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Theorem frlmsplit2 21811

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 1135	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑅 ∈ Ring)
2		simp2 1136	. . . . . 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 2735	. . . . . . 7 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)) = (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈))
6	3, 4, 5	frlmlss 21789	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋) → 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)))
7	1, 2, 6	syl2anc 584	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)))
8		eqid 2735	. . . . . 6 ⊢ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) = (Base‘((ringLMod‘𝑅) ↑_s 𝑈))
9	8, 5	lssss 20952	. . . . 5 ⊢ (𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)) → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)))
10		resmpt 6057	. . . . 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 18	. . . 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 2793	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) = 𝐹)
14		rlmlmod 21228	. . . . . 6 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
15		eqid 2735	. . . . . . 7 ⊢ ((ringLMod‘𝑅) ↑_s 𝑈) = ((ringLMod‘𝑅) ↑_s 𝑈)
16		eqid 2735	. . . . . . 7 ⊢ ((ringLMod‘𝑅) ↑_s 𝑉) = ((ringLMod‘𝑅) ↑_s 𝑉)
17		eqid 2735	. . . . . . 7 ⊢ (Base‘((ringLMod‘𝑅) ↑_s 𝑉)) = (Base‘((ringLMod‘𝑅) ↑_s 𝑉))
18		eqid 2735	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) = (𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉))
19	15, 16, 8, 17, 18	pwssplit3 21078	. . . . . 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 1162	. . . . 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 2735	. . . . . 6 ⊢ (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) = (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵)
22	5, 21	reslmhm 21069	. . . . 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 584	. . . 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 1132	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (ringLMod‘𝑅) ∈ LMod)
25		simp3 1137	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑉 ⊆ 𝑈)
26	2, 25	ssexd 5330	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑉 ∈ V)
27	16	pwslmod 20986	. . . . . 6 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝑉 ∈ V) → ((ringLMod‘𝑅) ↑_s 𝑉) ∈ LMod)
28	24, 26, 27	syl2anc 584	. . . . 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 2735	. . . . . . 7 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉)) = (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉))
32	29, 30, 31	frlmlss 21789	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ∈ V) → 𝐶 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉)))
33	1, 26, 32	syl2anc 584	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐶 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉)))
34	11	rneqd 5952	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) = ran (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)))
35		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
36	3, 35, 4	frlmbasf 21798	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑥 ∈ 𝐵) → 𝑥:𝑈⟶(Base‘𝑅))
37	2, 36	sylan 580	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → 𝑥:𝑈⟶(Base‘𝑅))
38		simpl3 1192	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → 𝑉 ⊆ 𝑈)
39	37, 38	fssresd 6776	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉):𝑉⟶(Base‘𝑅))
40		fvex 6920	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) ∈ V
41		elmapg 8878	. . . . . . . . . . . 12 ⊢ (((Base‘𝑅) ∈ V ∧ 𝑉 ∈ V) → ((𝑥 ↾ 𝑉) ∈ ((Base‘𝑅) ↑_m 𝑉) ↔ (𝑥 ↾ 𝑉):𝑉⟶(Base‘𝑅)))
42	40, 26, 41	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ((𝑥 ↾ 𝑉) ∈ ((Base‘𝑅) ↑_m 𝑉) ↔ (𝑥 ↾ 𝑉):𝑉⟶(Base‘𝑅)))
43	42	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ↾ 𝑉) ∈ ((Base‘𝑅) ↑_m 𝑉) ↔ (𝑥 ↾ 𝑉):𝑉⟶(Base‘𝑅)))
44	39, 43	mpbird 257	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉) ∈ ((Base‘𝑅) ↑_m 𝑉))
45		eqid 2735	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
46	3, 45, 4	frlmbasfsupp 21796	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑥 ∈ 𝐵) → 𝑥 finSupp (0_g‘𝑅))
47	2, 46	sylan 580	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → 𝑥 finSupp (0_g‘𝑅))
48		fvexd 6922	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (0_g‘𝑅) ∈ V)
49	47, 48	fsuppres 9431	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉) finSupp (0_g‘𝑅))
50	29, 35, 45, 30	frlmelbas 21794	. . . . . . . . . . 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 584	. . . . . . . . . 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 480	. . . . . . . . 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 713	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉) ∈ 𝐶)
54	53	fmpttd 7135	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)):𝐵⟶𝐶)
55	54	frnd 6745	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⊆ 𝐶)
56	34, 55	eqsstrd 4034	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ⊆ 𝐶)
57		eqid 2735	. . . . . 6 ⊢ (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶) = (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶)
58	57, 31	reslmhm2b 21071	. . . . 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 1370	. . . 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 232	. . 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 2840	. 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 21788	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋) → 𝑌 = (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵))
63	1, 2, 62	syl2anc 584	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑌 = (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵))
64	29, 30	frlmpws 21788	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ∈ V) → 𝑍 = (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶))
65	1, 26, 64	syl2anc 584	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑍 = (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶))
66	63, 65	oveq12d 7449	. 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 2842	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ↾ cres 5691 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 finSupp cfsupp 9399 Basecbs 17245 ↾_s cress 17274 0_gc0g 17486 ↑_s cpws 17493 Ringcrg 20251 LModclmod 20875 LSubSpclss 20947 LMHom clmhm 21036 ringLModcrglmod 21189 freeLMod cfrlm 21784

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lmhm 21039 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785

This theorem is referenced by: frlmsslss 21812

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