝑇)𝑏)))
51	27, 33, 50	3eqtrrd 2661	. . . . . . . . 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 702	. . . . . . . 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 3144	. . . . . . . . . . 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 3152	. . . . . . . . . 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 482	. . . . . . . . 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 3203	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
58		oveq1 6657	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (𝑎(+_g‘𝑇)𝑐) = (𝑏(+_g‘𝑇)𝑐))
59	58	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = (𝐻‘(𝑏(+_g‘𝑇)𝑐)))
60		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (𝐻‘𝑎) = (𝐻‘𝑏))
61	60	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐)))
62	59, 61	eqeq12d 2637	. . . . . . . . . . 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))))
63		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑏(+_g‘𝑇)𝑐) = (𝑏(+_g‘𝑇)𝑏))
64	63	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → (𝐻‘(𝑏(+_g‘𝑇)𝑐)) = (𝐻‘(𝑏(+_g‘𝑇)𝑏)))
65		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝐻‘𝑐) = (𝐻‘𝑏))
66	65	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)))
67	64, 66	eqeq12d 2637	. . . . . . . . . . 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))))
68	62, 67	2ralsng 4220	. . . . . . . . . 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))))
69	57, 57, 68	mp2an 708	. . . . . . . . 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)))
70	56, 69	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))))
71	52, 70	mpbird 247	. . . . . . 7 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ∀𝑎 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)))
72	17, 71	jca 554	. . . . . 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))))
73	72	ex 450	. . . . 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)))))
74	73	exlimdv 1861	. . . 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)))))
75	9, 74	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)))))
76	75	3impia 1261	. 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))))
77	5, 10, 38, 28	ismgmhm 41783	. 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)))))
78	4, 76, 77	sylanbrc 698	1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 {csn 4177 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1c1 9937 #chash 13117 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 Mgmcmgm 17240 Mndcmnd 17294 MgmHom cmgmhm 41777

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mgmhm 41779

This theorem is referenced by: c0snmhm 41915

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