Theorem nmotri 23031

Description: Triangle inequality for the operator norm. (Contributed by Mario Carneiro, 20-Oct-2015.)

Hypotheses

Ref	Expression
nmotri.1	⊢ 𝑁 = (𝑆 normOp 𝑇)
nmotri.p	⊢ + = (+g‘𝑇)

Assertion

Ref	Expression
nmotri	⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘𝑓 + 𝐺)) ≤ ((𝑁‘𝐹) + (𝑁‘𝐺)))

Proof of Theorem nmotri

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmotri.1	. 2 ⊢ 𝑁 = (𝑆 normOp 𝑇)
2		eqid 2794	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2794	. 2 ⊢ (norm‘𝑆) = (norm‘𝑆)
4		eqid 2794	. 2 ⊢ (norm‘𝑇) = (norm‘𝑇)
5		eqid 2794	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		nghmrcl1 23024	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
7	6	3ad2ant2 1127	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8		nghmrcl2 23025	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
9	8	3ad2ant2 1127	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑇 ∈ NrmGrp)
10		id 22	. . 3 ⊢ (𝑇 ∈ Abel → 𝑇 ∈ Abel)
11		nghmghm 23026	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
12		nghmghm 23026	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
13		nmotri.p	. . . 4 ⊢ + = (+g‘𝑇)
14	13	ghmplusg 18689	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘𝑓 + 𝐺) ∈ (𝑆 GrpHom 𝑇))
15	10, 11, 12, 14	syl3an 1153	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘𝑓 + 𝐺) ∈ (𝑆 GrpHom 𝑇))
16	1	nghmcl 23019	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐹) ∈ ℝ)
17	16	3ad2ant2 1127	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ)
18	1	nghmcl 23019	. . . 4 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐺) ∈ ℝ)
19	18	3ad2ant3 1128	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘𝐺) ∈ ℝ)
20	17, 19	readdcld 10519	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝑁‘𝐹) + (𝑁‘𝐺)) ∈ ℝ)
21	11	3ad2ant2 1127	. . . 4 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
22	1	nmoge0 23013	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
23	7, 9, 21, 22	syl3anc 1364	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
24	12	3ad2ant3 1128	. . . 4 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
25	1	nmoge0 23013	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐺))
26	7, 9, 24, 25	syl3anc 1364	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁‘𝐺))
27	17, 19, 23, 26	addge0d 11066	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ ((𝑁‘𝐹) + (𝑁‘𝐺)))
28	9	adantr 481	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmGrp)
29		ngpgrp 22891	. . . . . . 7 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
30	28, 29	syl 17	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑇 ∈ Grp)
31	21	adantr 481	. . . . . . . 8 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
32		eqid 2794	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	2, 32	ghmf 18103	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
34	31, 33	syl 17	. . . . . . 7 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
35		simprl 767	. . . . . . 7 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
36	34, 35	ffvelrnd 6720	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐹‘𝑥) ∈ (Base‘𝑇))
37	24	adantr 481	. . . . . . . 8 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
38	2, 32	ghmf 18103	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
39	37, 38	syl 17	. . . . . . 7 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
40	39, 35	ffvelrnd 6720	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐺‘𝑥) ∈ (Base‘𝑇))
41	32, 13	grpcl 17869	. . . . . 6 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇) ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((𝐹‘𝑥) + (𝐺‘𝑥)) ∈ (Base‘𝑇))
42	30, 36, 40, 41	syl3anc 1364	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝐹‘𝑥) + (𝐺‘𝑥)) ∈ (Base‘𝑇))
43	32, 4	nmcl 22908	. . . . 5 ⊢ ((𝑇 ∈ NrmGrp ∧ ((𝐹‘𝑥) + (𝐺‘𝑥)) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ ℝ)
44	28, 42, 43	syl2anc 584	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ ℝ)
45	32, 4	nmcl 22908	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹‘𝑥)) ∈ ℝ)
46	28, 36, 45	syl2anc 584	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘(𝐹‘𝑥)) ∈ ℝ)
47	32, 4	nmcl 22908	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
48	28, 40, 47	syl2anc 584	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
49	46, 48	readdcld 10519	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (((norm‘𝑇)‘(𝐹‘𝑥)) + ((norm‘𝑇)‘(𝐺‘𝑥))) ∈ ℝ)
50	17	adantr 481	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐹) ∈ ℝ)
51		simpl 483	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
52	2, 3	nmcl 22908	. . . . . . 7 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
53	7, 51, 52	syl2an 595	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
54	50, 53	remulcld 10520	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝑁‘𝐹) · ((norm‘𝑆)‘𝑥)) ∈ ℝ)
55	19	adantr 481	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐺) ∈ ℝ)
56	55, 53	remulcld 10520	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝑁‘𝐺) · ((norm‘𝑆)‘𝑥)) ∈ ℝ)
57	54, 56	readdcld 10519	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (((𝑁‘𝐹) · ((norm‘𝑆)‘𝑥)) + ((𝑁‘𝐺) · ((norm‘𝑆)‘𝑥))) ∈ ℝ)
58	32, 4, 13	nmtri 22918	. . . . 5 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇) ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹‘𝑥) + (𝐺‘𝑥))) ≤ (((norm‘𝑇)‘(𝐹‘𝑥)) + ((norm‘𝑇)‘(𝐺‘𝑥))))
59	28, 36, 40, 58	syl3anc 1364	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘((𝐹‘𝑥) + (𝐺‘𝑥))) ≤ (((norm‘𝑇)‘(𝐹‘𝑥)) + ((norm‘𝑇)‘(𝐺‘𝑥))))
60		simpl2 1185	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 ∈ (𝑆 NGHom 𝑇))
61	1, 2, 3, 4	nmoi 23020	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹‘𝑥)) ≤ ((𝑁‘𝐹) · ((norm‘𝑆)‘𝑥)))
62	60, 35, 61	syl2anc 584	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘(𝐹‘𝑥)) ≤ ((𝑁‘𝐹) · ((norm‘𝑆)‘𝑥)))
63		simpl3 1186	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺 ∈ (𝑆 NGHom 𝑇))
64	1, 2, 3, 4	nmoi 23020	. . . . . 6 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ≤ ((𝑁‘𝐺) · ((norm‘𝑆)‘𝑥)))
65	63, 35, 64	syl2anc 584	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘(𝐺‘𝑥)) ≤ ((𝑁‘𝐺) · ((norm‘𝑆)‘𝑥)))
66	46, 48, 54, 56, 62, 65	le2addd 11109	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (((norm‘𝑇)‘(𝐹‘𝑥)) + ((norm‘𝑇)‘(𝐺‘𝑥))) ≤ (((𝑁‘𝐹) · ((norm‘𝑆)‘𝑥)) + ((𝑁‘𝐺) · ((norm‘𝑆)‘𝑥))))
67	44, 49, 57, 59, 66	letrd 10646	. . 3 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘((𝐹‘𝑥) + (𝐺‘𝑥))) ≤ (((𝑁‘𝐹) · ((norm‘𝑆)‘𝑥)) + ((𝑁‘𝐺) · ((norm‘𝑆)‘𝑥))))
68	34	ffnd 6386	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 Fn (Base‘𝑆))
69	39	ffnd 6386	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺 Fn (Base‘𝑆))
70		fvexd 6556	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (Base‘𝑆) ∈ V)
71		fnfvof 7284	. . . . 5 ⊢ (((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) ∧ ((Base‘𝑆) ∈ V ∧ 𝑥 ∈ (Base‘𝑆))) → ((𝐹 ∘𝑓 + 𝐺)‘𝑥) = ((𝐹‘𝑥) + (𝐺‘𝑥)))
72	68, 69, 70, 35, 71	syl22anc 835	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝐹 ∘𝑓 + 𝐺)‘𝑥) = ((𝐹‘𝑥) + (𝐺‘𝑥)))
73	72	fveq2d 6545	. . 3 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘((𝐹 ∘𝑓 + 𝐺)‘𝑥)) = ((norm‘𝑇)‘((𝐹‘𝑥) + (𝐺‘𝑥))))
74	50	recnd 10518	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐹) ∈ ℂ)
75	55	recnd 10518	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐺) ∈ ℂ)
76	53	recnd 10518	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
77	74, 75, 76	adddird 10515	. . 3 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (((𝑁‘𝐹) + (𝑁‘𝐺)) · ((norm‘𝑆)‘𝑥)) = (((𝑁‘𝐹) · ((norm‘𝑆)‘𝑥)) + ((𝑁‘𝐺) · ((norm‘𝑆)‘𝑥))))
78	67, 73, 77	3brtr4d 4996	. 2 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘((𝐹 ∘𝑓 + 𝐺)‘𝑥)) ≤ (((𝑁‘𝐹) + (𝑁‘𝐺)) · ((norm‘𝑆)‘𝑥)))
79	1, 2, 3, 4, 5, 7, 9, 15, 20, 27, 78	nmolb2d 23010	1 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘𝑓 + 𝐺)) ≤ ((𝑁‘𝐹) + (𝑁‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2080   ≠ wne 2983  Vcvv 3436   class class class wbr 4964   Fn wfn 6223  ⟶wf 6224  ‘cfv 6228  (class class class)co 7019   ∘_𝑓 cof 7268  ℝcr 10385  0cc0 10386   + caddc 10389   · cmul 10391   ≤ cle 10525  Basecbs 16312  +_gcplusg 16394  0_gc0g 16542  Grpcgrp 17861   GrpHom cghm 18096  Abelcabl 18634  normcnm 22869  NrmGrpcngp 22870   normOp cnmo 22997   NGHom cnghm 22998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322  ax-cnex 10442  ax-resscn 10443  ax-1cn 10444  ax-icn 10445  ax-addcl 10446  ax-addrcl 10447  ax-mulcl 10448  ax-mulrcl 10449  ax-mulcom 10450  ax-addass 10451  ax-mulass 10452  ax-distr 10453  ax-i2m1 10454  ax-1ne0 10455  ax-1rid 10456  ax-rnegex 10457  ax-rrecex 10458  ax-cnre 10459  ax-pre-lttri 10460  ax-pre-lttrn 10461  ax-pre-ltadd 10462  ax-pre-mulgt0 10463  ax-pre-sup 10464

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-nel 3090  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-of 7270  df-om 7440  df-1st 7548  df-2nd 7549  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-er 8142  df-map 8261  df-en 8361  df-dom 8362  df-sdom 8363  df-sup 8755  df-inf 8756  df-pnf 10526  df-mnf 10527  df-xr 10528  df-ltxr 10529  df-le 10530  df-sub 10721  df-neg 10722  df-div 11148  df-nn 11489  df-2 11550  df-n0 11748  df-z 11832  df-uz 12094  df-q 12198  df-rp 12240  df-xneg 12357  df-xadd 12358  df-xmul 12359  df-ico 12594  df-0g 16544  df-topgen 16546  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-grp 17864  df-minusg 17865  df-sbg 17866  df-ghm 18097  df-cmn 18635  df-abl 18636  df-psmet 20219  df-xmet 20220  df-met 20221  df-bl 20222  df-mopn 20223  df-top 21186  df-topon 21203  df-topsp 21225  df-bases 21238  df-xms 22613  df-ms 22614  df-nm 22875  df-ngp 22876  df-nmo 23000  df-nghm 23001

This theorem is referenced by:  nghmplusg  23032