frlmsplit2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > frlmsplit2		Structured version Visualization version GIF version

Theorem frlmsplit2 22079

Description: Restriction is homomorphic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)

**Hypotheses**
Ref	Expression
frlmsplit2.y	⊢ 𝑌 = (𝑅 freeLMod 𝑈)
frlmsplit2.z	⊢ 𝑍 = (𝑅 freeLMod 𝑉)
frlmsplit2.b	⊢ 𝐵 = (Base‘𝑌)
frlmsplit2.c	⊢ 𝐶 = (Base‘𝑍)
frlmsplit2.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))

**Assertion**
Ref	Expression
frlmsplit2	⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍))

Distinct variable groups: 𝑥,𝑌 𝑥,𝑅 𝑥,𝑈 𝑥,𝑍 𝑥,𝑉 𝑥,𝐵 𝑥,𝐶 𝑥,𝑋 Allowed substitution hint: 𝐹(𝑥)

**Proof of Theorem frlmsplit2**
Step	Hyp	Ref	Expression
1		simp1 1154	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑅 ∈ Ring)
2		simp2 1155	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑈 ∈ 𝑋)
3		frlmsplit2.y	. . . . . . 7 ⊢ 𝑌 = (𝑅 freeLMod 𝑈)
4		frlmsplit2.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑌)
5		eqid 2761	. . . . . . 7 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)) = (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈))
6	3, 4, 5	frlmlss 22057	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋) → 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)))
7	1, 2, 6	syl2anc 596	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)))
8		eqid 2761	. . . . . 6 ⊢ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) = (Base‘((ringLMod‘𝑅) ↑_s 𝑈))
9	8, 5	lssss 21211	. . . . 5 ⊢ (𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)) → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)))
10		resmpt 6029	. . . . 5 ⊢ (𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) → ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)))
11	7, 9, 10	3syl 19	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)))
12		frlmsplit2.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))
13	11, 12	eqtr4di 2814	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) = 𝐹)
14		rlmlmod 21478	. . . . . 6 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
15		eqid 2761	. . . . . . 7 ⊢ ((ringLMod‘𝑅) ↑_s 𝑈) = ((ringLMod‘𝑅) ↑_s 𝑈)
16		eqid 2761	. . . . . . 7 ⊢ ((ringLMod‘𝑅) ↑_s 𝑉) = ((ringLMod‘𝑅) ↑_s 𝑉)
17		eqid 2761	. . . . . . 7 ⊢ (Base‘((ringLMod‘𝑅) ↑_s 𝑉)) = (Base‘((ringLMod‘𝑅) ↑_s 𝑉))
18		eqid 2761	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) = (𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉))
19	15, 16, 8, 17, 18	pwssplit3 21336	. . . . . 6 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ∈ (((ringLMod‘𝑅) ↑_s 𝑈) LMHom ((ringLMod‘𝑅) ↑_s 𝑉)))
20	14, 19	syl3an1 1181	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ∈ (((ringLMod‘𝑅) ↑_s 𝑈) LMHom ((ringLMod‘𝑅) ↑_s 𝑉)))
21		eqid 2761	. . . . . 6 ⊢ (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) = (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵)
22	5, 21	reslmhm 21327	. . . . 5 ⊢ (((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ∈ (((ringLMod‘𝑅) ↑_s 𝑈) LMHom ((ringLMod‘𝑅) ↑_s 𝑉)) ∧ 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈))) → ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ∈ ((((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) LMHom ((ringLMod‘𝑅) ↑_s 𝑉)))
23	20, 7, 22	syl2anc 596	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ∈ ((((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) LMHom ((ringLMod‘𝑅) ↑_s 𝑉)))
24	14	3ad2ant1 1151	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (ringLMod‘𝑅) ∈ LMod)
25		simp3 1156	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑉 ⊆ 𝑈)
26	2, 25	ssexd 5286	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑉 ∈ V)
27	16	pwslmod 21245	. . . . . 6 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝑉 ∈ V) → ((ringLMod‘𝑅) ↑_s 𝑉) ∈ LMod)
28	24, 26, 27	syl2anc 596	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ((ringLMod‘𝑅) ↑_s 𝑉) ∈ LMod)
29		frlmsplit2.z	. . . . . . 7 ⊢ 𝑍 = (𝑅 freeLMod 𝑉)
30		frlmsplit2.c	. . . . . . 7 ⊢ 𝐶 = (Base‘𝑍)
31		eqid 2761	. . . . . . 7 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉)) = (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉))
32	29, 30, 31	frlmlss 22057	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ∈ V) → 𝐶 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉)))
33	1, 26, 32	syl2anc 596	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐶 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉)))
34	11	rneqd 5920	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) = ran (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)))
35		eqid 2761	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
36	3, 35, 4	frlmbasf 22066	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑥 ∈ 𝐵) → 𝑥:𝑈⟶(Base‘𝑅))
37	2, 36	sylan 592	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → 𝑥:𝑈⟶(Base‘𝑅))
38		simpl3 1212	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → 𝑉 ⊆ 𝑈)
39	37, 38	fssresd 6749	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉):𝑉⟶(Base‘𝑅))
40		fvex 6898	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) ∈ V
41		elmapg 8859	. . . . . . . . . . . 12 ⊢ (((Base‘𝑅) ∈ V ∧ 𝑉 ∈ V) → ((𝑥 ↾ 𝑉) ∈ ((Base‘𝑅) ↑_m 𝑉) ↔ (𝑥 ↾ 𝑉):𝑉⟶(Base‘𝑅)))
42	40, 26, 41	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ((𝑥 ↾ 𝑉) ∈ ((Base‘𝑅) ↑_m 𝑉) ↔ (𝑥 ↾ 𝑉):𝑉⟶(Base‘𝑅)))
43	42	adantr 486	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ↾ 𝑉) ∈ ((Base‘𝑅) ↑_m 𝑉) ↔ (𝑥 ↾ 𝑉):𝑉⟶(Base‘𝑅)))
44	39, 43	mpbird 260	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉) ∈ ((Base‘𝑅) ↑_m 𝑉))
45		eqid 2761	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
46	3, 45, 4	frlmbasfsupp 22064	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑥 ∈ 𝐵) → 𝑥 finSupp (0_g‘𝑅))
47	2, 46	sylan 592	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → 𝑥 finSupp (0_g‘𝑅))
48		fvexd 6900	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (0_g‘𝑅) ∈ V)
49	47, 48	fsuppres 9385	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉) finSupp (0_g‘𝑅))
50	29, 35, 45, 30	frlmelbas 22062	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ∈ V) → ((𝑥 ↾ 𝑉) ∈ 𝐶 ↔ ((𝑥 ↾ 𝑉) ∈ ((Base‘𝑅) ↑_m 𝑉) ∧ (𝑥 ↾ 𝑉) finSupp (0_g‘𝑅))))
51	1, 26, 50	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ((𝑥 ↾ 𝑉) ∈ 𝐶 ↔ ((𝑥 ↾ 𝑉) ∈ ((Base‘𝑅) ↑_m 𝑉) ∧ (𝑥 ↾ 𝑉) finSupp (0_g‘𝑅))))
52	51	adantr 486	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ↾ 𝑉) ∈ 𝐶 ↔ ((𝑥 ↾ 𝑉) ∈ ((Base‘𝑅) ↑_m 𝑉) ∧ (𝑥 ↾ 𝑉) finSupp (0_g‘𝑅))))
53	44, 49, 52	mpbir2and 726	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉) ∈ 𝐶)
54	53	fmpttd 7115	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)):𝐵⟶𝐶)
55	54	frnd 6718	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⊆ 𝐶)
56	34, 55	eqsstrd 3965	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ⊆ 𝐶)
57		eqid 2761	. . . . . 6 ⊢ (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶) = (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶)
58	57, 31	reslmhm2b 21329	. . . . 5 ⊢ ((((ringLMod‘𝑅) ↑_s 𝑉) ∈ LMod ∧ 𝐶 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉)) ∧ ran ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ⊆ 𝐶) → (((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ∈ ((((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) LMHom ((ringLMod‘𝑅) ↑_s 𝑉)) ↔ ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ∈ ((((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) LMHom (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶))))
59	28, 33, 56, 58	syl3anc 1398	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ∈ ((((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) LMHom ((ringLMod‘𝑅) ↑_s 𝑉)) ↔ ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ∈ ((((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) LMHom (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶))))
60	23, 59	mpbid 235	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ∈ ((((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) LMHom (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶)))
61	13, 60	eqeltrrd 2862	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ ((((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) LMHom (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶)))
62	3, 4	frlmpws 22056	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋) → 𝑌 = (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵))
63	1, 2, 62	syl2anc 596	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑌 = (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵))
64	29, 30	frlmpws 22056	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ∈ V) → 𝑍 = (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶))
65	1, 26, 64	syl2anc 596	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑍 = (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶))
66	63, 65	oveq12d 7438	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (𝑌 LMHom 𝑍) = ((((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) LMHom (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶)))
67	61, 66	eleqtrrd 2864	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 Vcvv 3451 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 ran crn 5652 ↾ cres 5653 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ↑_m cmap 8847 finSupp cfsupp 9353 Basecbs 17387 ↾_s cress 17408 0_gc0g 17610 ↑_s cpws 17617 Ringcrg 20459 LModclmod 21135 LSubSpclss 21206 LMHom clmhm 21294 ringLModcrglmod 21447 freeLMod cfrlm 22052

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-sup 9434 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-fz 13640 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-hom 17452 df-cco 17453 df-0g 17612 df-prds 17618 df-pws 17620 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-submnd 18979 df-grp 19147 df-minusg 19148 df-sbg 19149 df-subg 19333 df-ghm 19428 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-subrg 20822 df-lmod 21137 df-lss 21207 df-lmhm 21297 df-sra 21448 df-rgmod 21449 df-dsmm 22038 df-frlm 22053

This theorem is used by: frlmsslss 22080

W3C validator