Theorem nmoco 25065

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 2760	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2760	. 2 ⊢ (norm‘𝑆) = (norm‘𝑆)
4		eqid 2760	. 2 ⊢ (norm‘𝑈) = (norm‘𝑈)
5		eqid 2760	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		nghmrcl1 25060	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
7	6	adantl 487	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8		nghmrcl2 25061	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑈 ∈ NrmGrp)
9	8	adantr 486	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑈 ∈ NrmGrp)
10		nghmghm 25062	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝐹 ∈ (𝑇 GrpHom 𝑈))
11		nghmghm 25062	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
12		ghmco 19459	. . 3 ⊢ ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈))
13	10, 11, 12	syl2an 608	. 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 25055	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → (𝐿‘𝐹) ∈ ℝ)
16		nmoco.3	. . . 4 ⊢ 𝑀 = (𝑆 normOp 𝑇)
17	16	nghmcl 25055	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑀‘𝐺) ∈ ℝ)
18		remulcl 11292	. . 3 ⊢ (((𝐿‘𝐹) ∈ ℝ ∧ (𝑀‘𝐺) ∈ ℝ) → ((𝐿‘𝐹) · (𝑀‘𝐺)) ∈ ℝ)
19	15, 17, 18	syl2an 608	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝐿‘𝐹) · (𝑀‘𝐺)) ∈ ℝ)
20		nghmrcl1 25060	. . . . 5 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑇 ∈ NrmGrp)
21	14	nmoge0 25049	. . . . 5 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑈 ∈ NrmGrp ∧ 𝐹 ∈ (𝑇 GrpHom 𝑈)) → 0 ≤ (𝐿‘𝐹))
22	20, 8, 10, 21	syl3anc 1398	. . . 4 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 0 ≤ (𝐿‘𝐹))
23	15, 22	jca 521	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → ((𝐿‘𝐹) ∈ ℝ ∧ 0 ≤ (𝐿‘𝐹)))
24		nghmrcl2 25061	. . . . 5 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
25	16	nmoge0 25049	. . . . 5 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑀‘𝐺))
26	6, 24, 11, 25	syl3anc 1398	. . . 4 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 0 ≤ (𝑀‘𝐺))
27	17, 26	jca 521	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → ((𝑀‘𝐺) ∈ ℝ ∧ 0 ≤ (𝑀‘𝐺)))
28		mulge0 11841	. . 3 ⊢ ((((𝐿‘𝐹) ∈ ℝ ∧ 0 ≤ (𝐿‘𝐹)) ∧ ((𝑀‘𝐺) ∈ ℝ ∧ 0 ≤ (𝑀‘𝐺))) → 0 ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))
29	23, 27, 28	syl2an 608	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))
30	8	ad2antrr 739	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑈 ∈ NrmGrp)
31	10	ad2antrr 739	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 ∈ (𝑇 GrpHom 𝑈))
32		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
33		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
34	32, 33	ghmf 19443	. . . . . . 7 ⊢ (𝐹 ∈ (𝑇 GrpHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
35	31, 34	syl 18	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
36	11	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
37	2, 32	ghmf 19443	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
38	36, 37	syl 18	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
39		simprl 783	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
40	38, 39	ffvelcdmd 7090	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐺‘𝑥) ∈ (Base‘𝑇))
41	35, 40	ffvelcdmd 7090	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐹‘(𝐺‘𝑥)) ∈ (Base‘𝑈))
42	33, 4	nmcl 24944	. . . . 5 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐹‘(𝐺‘𝑥)) ∈ (Base‘𝑈)) → ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))) ∈ ℝ)
43	30, 41, 42	syl2anc 596	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))) ∈ ℝ)
44	15	ad2antrr 739	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐿‘𝐹) ∈ ℝ)
45	20	ad2antrr 739	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmGrp)
46		eqid 2760	. . . . . . 7 ⊢ (norm‘𝑇) = (norm‘𝑇)
47	32, 46	nmcl 24944	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
48	45, 40, 47	syl2anc 596	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
49	44, 48	remulcld 11346	. . . 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 740	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑀‘𝐺) ∈ ℝ)
51	2, 3	nmcl 24944	. . . . . . . 8 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
52	6, 51	sylan 592	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
53	52	ad2ant2lr 761	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
54	50, 53	remulcld 11346	. . . . 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 11346	. . . 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 779	. . . . 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 25056	. . . . 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 596	. . . 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 739	. . . . 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 25056	. . . . . 6 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ≤ ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥)))
61	60	ad2ant2lr 761	. . . . 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 12179	. . . . 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 1400	. . . 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 11474	. . 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 6990	. . . . 5 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
66	38, 39, 65	syl2anc 596	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
67	66	fveq2d 6894	. . 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 11344	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐿‘𝐹) ∈ ℂ)
69	50	recnd 11344	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑀‘𝐺) ∈ ℂ)
70	53	recnd 11344	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
71	68, 69, 70	mulassd 11339	. . 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 5136	. 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 25046	1 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘ 𝐺)) ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   class class class wbr 5102   ∘ ccom 5655  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424  ℝcr 11206  0cc0 11207   · cmul 11212   ≤ cle 11351  Basecbs 17396  0_gc0g 17619   GrpHom cghm 19436  normcnm 24904  NrmGrpcngp 24905   normOp cnmo 25033   NGHom cnghm 25034

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-er 8724  df-map 8856  df-en 8981  df-dom 8982  df-sdom 8983  df-sup 9441  df-inf 9442  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-div 11981  df-nn 12343  df-2 12412  df-n0 12614  df-z 12701  df-uz 12973  df-q 13083  df-rp 13128  df-xneg 13248  df-xadd 13249  df-xmul 13250  df-ico 13489  df-0g 17621  df-topgen 17623  df-mgm 18825  df-sgrp 18917  df-mnd 18933  df-mhm 18987  df-grp 19156  df-ghm 19437  df-psmet 21679  df-xmet 21680  df-met 21681  df-bl 21682  df-mopn 21683  df-top 23221  df-topon 23238  df-topsp 23260  df-bases 23273  df-xms 24648  df-ms 24649  df-nm 24910  df-ngp 24911  df-nmo 25036  df-nghm 25037

This theorem is used by:  nghmco  25066