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Theorem nmoix 24131

Description: The operator norm is a bound on the size of an operator, even when it is infinite (using extended real multiplication). (Contributed by Mario Carneiro, 18-Oct-2015.)

**Hypotheses**
Ref	Expression
nmofval.1	⊢ 𝑁 = (𝑆 normOp 𝑇)
nmoi.2	⊢ 𝑉 = (Base‘𝑆)
nmoi.3	⊢ 𝐿 = (norm‘𝑆)
nmoi.4	⊢ 𝑀 = (norm‘𝑇)

**Assertion**
Ref	Expression
nmoix	⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) ·_e (𝐿‘𝑋)))

**Proof of Theorem nmoix**
Step	Hyp	Ref	Expression
1		nmofval.1	. . . . . . 7 ⊢ 𝑁 = (𝑆 normOp 𝑇)
2	1	isnghm2 24126	. . . . . 6 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ (𝑁‘𝐹) ∈ ℝ))
3	2	biimpar 479	. . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑁‘𝐹) ∈ ℝ) → 𝐹 ∈ (𝑆 NGHom 𝑇))
4		nmoi.2	. . . . . 6 ⊢ 𝑉 = (Base‘𝑆)
5		nmoi.3	. . . . . 6 ⊢ 𝐿 = (norm‘𝑆)
6		nmoi.4	. . . . . 6 ⊢ 𝑀 = (norm‘𝑇)
7	1, 4, 5, 6	nmoi 24130	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) · (𝐿‘𝑋)))
8	3, 7	sylan 581	. . . 4 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑁‘𝐹) ∈ ℝ) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) · (𝐿‘𝑋)))
9	8	an32s 651	. . 3 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ (𝑁‘𝐹) ∈ ℝ) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) · (𝐿‘𝑋)))
10		id 22	. . . 4 ⊢ ((𝑁‘𝐹) ∈ ℝ → (𝑁‘𝐹) ∈ ℝ)
11	4, 5	nmcl 24010	. . . . 5 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑋 ∈ 𝑉) → (𝐿‘𝑋) ∈ ℝ)
12	11	3ad2antl1 1186	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝐿‘𝑋) ∈ ℝ)
13		rexmul 13201	. . . 4 ⊢ (((𝑁‘𝐹) ∈ ℝ ∧ (𝐿‘𝑋) ∈ ℝ) → ((𝑁‘𝐹) ·_e (𝐿‘𝑋)) = ((𝑁‘𝐹) · (𝐿‘𝑋)))
14	10, 12, 13	syl2anr 598	. . 3 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ (𝑁‘𝐹) ∈ ℝ) → ((𝑁‘𝐹) ·_e (𝐿‘𝑋)) = ((𝑁‘𝐹) · (𝐿‘𝑋)))
15	9, 14	breqtrrd 5139	. 2 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ (𝑁‘𝐹) ∈ ℝ) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) ·_e (𝐿‘𝑋)))
16		fveq2 6848	. . . . . . 7 ⊢ (𝑋 = (0_g‘𝑆) → (𝐹‘𝑋) = (𝐹‘(0_g‘𝑆)))
17	16	fveq2d 6852	. . . . . 6 ⊢ (𝑋 = (0_g‘𝑆) → (𝑀‘(𝐹‘𝑋)) = (𝑀‘(𝐹‘(0_g‘𝑆))))
18		fveq2 6848	. . . . . . 7 ⊢ (𝑋 = (0_g‘𝑆) → (𝐿‘𝑋) = (𝐿‘(0_g‘𝑆)))
19	18	oveq2d 7379	. . . . . 6 ⊢ (𝑋 = (0_g‘𝑆) → (+∞ ·_e (𝐿‘𝑋)) = (+∞ ·_e (𝐿‘(0_g‘𝑆))))
20	17, 19	breq12d 5124	. . . . 5 ⊢ (𝑋 = (0_g‘𝑆) → ((𝑀‘(𝐹‘𝑋)) ≤ (+∞ ·_e (𝐿‘𝑋)) ↔ (𝑀‘(𝐹‘(0_g‘𝑆))) ≤ (+∞ ·_e (𝐿‘(0_g‘𝑆)))))
21		simpl2 1193	. . . . . . . . . 10 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → 𝑇 ∈ NrmGrp)
22		eqid 2732	. . . . . . . . . . . . 13 ⊢ (Base‘𝑇) = (Base‘𝑇)
23	4, 22	ghmf 19027	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑉⟶(Base‘𝑇))
24	23	ffvelcdmda 7041	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ (Base‘𝑇))
25	24	3ad2antl3 1188	. . . . . . . . . 10 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ (Base‘𝑇))
26	22, 6	nmcl 24010	. . . . . . . . . 10 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑋) ∈ (Base‘𝑇)) → (𝑀‘(𝐹‘𝑋)) ∈ ℝ)
27	21, 25, 26	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ∈ ℝ)
28	27	adantr 482	. . . . . . . 8 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝑀‘(𝐹‘𝑋)) ∈ ℝ)
29	28	rexrd 11215	. . . . . . 7 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝑀‘(𝐹‘𝑋)) ∈ ℝ^*)
30		pnfge 13061	. . . . . . 7 ⊢ ((𝑀‘(𝐹‘𝑋)) ∈ ℝ^* → (𝑀‘(𝐹‘𝑋)) ≤ +∞)
31	29, 30	syl 17	. . . . . 6 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝑀‘(𝐹‘𝑋)) ≤ +∞)
32		simp1 1137	. . . . . . . 8 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 𝑆 ∈ NrmGrp)
33		eqid 2732	. . . . . . . . . 10 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
34	4, 5, 33	nmrpcl 24014	. . . . . . . . 9 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝐿‘𝑋) ∈ ℝ⁺)
35	34	3expa 1119	. . . . . . . 8 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝐿‘𝑋) ∈ ℝ⁺)
36	32, 35	sylanl1 679	. . . . . . 7 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝐿‘𝑋) ∈ ℝ⁺)
37		rpxr 12934	. . . . . . . 8 ⊢ ((𝐿‘𝑋) ∈ ℝ⁺ → (𝐿‘𝑋) ∈ ℝ^*)
38		rpgt0 12937	. . . . . . . 8 ⊢ ((𝐿‘𝑋) ∈ ℝ⁺ → 0 < (𝐿‘𝑋))
39		xmulpnf2 13205	. . . . . . . 8 ⊢ (((𝐿‘𝑋) ∈ ℝ^* ∧ 0 < (𝐿‘𝑋)) → (+∞ ·_e (𝐿‘𝑋)) = +∞)
40	37, 38, 39	syl2anc 585	. . . . . . 7 ⊢ ((𝐿‘𝑋) ∈ ℝ⁺ → (+∞ ·_e (𝐿‘𝑋)) = +∞)
41	36, 40	syl 17	. . . . . 6 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (+∞ ·_e (𝐿‘𝑋)) = +∞)
42	31, 41	breqtrrd 5139	. . . . 5 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝑀‘(𝐹‘𝑋)) ≤ (+∞ ·_e (𝐿‘𝑋)))
43		0le0 12264	. . . . . 6 ⊢ 0 ≤ 0
44		simpl3 1194	. . . . . . . . . 10 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
45		eqid 2732	. . . . . . . . . . 11 ⊢ (0_g‘𝑇) = (0_g‘𝑇)
46	33, 45	ghmid 19029	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0_g‘𝑆)) = (0_g‘𝑇))
47	44, 46	syl 17	. . . . . . . . 9 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝐹‘(0_g‘𝑆)) = (0_g‘𝑇))
48	47	fveq2d 6852	. . . . . . . 8 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘(0_g‘𝑆))) = (𝑀‘(0_g‘𝑇)))
49	6, 45	nm0 24023	. . . . . . . . 9 ⊢ (𝑇 ∈ NrmGrp → (𝑀‘(0_g‘𝑇)) = 0)
50	21, 49	syl 17	. . . . . . . 8 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(0_g‘𝑇)) = 0)
51	48, 50	eqtrd 2772	. . . . . . 7 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘(0_g‘𝑆))) = 0)
52		simpl1 1192	. . . . . . . . . 10 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → 𝑆 ∈ NrmGrp)
53	5, 33	nm0 24023	. . . . . . . . . 10 ⊢ (𝑆 ∈ NrmGrp → (𝐿‘(0_g‘𝑆)) = 0)
54	52, 53	syl 17	. . . . . . . . 9 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝐿‘(0_g‘𝑆)) = 0)
55	54	oveq2d 7379	. . . . . . . 8 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (+∞ ·_e (𝐿‘(0_g‘𝑆))) = (+∞ ·_e 0))
56		pnfxr 11219	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
57		xmul01 13197	. . . . . . . . 9 ⊢ (+∞ ∈ ℝ^* → (+∞ ·_e 0) = 0)
58	56, 57	ax-mp 5	. . . . . . . 8 ⊢ (+∞ ·_e 0) = 0
59	55, 58	eqtrdi 2788	. . . . . . 7 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (+∞ ·_e (𝐿‘(0_g‘𝑆))) = 0)
60	51, 59	breq12d 5124	. . . . . 6 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → ((𝑀‘(𝐹‘(0_g‘𝑆))) ≤ (+∞ ·_e (𝐿‘(0_g‘𝑆))) ↔ 0 ≤ 0))
61	43, 60	mpbiri 258	. . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘(0_g‘𝑆))) ≤ (+∞ ·_e (𝐿‘(0_g‘𝑆))))
62	20, 42, 61	pm2.61ne 3027	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ (+∞ ·_e (𝐿‘𝑋)))
63	62	adantr 482	. . 3 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ (𝑁‘𝐹) = +∞) → (𝑀‘(𝐹‘𝑋)) ≤ (+∞ ·_e (𝐿‘𝑋)))
64		simpr 486	. . . 4 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ (𝑁‘𝐹) = +∞) → (𝑁‘𝐹) = +∞)
65	64	oveq1d 7378	. . 3 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ (𝑁‘𝐹) = +∞) → ((𝑁‘𝐹) ·_e (𝐿‘𝑋)) = (+∞ ·_e (𝐿‘𝑋)))
66	63, 65	breqtrrd 5139	. 2 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ (𝑁‘𝐹) = +∞) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) ·_e (𝐿‘𝑋)))
67	1	nmocl 24122	. . . . 5 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ^*)
68	1	nmoge0 24123	. . . . . 6 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
69		ge0nemnf 13103	. . . . . 6 ⊢ (((𝑁‘𝐹) ∈ ℝ^* ∧ 0 ≤ (𝑁‘𝐹)) → (𝑁‘𝐹) ≠ -∞)
70	67, 68, 69	syl2anc 585	. . . . 5 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) ≠ -∞)
71	67, 70	jca 513	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → ((𝑁‘𝐹) ∈ ℝ^* ∧ (𝑁‘𝐹) ≠ -∞))
72		xrnemnf 13048	. . . 4 ⊢ (((𝑁‘𝐹) ∈ ℝ^* ∧ (𝑁‘𝐹) ≠ -∞) ↔ ((𝑁‘𝐹) ∈ ℝ ∨ (𝑁‘𝐹) = +∞))
73	71, 72	sylib 217	. . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → ((𝑁‘𝐹) ∈ ℝ ∨ (𝑁‘𝐹) = +∞))
74	73	adantr 482	. 2 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → ((𝑁‘𝐹) ∈ ℝ ∨ (𝑁‘𝐹) = +∞))
75	15, 66, 74	mpjaodan 958	1 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) ·_e (𝐿‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 class class class wbr 5111 ‘cfv 6502 (class class class)co 7363 ℝcr 11060 0cc0 11061 · cmul 11066 +∞cpnf 11196 -∞cmnf 11197 ℝ^*cxr 11198 < clt 11199 ≤ cle 11200 ℝ⁺crp 12925 ·_e cxmu 13042 Basecbs 17095 0_gc0g 17336 GrpHom cghm 19020 normcnm 23970 NrmGrpcngp 23971 normOp cnmo 24107 NGHom cnghm 24108

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2703 ax-rep 5248 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-resscn 11118 ax-1cn 11119 ax-icn 11120 ax-addcl 11121 ax-addrcl 11122 ax-mulcl 11123 ax-mulrcl 11124 ax-mulcom 11125 ax-addass 11126 ax-mulass 11127 ax-distr 11128 ax-i2m1 11129 ax-1ne0 11130 ax-1rid 11131 ax-rnegex 11132 ax-rrecex 11133 ax-cnre 11134 ax-pre-lttri 11135 ax-pre-lttrn 11136 ax-pre-ltadd 11137 ax-pre-mulgt0 11138 ax-pre-sup 11139

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4872 df-iun 4962 df-br 5112 df-opab 5174 df-mpt 5195 df-tr 5229 df-id 5537 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5594 df-we 5596 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-pred 6259 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7319 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7809 df-1st 7927 df-2nd 7928 df-frecs 8218 df-wrecs 8249 df-recs 8323 df-rdg 8362 df-er 8656 df-map 8775 df-en 8892 df-dom 8893 df-sdom 8894 df-sup 9388 df-inf 9389 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 df-sub 11397 df-neg 11398 df-div 11823 df-nn 12164 df-2 12226 df-n0 12424 df-z 12510 df-uz 12774 df-q 12884 df-rp 12926 df-xneg 13043 df-xadd 13044 df-xmul 13045 df-ico 13281 df-0g 17338 df-topgen 17340 df-mgm 18512 df-sgrp 18561 df-mnd 18572 df-grp 18766 df-ghm 19021 df-psmet 20826 df-xmet 20827 df-met 20828 df-bl 20829 df-mopn 20830 df-top 22281 df-topon 22298 df-topsp 22320 df-bases 22334 df-xms 23711 df-ms 23712 df-nm 23976 df-ngp 23977 df-nmo 24110 df-nghm 24111

This theorem is referenced by: nmoi2 24132 nmoleub2lem 24515

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