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Theorem frlmsplit2 22134

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 2763	. . . . . . 7 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)) = (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈))
6	3, 4, 5	frlmlss 22112	. . . . . 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 2763	. . . . . 6 ⊢ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) = (Base‘((ringLMod‘𝑅) ↑_s 𝑈))
9	8, 5	lssss 21262	. . . . 5 ⊢ (𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)) → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)))
10		resmpt 6047	. . . . 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 2816	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) = 𝐹)
14		rlmlmod 21529	. . . . . 6 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
15		eqid 2763	. . . . . . 7 ⊢ ((ringLMod‘𝑅) ↑_s 𝑈) = ((ringLMod‘𝑅) ↑_s 𝑈)
16		eqid 2763	. . . . . . 7 ⊢ ((ringLMod‘𝑅) ↑_s 𝑉) = ((ringLMod‘𝑅) ↑_s 𝑉)
17		eqid 2763	. . . . . . 7 ⊢ (Base‘((ringLMod‘𝑅) ↑_s 𝑉)) = (Base‘((ringLMod‘𝑅) ↑_s 𝑉))
18		eqid 2763	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) = (𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉))
19	15, 16, 8, 17, 18	pwssplit3 21387	. . . . . 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 2763	. . . . . 6 ⊢ (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) = (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵)
22	5, 21	reslmhm 21378	. . . . 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 5293	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑉 ∈ V)
27	16	pwslmod 21296	. . . . . 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 2763	. . . . . . 7 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉)) = (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉))
32	29, 30, 31	frlmlss 22112	. . . . . 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 5932	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) = ran (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)))
35		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
36	3, 35, 4	frlmbasf 22121	. . . . . . . . . . . 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 6779	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉):𝑉⟶(Base‘𝑅))
40		fvex 6929	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) ∈ V
41		elmapg 8897	. . . . . . . . . . . 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 2763	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
46	3, 45, 4	frlmbasfsupp 22119	. . . . . . . . . . 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 6931	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (0_g‘𝑅) ∈ V)
49	47, 48	fsuppres 9422	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉) finSupp (0_g‘𝑅))
50	29, 35, 45, 30	frlmelbas 22117	. . . . . . . . . . 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 7147	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)):𝐵⟶𝐶)
55	54	frnd 6748	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⊆ 𝐶)
56	34, 55	eqsstrd 3947	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ⊆ 𝐶)
57		eqid 2763	. . . . . 6 ⊢ (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶) = (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶)
58	57, 31	reslmhm2b 21380	. . . . 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 1399	. . . 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 2865	. 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 22111	. . . 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 22111	. . . 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 7469	. 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 2867	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 2147 Vcvv 3454 ⊆ wss 3900 class class class wbr 5108 ↦ cmpt 5192 ran crn 5659 ↾ cres 5660 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ↑_m cmap 8885 finSupp cfsupp 9391 Basecbs 17431 ↾_s cress 17452 0_gc0g 17660 ↑_s cpws 17667 Ringcrg 20510 LModclmod 21186 LSubSpclss 21257 LMHom clmhm 21345 ringLModcrglmod 21498 freeLMod cfrlm 22107

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-map 8887 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-sup 9471 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-fz 13680 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-hom 17497 df-cco 17498 df-0g 17662 df-prds 17668 df-pws 17670 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-mhm 19028 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-subg 19383 df-ghm 19478 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-ring 20512 df-subrg 20873 df-lmod 21188 df-lss 21258 df-lmhm 21348 df-sra 21499 df-rgmod 21500 df-dsmm 22093 df-frlm 22108

This theorem is used by: frlmsslss 22135

W3C validator