Theorem nmotri 24949

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 2765	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2765	. 2 ⊢ (norm‘𝑆) = (norm‘𝑆)
4		eqid 2765	. 2 ⊢ (norm‘𝑇) = (norm‘𝑇)
5		eqid 2765	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		nghmrcl1 24942	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
7	6	3ad2ant2 1152	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8		nghmrcl2 24943	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
9	8	3ad2ant2 1152	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑇 ∈ NrmGrp)
10		id 23	. . 3 ⊢ (𝑇 ∈ Abel → 𝑇 ∈ Abel)
11		nghmghm 24944	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
12		nghmghm 24944	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
13		nmotri.p	. . . 4 ⊢ + = (+g‘𝑇)
14	13	ghmplusg 19962	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
15	10, 11, 12, 14	syl3an 1178	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
16	1	nghmcl 24937	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐹) ∈ ℝ)
17	16	3ad2ant2 1152	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ)
18	1	nghmcl 24937	. . . 4 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐺) ∈ ℝ)
19	18	3ad2ant3 1153	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘𝐺) ∈ ℝ)
20	17, 19	readdcld 11255	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝑁‘𝐹) + (𝑁‘𝐺)) ∈ ℝ)
21	11	3ad2ant2 1152	. . . 4 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
22	1	nmoge0 24931	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
23	7, 9, 21, 22	syl3anc 1398	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
24	12	3ad2ant3 1153	. . . 4 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
25	1	nmoge0 24931	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐺))
26	7, 9, 24, 25	syl3anc 1398	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁‘𝐺))
27	17, 19, 23, 26	addge0d 11807	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ ((𝑁‘𝐹) + (𝑁‘𝐺)))
28	9	adantr 486	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmGrp)
29		ngpgrp 24809	. . . . . . 7 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
30	28, 29	syl 18	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑇 ∈ Grp)
31	21	adantr 486	. . . . . . . 8 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
32		eqid 2765	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	2, 32	ghmf 19336	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
34	31, 33	syl 18	. . . . . . 7 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
35		simprl 783	. . . . . . 7 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
36	34, 35	ffvelcdmd 7084	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐹‘𝑥) ∈ (Base‘𝑇))
37	24	adantr 486	. . . . . . . 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 19336	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
39	37, 38	syl 18	. . . . . . 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	ffvelcdmd 7084	. . . . . 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 19054	. . . . . 6 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇) ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((𝐹‘𝑥) + (𝐺‘𝑥)) ∈ (Base‘𝑇))
42	30, 36, 40, 41	syl3anc 1398	. . . . 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 24826	. . . . 5 ⊢ ((𝑇 ∈ NrmGrp ∧ ((𝐹‘𝑥) + (𝐺‘𝑥)) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ ℝ)
44	28, 42, 43	syl2anc 596	. . . 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 24826	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹‘𝑥)) ∈ ℝ)
46	28, 36, 45	syl2anc 596	. . . . 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 24826	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
48	28, 40, 47	syl2anc 596	. . . . 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 11255	. . . 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 486	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐹) ∈ ℝ)
51		simpl 488	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
52	2, 3	nmcl 24826	. . . . . . 7 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
53	7, 51, 52	syl2an 608	. . . . . 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 11256	. . . . 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 486	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐺) ∈ ℝ)
56	55, 53	remulcld 11256	. . . . 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 11255	. . . 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 24836	. . . . 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 1398	. . . 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 1211	. . . . . 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 24938	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹‘𝑥)) ≤ ((𝑁‘𝐹) · ((norm‘𝑆)‘𝑥)))
62	60, 35, 61	syl2anc 596	. . . . 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 1212	. . . . . 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 24938	. . . . . 6 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ≤ ((𝑁‘𝐺) · ((norm‘𝑆)‘𝑥)))
65	63, 35, 64	syl2anc 596	. . . . 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 11850	. . . 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 11384	. . 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 6710	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 Fn (Base‘𝑆))
69	39	ffnd 6710	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺 Fn (Base‘𝑆))
70		fvexd 6900	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (Base‘𝑆) ∈ V)
71		fnfvof 7701	. . . . 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 852	. . . 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 6889	. . 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 11254	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐹) ∈ ℂ)
75	55	recnd 11254	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐺) ∈ ℂ)
76	53	recnd 11254	. . . 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 11251	. . 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 5145	. 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 24928	1 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘f + 𝐺)) ≤ ((𝑁‘𝐹) + (𝑁‘𝐺)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457   class class class wbr 5111   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ∘_f cof 7682  ℝcr 11116  0cc0 11117   + caddc 11120   · cmul 11122   ≤ cle 11261  Basecbs 17293  +_gcplusg 17334  0_gc0g 17516  Grpcgrp 19046   GrpHom cghm 19329  Abelcabl 19897  normcnm 24786  NrmGrpcngp 24787   normOp cnmo 24915   NGHom cnghm 24916

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194  ax-pre-sup 11195

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-sup 9409  df-inf 9410  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-div 11889  df-nn 12251  df-2 12320  df-n0 12522  df-z 12609  df-uz 12881  df-q 12991  df-rp 13035  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ico 13396  df-0g 17518  df-topgen 17520  df-mgm 18722  df-sgrp 18811  df-mnd 18827  df-grp 19049  df-minusg 19050  df-sbg 19051  df-ghm 19330  df-cmn 19898  df-abl 19899  df-psmet 21566  df-xmet 21567  df-met 21568  df-bl 21569  df-mopn 21570  df-top 23103  df-topon 23120  df-topsp 23142  df-bases 23155  df-xms 24530  df-ms 24531  df-nm 24792  df-ngp 24793  df-nmo 24918  df-nghm 24919

This theorem is used by:  nghmplusg  24950