c0snmgmhm - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > c0snmgmhm		Structured version Visualization version GIF version

Theorem c0snmgmhm 20433

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 18700	. . . . 5 ⊢ (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
2	1	anim1i 616	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3	2	3adant3 1133	. . 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 6848	. . . . 5 ⊢ 𝐵 ∈ V
7		hash1snb 14372	. . . . 5 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏}))
8	6, 7	ax-mp 5	. . . 4 ⊢ ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏})
9		eqid 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		zrrhm.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑆)
11	9, 10	mndidcl 18708	. . . . . . . . . . 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 7060	. . . . . . 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 2738	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 = 𝑏) → 0 = 0 )
19		vsnid 4608	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ {𝑏}
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐵 = {𝑏} → 𝑏 ∈ {𝑏})
21		eleq2 2826	. . . . . . . . . . . 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 6949	. . . . . . . . 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 7378	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)) = ( 0 (+_g‘𝑆) 0 ))
27		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
28	9, 27, 10	mndlid 18713	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Mnd ∧ 0 ∈ (Base‘𝑆)) → ( 0 (+_g‘𝑆) 0 ) = 0 )
29	11, 28	mpdan 688	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝑇) = (+_g‘𝑇)
38	5, 37	mgmcl 18602	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ Mgm ∧ 𝑏 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) ∈ 𝐵)
39	35, 36, 36, 38	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏 ∈ 𝐵) → (𝑏(+_g‘𝑇)𝑏) ∈ 𝐵)
40		eleq2 2826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = {𝑏} → ((𝑏(+_g‘𝑇)𝑏) ∈ 𝐵 ↔ (𝑏(+_g‘𝑇)𝑏) ∈ {𝑏}))
41		elsni 4585	. . . . . . . . . . . . . . . . . 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 688	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝑏(+_g‘𝑇)𝑏) = 𝑏)
47	46	fveq2d 6838	. . . . . . . . . . . 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 2773	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻‘𝑏) = 0 ) → 0 = (𝐻‘(𝑏(+_g‘𝑇)𝑏)))
50	26, 32, 49	3eqtrrd 2777	. . . . . . . . 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 688	. . . . . . . 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 3296	. . . . . . . . . . 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 7383	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (𝐻‘(𝑎(+_g‘𝑇)𝑐)) = (𝐻‘(𝑏(+_g‘𝑇)𝑐)))
57		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (𝐻‘𝑎) = (𝐻‘𝑏))
58	57	oveq1d 7375	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ((𝐻‘𝑎)(+_g‘𝑆)(𝐻‘𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐)))
59	56, 58	eqeq12d 2753	. . . . . . . . . . 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 7368	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑏(+_g‘𝑇)𝑐) = (𝑏(+_g‘𝑇)𝑏))
61	60	fveq2d 6838	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → (𝐻‘(𝑏(+_g‘𝑇)𝑐)) = (𝐻‘(𝑏(+_g‘𝑇)𝑏)))
62		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝐻‘𝑐) = (𝐻‘𝑏))
63	62	oveq2d 7376	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑐)) = ((𝐻‘𝑏)(+_g‘𝑆)(𝐻‘𝑏)))
64	61, 63	eqeq12d 2753	. . . . . . . . . . 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 4623	. . . . . . . . . 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 3437	. . . . . . . . 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 1935	. . . 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 1118	. 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 18655	. 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 584	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 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 {csn 4568 ↦ cmpt 5167 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 1c1 11030 ♯chash 14283 Basecbs 17170 +_gcplusg 17211 0_gc0g 17393 Mgmcmgm 18597 MgmHom cmgmhm 18649 Mndcmnd 18693

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-oadd 8402 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-hash 14284 df-0g 17395 df-mgm 18599 df-mgmhm 18651 df-sgrp 18678 df-mnd 18694

This theorem is referenced by: c0snmhm 20434

W3C validator