Theorem nmoco 25049

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 2761	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2761	. 2 ⊢ (norm‘𝑆) = (norm‘𝑆)
4		eqid 2761	. 2 ⊢ (norm‘𝑈) = (norm‘𝑈)
5		eqid 2761	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		nghmrcl1 25044	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
7	6	adantl 487	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8		nghmrcl2 25045	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑈 ∈ NrmGrp)
9	8	adantr 486	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑈 ∈ NrmGrp)
10		nghmghm 25046	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝐹 ∈ (𝑇 GrpHom 𝑈))
11		nghmghm 25046	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
12		ghmco 19443	. . 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 25039	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → (𝐿‘𝐹) ∈ ℝ)
16		nmoco.3	. . . 4 ⊢ 𝑀 = (𝑆 normOp 𝑇)
17	16	nghmcl 25039	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑀‘𝐺) ∈ ℝ)
18		remulcl 11278	. . 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 25044	. . . . 5 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑇 ∈ NrmGrp)
21	14	nmoge0 25033	. . . . 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 25045	. . . . 5 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
25	16	nmoge0 25033	. . . . 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 11827	. . 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 2761	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
33		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
34	32, 33	ghmf 19427	. . . . . . 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 19427	. . . . . . . 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 7083	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐺‘𝑥) ∈ (Base‘𝑇))
41	35, 40	ffvelcdmd 7083	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐹‘(𝐺‘𝑥)) ∈ (Base‘𝑈))
42	33, 4	nmcl 24928	. . . . 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 2761	. . . . . . 7 ⊢ (norm‘𝑇) = (norm‘𝑇)
47	32, 46	nmcl 24928	. . . . . 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 11332	. . . 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 24928	. . . . . . . 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 11332	. . . . 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 11332	. . . 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 25040	. . . . 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 25040	. . . . . 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 12165	. . . . 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 11460	. . 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 6983	. . . . 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 6887	. . 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 11330	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐿‘𝐹) ∈ ℂ)
69	50	recnd 11330	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑀‘𝐺) ∈ ℂ)
70	53	recnd 11330	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
71	68, 69, 70	mulassd 11325	. . 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 5137	. 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 25030	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 1570   ∈ wcel 2145   ≠ wne 2956   class class class wbr 5103   ∘ ccom 5655  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  ℝcr 11192  0cc0 11193   · cmul 11198   ≤ cle 11337  Basecbs 17380  0_gc0g 17603   GrpHom cghm 19420  normcnm 24888  NrmGrpcngp 24889   normOp cnmo 25017   NGHom cnghm 25018

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-sup 9427  df-inf 9428  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-n0 12600  df-z 12687  df-uz 12959  df-q 13069  df-rp 13114  df-xneg 13234  df-xadd 13235  df-xmul 13236  df-ico 13475  df-0g 17605  df-topgen 17607  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-mhm 18971  df-grp 19140  df-ghm 19421  df-psmet 21663  df-xmet 21664  df-met 21665  df-bl 21666  df-mopn 21667  df-top 23205  df-topon 23222  df-topsp 23244  df-bases 23257  df-xms 24632  df-ms 24633  df-nm 24894  df-ngp 24895  df-nmo 25020  df-nghm 25021

This theorem is used by:  nghmco  25050