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 21658

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 21636	. . . . . 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 20818	. . . . 5 ⊢ (𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑_s 𝑈)) → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑_s 𝑈)))
10		resmpt 5997	. . . . 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 21086	. . . . . 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 20944	. . . . . 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 20935	. . . . 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 5274	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑉 ∈ V)
27	16	pwslmod 20852	. . . . . 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 21636	. . . . . 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 5891	. . . . . 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 21645	. . . . . . . . . . . 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 6709	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉):𝑉⟶(Base‘𝑅))
40		fvex 6853	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) ∈ V
41		elmapg 8789	. . . . . . . . . . . 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 21643	. . . . . . . . . . 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 6855	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (0_g‘𝑅) ∈ V)
49	47, 48	fsuppres 9320	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ 𝐵) → (𝑥 ↾ 𝑉) finSupp (0_g‘𝑅))
50	29, 35, 45, 30	frlmelbas 21641	. . . . . . . . . . 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 7069	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)):𝐵⟶𝐶)
55	54	frnd 6678	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ran (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⊆ 𝐶)
56	34, 55	eqsstrd 3978	. . . . 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 20937	. . . . 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 21635	. . . 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 21635	. . . 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 7387	. 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 3444 ⊆ wss 3911 class class class wbr 5102 ↦ cmpt 5183 ran crn 5632 ↾ cres 5633 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ↑_m cmap 8776 finSupp cfsupp 9288 Basecbs 17155 ↾_s cress 17176 0_gc0g 17378 ↑_s cpws 17385 Ringcrg 20118 LModclmod 20742 LSubSpclss 20813 LMHom clmhm 20902 ringLModcrglmod 21055 freeLMod cfrlm 21631

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-fz 13445 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17380 df-prds 17386 df-pws 17388 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-mhm 18686 df-submnd 18687 df-grp 18844 df-minusg 18845 df-sbg 18846 df-subg 19031 df-ghm 19121 df-cmn 19688 df-abl 19689 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-subrg 20455 df-lmod 20744 df-lss 20814 df-lmhm 20905 df-sra 21056 df-rgmod 21057 df-dsmm 21617 df-frlm 21632

This theorem is referenced by: frlmsslss 21659

W3C validator