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Theorem nmoix 24568

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 24563	. . . . . 6 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ (𝑁‘𝐹) ∈ ℝ))
3	2	biimpar 477	. . . . 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 24567	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) · (𝐿‘𝑋)))
8	3, 7	sylan 579	. . . 4 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑁‘𝐹) ∈ ℝ) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) · (𝐿‘𝑋)))
9	8	an32s 649	. . 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 24447	. . . . 5 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑋 ∈ 𝑉) → (𝐿‘𝑋) ∈ ℝ)
12	11	3ad2antl1 1182	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝐿‘𝑋) ∈ ℝ)
13		rexmul 13247	. . . 4 ⊢ (((𝑁‘𝐹) ∈ ℝ ∧ (𝐿‘𝑋) ∈ ℝ) → ((𝑁‘𝐹) ·_e (𝐿‘𝑋)) = ((𝑁‘𝐹) · (𝐿‘𝑋)))
14	10, 12, 13	syl2anr 596	. . 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 5166	. 2 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ (𝑁‘𝐹) ∈ ℝ) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) ·_e (𝐿‘𝑋)))
16		fveq2 6881	. . . . . . 7 ⊢ (𝑋 = (0_g‘𝑆) → (𝐹‘𝑋) = (𝐹‘(0_g‘𝑆)))
17	16	fveq2d 6885	. . . . . 6 ⊢ (𝑋 = (0_g‘𝑆) → (𝑀‘(𝐹‘𝑋)) = (𝑀‘(𝐹‘(0_g‘𝑆))))
18		fveq2 6881	. . . . . . 7 ⊢ (𝑋 = (0_g‘𝑆) → (𝐿‘𝑋) = (𝐿‘(0_g‘𝑆)))
19	18	oveq2d 7417	. . . . . 6 ⊢ (𝑋 = (0_g‘𝑆) → (+∞ ·_e (𝐿‘𝑋)) = (+∞ ·_e (𝐿‘(0_g‘𝑆))))
20	17, 19	breq12d 5151	. . . . 5 ⊢ (𝑋 = (0_g‘𝑆) → ((𝑀‘(𝐹‘𝑋)) ≤ (+∞ ·_e (𝐿‘𝑋)) ↔ (𝑀‘(𝐹‘(0_g‘𝑆))) ≤ (+∞ ·_e (𝐿‘(0_g‘𝑆)))))
21		simpl2 1189	. . . . . . . . . 10 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → 𝑇 ∈ NrmGrp)
22		eqid 2724	. . . . . . . . . . . . 13 ⊢ (Base‘𝑇) = (Base‘𝑇)
23	4, 22	ghmf 19135	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑉⟶(Base‘𝑇))
24	23	ffvelcdmda 7076	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ (Base‘𝑇))
25	24	3ad2antl3 1184	. . . . . . . . . 10 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ (Base‘𝑇))
26	22, 6	nmcl 24447	. . . . . . . . . 10 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑋) ∈ (Base‘𝑇)) → (𝑀‘(𝐹‘𝑋)) ∈ ℝ)
27	21, 25, 26	syl2anc 583	. . . . . . . . 9 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ∈ ℝ)
28	27	adantr 480	. . . . . . . 8 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝑀‘(𝐹‘𝑋)) ∈ ℝ)
29	28	rexrd 11261	. . . . . . 7 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝑀‘(𝐹‘𝑋)) ∈ ℝ^*)
30		pnfge 13107	. . . . . . 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 1133	. . . . . . . 8 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 𝑆 ∈ NrmGrp)
33		eqid 2724	. . . . . . . . . 10 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
34	4, 5, 33	nmrpcl 24451	. . . . . . . . 9 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝐿‘𝑋) ∈ ℝ⁺)
35	34	3expa 1115	. . . . . . . 8 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝐿‘𝑋) ∈ ℝ⁺)
36	32, 35	sylanl1 677	. . . . . . 7 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝐿‘𝑋) ∈ ℝ⁺)
37		rpxr 12980	. . . . . . . 8 ⊢ ((𝐿‘𝑋) ∈ ℝ⁺ → (𝐿‘𝑋) ∈ ℝ^*)
38		rpgt0 12983	. . . . . . . 8 ⊢ ((𝐿‘𝑋) ∈ ℝ⁺ → 0 < (𝐿‘𝑋))
39		xmulpnf2 13251	. . . . . . . 8 ⊢ (((𝐿‘𝑋) ∈ ℝ^* ∧ 0 < (𝐿‘𝑋)) → (+∞ ·_e (𝐿‘𝑋)) = +∞)
40	37, 38, 39	syl2anc 583	. . . . . . 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 5166	. . . . 5 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑋 ≠ (0_g‘𝑆)) → (𝑀‘(𝐹‘𝑋)) ≤ (+∞ ·_e (𝐿‘𝑋)))
43		0le0 12310	. . . . . 6 ⊢ 0 ≤ 0
44		simpl3 1190	. . . . . . . . . 10 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
45		eqid 2724	. . . . . . . . . . 11 ⊢ (0_g‘𝑇) = (0_g‘𝑇)
46	33, 45	ghmid 19137	. . . . . . . . . 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 6885	. . . . . . . 8 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘(0_g‘𝑆))) = (𝑀‘(0_g‘𝑇)))
49	6, 45	nm0 24460	. . . . . . . . 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 2764	. . . . . . 7 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘(0_g‘𝑆))) = 0)
52		simpl1 1188	. . . . . . . . . 10 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → 𝑆 ∈ NrmGrp)
53	5, 33	nm0 24460	. . . . . . . . . 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 7417	. . . . . . . 8 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (+∞ ·_e (𝐿‘(0_g‘𝑆))) = (+∞ ·_e 0))
56		pnfxr 11265	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
57		xmul01 13243	. . . . . . . . 9 ⊢ (+∞ ∈ ℝ^* → (+∞ ·_e 0) = 0)
58	56, 57	ax-mp 5	. . . . . . . 8 ⊢ (+∞ ·_e 0) = 0
59	55, 58	eqtrdi 2780	. . . . . . 7 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (+∞ ·_e (𝐿‘(0_g‘𝑆))) = 0)
60	51, 59	breq12d 5151	. . . . . 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 3019	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ (+∞ ·_e (𝐿‘𝑋)))
63	62	adantr 480	. . 3 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ (𝑁‘𝐹) = +∞) → (𝑀‘(𝐹‘𝑋)) ≤ (+∞ ·_e (𝐿‘𝑋)))
64		simpr 484	. . . 4 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ (𝑁‘𝐹) = +∞) → (𝑁‘𝐹) = +∞)
65	64	oveq1d 7416	. . 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 5166	. 2 ⊢ ((((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) ∧ (𝑁‘𝐹) = +∞) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) ·_e (𝐿‘𝑋)))
67	1	nmocl 24559	. . . . 5 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ^*)
68	1	nmoge0 24560	. . . . . 6 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
69		ge0nemnf 13149	. . . . . 6 ⊢ (((𝑁‘𝐹) ∈ ℝ^* ∧ 0 ≤ (𝑁‘𝐹)) → (𝑁‘𝐹) ≠ -∞)
70	67, 68, 69	syl2anc 583	. . . . 5 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) ≠ -∞)
71	67, 70	jca 511	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → ((𝑁‘𝐹) ∈ ℝ^* ∧ (𝑁‘𝐹) ≠ -∞))
72		xrnemnf 13094	. . . 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 480	. 2 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → ((𝑁‘𝐹) ∈ ℝ ∨ (𝑁‘𝐹) = +∞))
75	15, 66, 74	mpjaodan 955	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 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 ℝcr 11105 0cc0 11106 · cmul 11111 +∞cpnf 11242 -∞cmnf 11243 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 ℝ⁺crp 12971 ·_e cxmu 13088 Basecbs 17143 0_gc0g 17384 GrpHom cghm 19128 normcnm 24407 NrmGrpcngp 24408 normOp cnmo 24544 NGHom cnghm 24545

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ico 13327 df-0g 17386 df-topgen 17388 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-ghm 19129 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-xms 24148 df-ms 24149 df-nm 24413 df-ngp 24414 df-nmo 24547 df-nghm 24548

This theorem is referenced by: nmoi2 24569 nmoleub2lem 24963

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