Theorem nmoco 23807

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 2738	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2738	. 2 ⊢ (norm‘𝑆) = (norm‘𝑆)
4		eqid 2738	. 2 ⊢ (norm‘𝑈) = (norm‘𝑈)
5		eqid 2738	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		nghmrcl1 23802	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
7	6	adantl 481	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8		nghmrcl2 23803	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑈 ∈ NrmGrp)
9	8	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑈 ∈ NrmGrp)
10		nghmghm 23804	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝐹 ∈ (𝑇 GrpHom 𝑈))
11		nghmghm 23804	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
12		ghmco 18769	. . 3 ⊢ ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈))
13	10, 11, 12	syl2an 595	. 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 23797	. . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → (𝐿‘𝐹) ∈ ℝ)
16		nmoco.3	. . . 4 ⊢ 𝑀 = (𝑆 normOp 𝑇)
17	16	nghmcl 23797	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑀‘𝐺) ∈ ℝ)
18		remulcl 10887	. . 3 ⊢ (((𝐿‘𝐹) ∈ ℝ ∧ (𝑀‘𝐺) ∈ ℝ) → ((𝐿‘𝐹) · (𝑀‘𝐺)) ∈ ℝ)
19	15, 17, 18	syl2an 595	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝐿‘𝐹) · (𝑀‘𝐺)) ∈ ℝ)
20		nghmrcl1 23802	. . . . 5 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑇 ∈ NrmGrp)
21	14	nmoge0 23791	. . . . 5 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑈 ∈ NrmGrp ∧ 𝐹 ∈ (𝑇 GrpHom 𝑈)) → 0 ≤ (𝐿‘𝐹))
22	20, 8, 10, 21	syl3anc 1369	. . . 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 23803	. . . . 5 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
25	16	nmoge0 23791	. . . . 5 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑀‘𝐺))
26	6, 24, 11, 25	syl3anc 1369	. . . 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 11423	. . 3 ⊢ ((((𝐿‘𝐹) ∈ ℝ ∧ 0 ≤ (𝐿‘𝐹)) ∧ ((𝑀‘𝐺) ∈ ℝ ∧ 0 ≤ (𝑀‘𝐺))) → 0 ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))
29	23, 27, 28	syl2an 595	. 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))
30	8	ad2antrr 722	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑈 ∈ NrmGrp)
31	10	ad2antrr 722	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 ∈ (𝑇 GrpHom 𝑈))
32		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
33		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
34	32, 33	ghmf 18753	. . . . . . 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 723	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
37	2, 32	ghmf 18753	. . . . . . . 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 767	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
40	38, 39	ffvelrnd 6944	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐺‘𝑥) ∈ (Base‘𝑇))
41	35, 40	ffvelrnd 6944	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐹‘(𝐺‘𝑥)) ∈ (Base‘𝑈))
42	33, 4	nmcl 23678	. . . . 5 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐹‘(𝐺‘𝑥)) ∈ (Base‘𝑈)) → ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))) ∈ ℝ)
43	30, 41, 42	syl2anc 583	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑈)‘(𝐹‘(𝐺‘𝑥))) ∈ ℝ)
44	15	ad2antrr 722	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐿‘𝐹) ∈ ℝ)
45	20	ad2antrr 722	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmGrp)
46		eqid 2738	. . . . . . 7 ⊢ (norm‘𝑇) = (norm‘𝑇)
47	32, 46	nmcl 23678	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
48	45, 40, 47	syl2anc 583	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
49	44, 48	remulcld 10936	. . . 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 723	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑀‘𝐺) ∈ ℝ)
51	2, 3	nmcl 23678	. . . . . . . 8 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
52	6, 51	sylan 579	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
53	52	ad2ant2lr 744	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
54	50, 53	remulcld 10936	. . . . 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 10936	. . . 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 763	. . . . 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 23798	. . . . 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 583	. . . 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 722	. . . . 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 23798	. . . . . 6 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ≤ ((𝑀‘𝐺) · ((norm‘𝑆)‘𝑥)))
61	60	ad2ant2lr 744	. . . . 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 11760	. . . . 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 1371	. . . 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 11062	. . 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 6849	. . . . 5 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
66	38, 39, 65	syl2anc 583	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
67	66	fveq2d 6760	. . 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 10934	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐿‘𝐹) ∈ ℂ)
69	50	recnd 10934	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑀‘𝐺) ∈ ℂ)
70	53	recnd 10934	. . . 4 ⊢ (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
71	68, 69, 70	mulassd 10929	. . 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 5102	. 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 23788	1 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘ 𝐺)) ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   class class class wbr 5070   ∘ ccom 5584  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ℝcr 10801  0cc0 10802   · cmul 10807   ≤ cle 10941  Basecbs 16840  0_gc0g 17067   GrpHom cghm 18746  normcnm 23638  NrmGrpcngp 23639   normOp cnmo 23775   NGHom cnghm 23776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ico 13014  df-0g 17069  df-topgen 17071  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-grp 18495  df-ghm 18747  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-xms 23381  df-ms 23382  df-nm 23644  df-ngp 23645  df-nmo 23778  df-nghm 23779

This theorem is referenced by:  nghmco  23808