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Theorem frlmsplit2 21682

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 1136	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑅 ∈ Ring)
2		simp2 1137	. . . . . 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 2729	. . . . . . 7 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)) = (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈))
6	3, 4, 5	frlmlss 21660	. . . . . 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 2729	. . . . . 6 ⊢ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) = (Base‘((ringLMod‘𝑅) ↑_s 𝑈))
9	8, 5	lssss 20842	. . . . 5 ⊢ (𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)) → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)))
10		resmpt 6008	. . . . 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 2782	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) = 𝐹)
14		rlmlmod 21110	. . . . . 6 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
15		eqid 2729	. . . . . . 7 ⊢ ((ringLMod‘𝑅) ↑_s 𝑈) = ((ringLMod‘𝑅) ↑_s 𝑈)
16		eqid 2729	. . . . . . 7 ⊢ ((ringLMod‘𝑅) ↑_s 𝑉) = ((ringLMod‘𝑅) ↑_s 𝑉)
17		eqid 2729	. . . . . . 7 ⊢ (Base‘((ringLMod‘𝑅) ↑_s 𝑉)) = (Base‘((ringLMod‘𝑅) ↑_s 𝑉))
18		eqid 2729	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) = (𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉))
19	15, 16, 8, 17, 18	pwssplit3 20968	. . . . . 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 1163	. . . . 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 2729	. . . . . 6 ⊢ (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵) = (((ringLMod‘𝑅) ↑_s 𝑈) ↾_s 𝐵)
22	5, 21	reslmhm 20959	. . . . 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 1133	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (ringLMod‘𝑅) ∈ LMod)
25		simp3 1138	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑉 ⊆ 𝑈)
26	2, 25	ssexd 5279	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑉 ∈ V)
27	16	pwslmod 20876	. . . . . 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 2729	. . . . . . 7 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉)) = (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑉))
32	29, 30, 31	frlmlss 21660	. . . . . 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 5902	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) = ran (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)))
35		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
36	3, 35, 4	frlmbasf 21669	. . . . . . . . . . . 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 1194	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → 𝑉 ⊆ 𝑈)
39	37, 38	fssresd 6727	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉):𝑉⟶(Base‘𝑅))
40		fvex 6871	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) ∈ V
41		elmapg 8812	. . . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
46	3, 45, 4	frlmbasfsupp 21667	. . . . . . . . . . 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 6873	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (0_g‘𝑅) ∈ V)
49	47, 48	fsuppres 9344	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉) finSupp (0_g‘𝑅))
50	29, 35, 45, 30	frlmelbas 21665	. . . . . . . . . . 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 7087	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)):𝐵⟶𝐶)
55	54	frnd 6696	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⊆ 𝐶)
56	34, 55	eqsstrd 3981	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran ((𝑥 ∈ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)) ↦ (𝑥 ↾ 𝑉)) ↾ 𝐵) ⊆ 𝐶)
57		eqid 2729	. . . . . 6 ⊢ (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶) = (((ringLMod‘𝑅) ↑_s 𝑉) ↾_s 𝐶)
58	57, 31	reslmhm2b 20961	. . . . 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 1373	. . . 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 2829	. 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 21659	. . . 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 21659	. . . 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 7405	. 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 2831	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 1540 ∈ wcel 2109 Vcvv 3447 ⊆ wss 3914 class class class wbr 5107 ↦ cmpt 5188 ran crn 5639 ↾ cres 5640 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ↑_m cmap 8799 finSupp cfsupp 9312 Basecbs 17179 ↾_s cress 17200 0_gc0g 17402 ↑_s cpws 17409 Ringcrg 20142 LModclmod 20766 LSubSpclss 20837 LMHom clmhm 20926 ringLModcrglmod 21079 freeLMod cfrlm 21655

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-hom 17244 df-cco 17245 df-0g 17404 df-prds 17410 df-pws 17412 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-ghm 19145 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-subrg 20479 df-lmod 20768 df-lss 20838 df-lmhm 20929 df-sra 21080 df-rgmod 21081 df-dsmm 21641 df-frlm 21656

This theorem is referenced by: frlmsslss 21683

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