Theorem nmoco 23338

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 2819	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2819	. 2 ⊢ (norm‘𝑆) = (norm‘𝑆)
4		eqid 2819	. 2 ⊢ (norm‘𝑈) = (norm‘𝑈)
5		eqid 2819	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		nghmrcl1 23333	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
7	6	adantl 484	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8		nghmrcl2 23334	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑈 ∈ NrmGrp)
9	8	adantr 483	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑈 ∈ NrmGrp)
10		nghmghm 23335	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝐹 ∈ (𝑇 GrpHom 𝑈))
11		nghmghm 23335	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
12		ghmco 18370	. . 3 ⊢ ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈))
13	10, 11, 12	syl2an 597	. 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 23328	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → (𝐿‘𝐹) ∈ ℝ)
16		nmoco.3	. . . 4 ⊢ 𝑀 = (𝑆 normOp 𝑇)
17	16	nghmcl 23328	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑀‘𝐺) ∈ ℝ)
18		remulcl 10614	. . 3 ⊢ (((𝐿‘𝐹) ∈ ℝ ∧ (𝑀‘𝐺) ∈ ℝ) → ((𝐿‘𝐹) · (𝑀‘𝐺)) ∈ ℝ)
19	15, 17, 18	syl2an 597	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝐿‘𝐹) · (𝑀‘𝐺)) ∈ ℝ)
20		nghmrcl1 23333	. . . . 5 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑇 ∈ NrmGrp)
21	14	nmoge0 23322	. . . . 5 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑈 ∈ NrmGrp ∧ 𝐹 ∈ (𝑇 GrpHom 𝑈)) → 0 ≤ (𝐿‘𝐹))
22	20, 8, 10, 21	syl3anc 1366	. . . 4 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 0 ≤ (𝐿‘𝐹))
23	15, 22	jca 514	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → ((𝐿‘𝐹) ∈ ℝ ∧ 0 ≤ (𝐿‘𝐹)))
24		nghmrcl2 23334	. . . . 5 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
25	16	nmoge0 23322	. . . . 5 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑀‘𝐺))
26	6, 24, 11, 25	syl3anc 1366	. . . 4 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 0 ≤ (𝑀‘𝐺))
27	17, 26	jca 514	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → ((𝑀‘𝐺) ∈ ℝ ∧ 0 ≤ (𝑀‘𝐺)))
28		mulge0 11150	. . 3 ⊢ ((((𝐿‘𝐹) ∈ ℝ ∧ 0 ≤ (𝐿‘𝐹)) ∧ ((𝑀‘𝐺) ∈ ℝ ∧ 0 ≤ (𝑀‘𝐺))) → 0 ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))
29	23, 27, 28	syl2an 597	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))
30	8	ad2antrr 724	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑈 ∈ NrmGrp)
31	10	ad2antrr 724	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 ∈ (𝑇 GrpHom 𝑈))
32		eqid 2819	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
33		eqid 2819	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
34	32, 33	ghmf 18354	. . . . . . 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 725	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
37	2, 32	ghmf 18354	. . . . . . . 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 769	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
40	38, 39	ffvelrnd 6845	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐺‘𝑥) ∈ (Base‘𝑇))
41	35, 40	ffvelrnd 6845	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐹‘(𝐺‘𝑥)) ∈ (Base‘𝑈))
42	33, 4	nmcl 23217	. . . . 5 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐹‘(𝐺‘𝑥)) ∈ (Base‘𝑈)) → ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))) ∈ ℝ)
43	30, 41, 42	syl2anc 586	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))) ∈ ℝ)
44	15	ad2antrr 724	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐿‘𝐹) ∈ ℝ)
45	20	ad2antrr 724	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmGrp)
46		eqid 2819	. . . . . . 7 ⊢ (norm‘𝑇) = (norm‘𝑇)
47	32, 46	nmcl 23217	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
48	45, 40, 47	syl2anc 586	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
49	44, 48	remulcld 10663	. . . 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 725	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑀‘𝐺) ∈ ℝ)
51	2, 3	nmcl 23217	. . . . . . . 8 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
52	6, 51	sylan 582	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
53	52	ad2ant2lr 746	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
54	50, 53	remulcld 10663	. . . . 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 10663	. . . 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 765	. . . . 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 23329	. . . . 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 586	. . . 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 724	. . . . 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 23329	. . . . . 6 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ≤ ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥)))
61	60	ad2ant2lr 746	. . . . 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 11487	. . . . 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 1368	. . . 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 10789	. . 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 6753	. . . . 5 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
66	38, 39, 65	syl2anc 586	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
67	66	fveq2d 6667	. . 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 10661	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐿‘𝐹) ∈ ℂ)
69	50	recnd 10661	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑀‘𝐺) ∈ ℂ)
70	53	recnd 10661	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
71	68, 69, 70	mulassd 10656	. . 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 5089	. 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 23319	1 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘ 𝐺)) ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108   ≠ wne 3014   class class class wbr 5057   ∘ ccom 5552  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148  ℝcr 10528  0cc0 10529   · cmul 10534   ≤ cle 10668  Basecbs 16475  0_gc0g 16705   GrpHom cghm 18347  normcnm 23178  NrmGrpcngp 23179   normOp cnmo 23306   NGHom cnghm 23307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-sup 8898  df-inf 8899  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ico 12736  df-0g 16707  df-topgen 16709  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-grp 18098  df-ghm 18348  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-top 21494  df-topon 21511  df-topsp 21533  df-bases 21546  df-xms 22922  df-ms 22923  df-nm 23184  df-ngp 23185  df-nmo 23309  df-nghm 23310

This theorem is referenced by:  nghmco  23339