Theorem nmotri 24103

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 2736	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2736	. 2 ⊢ (norm‘𝑆) = (norm‘𝑆)
4		eqid 2736	. 2 ⊢ (norm‘𝑇) = (norm‘𝑇)
5		eqid 2736	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		nghmrcl1 24096	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
7	6	3ad2ant2 1134	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8		nghmrcl2 24097	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
9	8	3ad2ant2 1134	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑇 ∈ NrmGrp)
10		id 22	. . 3 ⊢ (𝑇 ∈ Abel → 𝑇 ∈ Abel)
11		nghmghm 24098	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
12		nghmghm 24098	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
13		nmotri.p	. . . 4 ⊢ + = (+g‘𝑇)
14	13	ghmplusg 19624	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
15	10, 11, 12, 14	syl3an 1160	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
16	1	nghmcl 24091	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐹) ∈ ℝ)
17	16	3ad2ant2 1134	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ)
18	1	nghmcl 24091	. . . 4 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐺) ∈ ℝ)
19	18	3ad2ant3 1135	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘𝐺) ∈ ℝ)
20	17, 19	readdcld 11184	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝑁‘𝐹) + (𝑁‘𝐺)) ∈ ℝ)
21	11	3ad2ant2 1134	. . . 4 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
22	1	nmoge0 24085	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
23	7, 9, 21, 22	syl3anc 1371	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
24	12	3ad2ant3 1135	. . . 4 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
25	1	nmoge0 24085	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐺))
26	7, 9, 24, 25	syl3anc 1371	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁‘𝐺))
27	17, 19, 23, 26	addge0d 11731	. 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 23955	. . . . . . 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 2736	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	2, 32	ghmf 19012	. . . . . . . 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 769	. . . . . . 7 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
36	34, 35	ffvelcdmd 7036	. . . . . 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 19012	. . . . . . . 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	ffvelcdmd 7036	. . . . . 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 18756	. . . . . 6 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇) ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((𝐹‘𝑥) + (𝐺‘𝑥)) ∈ (Base‘𝑇))
42	30, 36, 40, 41	syl3anc 1371	. . . . 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 23972	. . . . 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 23972	. . . . . 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 23972	. . . . . 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 11184	. . . 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 23972	. . . . . . 7 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
53	7, 51, 52	syl2an 596	. . . . . 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 11185	. . . . 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 11185	. . . . 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 11184	. . . 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 23982	. . . . 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 1371	. . . 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 1192	. . . . . 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 24092	. . . . . 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 1193	. . . . . 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 24092	. . . . . 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 11774	. . . 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 11312	. . 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 6669	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 Fn (Base‘𝑆))
69	39	ffnd 6669	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺 Fn (Base‘𝑆))
70		fvexd 6857	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (Base‘𝑆) ∈ V)
71		fnfvof 7634	. . . . 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 837	. . . 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 6846	. . 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 11183	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐹) ∈ ℂ)
75	55	recnd 11183	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐺) ∈ ℂ)
76	53	recnd 11183	. . . 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 11180	. . 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 5137	. 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 24082	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 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  Vcvv 3445   class class class wbr 5105   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∘_f cof 7615  ℝcr 11050  0cc0 11051   + caddc 11054   · cmul 11056   ≤ cle 11190  Basecbs 17083  +_gcplusg 17133  0_gc0g 17321  Grpcgrp 18748   GrpHom cghm 19005  Abelcabl 19563  normcnm 23932  NrmGrpcngp 23933   normOp cnmo 24069   NGHom cnghm 24070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ico 13270  df-0g 17323  df-topgen 17325  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-ghm 19006  df-cmn 19564  df-abl 19565  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-xms 23673  df-ms 23674  df-nm 23938  df-ngp 23939  df-nmo 24072  df-nghm 24073

This theorem is referenced by:  nghmplusg  24104