Theorem nmoco 24758

Description: An upper bound on the operator norm of a composition. (Contributed by Mario Carneiro, 20-Oct-2015.)

Hypotheses

Ref	Expression
nmoco.1	⊢ 𝑁 = (𝑆 normOp 𝑈)
nmoco.2	⊢ 𝐿 = (𝑇 normOp 𝑈)
nmoco.3	⊢ 𝑀 = (𝑆 normOp 𝑇)

Assertion

Ref	Expression
nmoco	⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘ 𝐺)) ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))

Proof of Theorem nmoco

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmoco.1	. 2 ⊢ 𝑁 = (𝑆 normOp 𝑈)
2		eqid 2737	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2737	. 2 ⊢ (norm‘𝑆) = (norm‘𝑆)
4		eqid 2737	. 2 ⊢ (norm‘𝑈) = (norm‘𝑈)
5		eqid 2737	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		nghmrcl1 24753	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
7	6	adantl 481	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8		nghmrcl2 24754	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑈 ∈ NrmGrp)
9	8	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑈 ∈ NrmGrp)
10		nghmghm 24755	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝐹 ∈ (𝑇 GrpHom 𝑈))
11		nghmghm 24755	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
12		ghmco 19254	. . 3 ⊢ ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈))
13	10, 11, 12	syl2an 596	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈))
14		nmoco.2	. . . 4 ⊢ 𝐿 = (𝑇 normOp 𝑈)
15	14	nghmcl 24748	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → (𝐿‘𝐹) ∈ ℝ)
16		nmoco.3	. . . 4 ⊢ 𝑀 = (𝑆 normOp 𝑇)
17	16	nghmcl 24748	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑀‘𝐺) ∈ ℝ)
18		remulcl 11240	. . 3 ⊢ (((𝐿‘𝐹) ∈ ℝ ∧ (𝑀‘𝐺) ∈ ℝ) → ((𝐿‘𝐹) · (𝑀‘𝐺)) ∈ ℝ)
19	15, 17, 18	syl2an 596	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝐿‘𝐹) · (𝑀‘𝐺)) ∈ ℝ)
20		nghmrcl1 24753	. . . . 5 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑇 ∈ NrmGrp)
21	14	nmoge0 24742	. . . . 5 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑈 ∈ NrmGrp ∧ 𝐹 ∈ (𝑇 GrpHom 𝑈)) → 0 ≤ (𝐿‘𝐹))
22	20, 8, 10, 21	syl3anc 1373	. . . 4 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 0 ≤ (𝐿‘𝐹))
23	15, 22	jca 511	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → ((𝐿‘𝐹) ∈ ℝ ∧ 0 ≤ (𝐿‘𝐹)))
24		nghmrcl2 24754	. . . . 5 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
25	16	nmoge0 24742	. . . . 5 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑀‘𝐺))
26	6, 24, 11, 25	syl3anc 1373	. . . 4 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 0 ≤ (𝑀‘𝐺))
27	17, 26	jca 511	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → ((𝑀‘𝐺) ∈ ℝ ∧ 0 ≤ (𝑀‘𝐺)))
28		mulge0 11781	. . 3 ⊢ ((((𝐿‘𝐹) ∈ ℝ ∧ 0 ≤ (𝐿‘𝐹)) ∧ ((𝑀‘𝐺) ∈ ℝ ∧ 0 ≤ (𝑀‘𝐺))) → 0 ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))
29	23, 27, 28	syl2an 596	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))
30	8	ad2antrr 726	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑈 ∈ NrmGrp)
31	10	ad2antrr 726	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 ∈ (𝑇 GrpHom 𝑈))
32		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
33		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
34	32, 33	ghmf 19238	. . . . . . 7 ⊢ (𝐹 ∈ (𝑇 GrpHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
35	31, 34	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
36	11	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
37	2, 32	ghmf 19238	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
38	36, 37	syl 17	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
39		simprl 771	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
40	38, 39	ffvelcdmd 7105	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐺‘𝑥) ∈ (Base‘𝑇))
41	35, 40	ffvelcdmd 7105	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐹‘(𝐺‘𝑥)) ∈ (Base‘𝑈))
42	33, 4	nmcl 24629	. . . . 5 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐹‘(𝐺‘𝑥)) ∈ (Base‘𝑈)) → ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))) ∈ ℝ)
43	30, 41, 42	syl2anc 584	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))) ∈ ℝ)
44	15	ad2antrr 726	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐿‘𝐹) ∈ ℝ)
45	20	ad2antrr 726	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmGrp)
46		eqid 2737	. . . . . . 7 ⊢ (norm‘𝑇) = (norm‘𝑇)
47	32, 46	nmcl 24629	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
48	45, 40, 47	syl2anc 584	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
49	44, 48	remulcld 11291	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝐿‘𝐹) · ((norm‘𝑇)‘(𝐺‘𝑥))) ∈ ℝ)
50	17	ad2antlr 727	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑀‘𝐺) ∈ ℝ)
51	2, 3	nmcl 24629	. . . . . . . 8 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
52	6, 51	sylan 580	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
53	52	ad2ant2lr 748	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
54	50, 53	remulcld 11291	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥)) ∈ ℝ)
55	44, 54	remulcld 11291	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝐿‘𝐹) · ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥))) ∈ ℝ)
56		simpll 767	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 ∈ (𝑇 NGHom 𝑈))
57	14, 32, 46, 4	nmoi 24749	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))) ≤ ((𝐿‘𝐹) · ((norm‘𝑇)‘(𝐺‘𝑥))))
58	56, 40, 57	syl2anc 584	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))) ≤ ((𝐿‘𝐹) · ((norm‘𝑇)‘(𝐺‘𝑥))))
59	23	ad2antrr 726	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝐿‘𝐹) ∈ ℝ ∧ 0 ≤ (𝐿‘𝐹)))
60	16, 2, 3, 46	nmoi 24749	. . . . . 6 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ≤ ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥)))
61	60	ad2ant2lr 748	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘(𝐺‘𝑥)) ≤ ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥)))
62		lemul2a 12122	. . . . 5 ⊢ (((((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ ∧ ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥)) ∈ ℝ ∧ ((𝐿‘𝐹) ∈ ℝ ∧ 0 ≤ (𝐿‘𝐹))) ∧ ((norm‘𝑇)‘(𝐺‘𝑥)) ≤ ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥))) → ((𝐿‘𝐹) · ((norm‘𝑇)‘(𝐺‘𝑥))) ≤ ((𝐿‘𝐹) · ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥))))
63	48, 54, 59, 61, 62	syl31anc 1375	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝐿‘𝐹) · ((norm‘𝑇)‘(𝐺‘𝑥))) ≤ ((𝐿‘𝐹) · ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥))))
64	43, 49, 55, 58, 63	letrd 11418	. . 3 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))) ≤ ((𝐿‘𝐹) · ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥))))
65		fvco3 7008	. . . . 5 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
66	38, 39, 65	syl2anc 584	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
67	66	fveq2d 6910	. . 3 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑈)‘((𝐹 ∘ 𝐺)‘𝑥)) = ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))))
68	44	recnd 11289	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐿‘𝐹) ∈ ℂ)
69	50	recnd 11289	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑀‘𝐺) ∈ ℂ)
70	53	recnd 11289	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
71	68, 69, 70	mulassd 11284	. . 3 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (((𝐿‘𝐹) · (𝑀‘𝐺)) · ((norm‘𝑆)‘𝑥)) = ((𝐿‘𝐹) · ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥))))
72	64, 67, 71	3brtr4d 5175	. 2 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑈)‘((𝐹 ∘ 𝐺)‘𝑥)) ≤ (((𝐿‘𝐹) · (𝑀‘𝐺)) · ((norm‘𝑆)‘𝑥)))
73	1, 2, 3, 4, 5, 7, 9, 13, 19, 29, 72	nmolb2d 24739	1 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘ 𝐺)) ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   class class class wbr 5143   ∘ ccom 5689  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℝcr 11154  0cc0 11155   · cmul 11160   ≤ cle 11296  Basecbs 17247  0_gc0g 17484   GrpHom cghm 19230  normcnm 24589  NrmGrpcngp 24590   normOp cnmo 24726   NGHom cnghm 24727

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ico 13393  df-0g 17486  df-topgen 17488  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-ghm 19231  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-xms 24330  df-ms 24331  df-nm 24595  df-ngp 24596  df-nmo 24729  df-nghm 24730

This theorem is referenced by:  nghmco  24759