Theorem nmotri 24865

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 2769	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2769	. 2 ⊢ (norm‘𝑆) = (norm‘𝑆)
4		eqid 2769	. 2 ⊢ (norm‘𝑇) = (norm‘𝑇)
5		eqid 2769	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		nghmrcl1 24858	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
7	6	3ad2ant2 1150	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8		nghmrcl2 24859	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
9	8	3ad2ant2 1150	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑇 ∈ NrmGrp)
10		id 23	. . 3 ⊢ (𝑇 ∈ Abel → 𝑇 ∈ Abel)
11		nghmghm 24860	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
12		nghmghm 24860	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
13		nmotri.p	. . . 4 ⊢ + = (+g‘𝑇)
14	13	ghmplusg 19916	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
15	10, 11, 12, 14	syl3an 1176	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
16	1	nghmcl 24853	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐹) ∈ ℝ)
17	16	3ad2ant2 1150	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ)
18	1	nghmcl 24853	. . . 4 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐺) ∈ ℝ)
19	18	3ad2ant3 1151	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘𝐺) ∈ ℝ)
20	17, 19	readdcld 11238	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝑁‘𝐹) + (𝑁‘𝐺)) ∈ ℝ)
21	11	3ad2ant2 1150	. . . 4 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
22	1	nmoge0 24847	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
23	7, 9, 21, 22	syl3anc 1396	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
24	12	3ad2ant3 1151	. . . 4 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
25	1	nmoge0 24847	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐺))
26	7, 9, 24, 25	syl3anc 1396	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁‘𝐺))
27	17, 19, 23, 26	addge0d 11790	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ ((𝑁‘𝐹) + (𝑁‘𝐺)))
28	9	adantr 485	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmGrp)
29		ngpgrp 24725	. . . . . . 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 485	. . . . . . . 8 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
32		eqid 2769	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	2, 32	ghmf 19290	. . . . . . . 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 782	. . . . . . 7 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
36	34, 35	ffvelcdmd 7081	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐹‘𝑥) ∈ (Base‘𝑇))
37	24	adantr 485	. . . . . . . 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 19290	. . . . . . . 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 7081	. . . . . 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 19008	. . . . . 6 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇) ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((𝐹‘𝑥) + (𝐺‘𝑥)) ∈ (Base‘𝑇))
42	30, 36, 40, 41	syl3anc 1396	. . . . 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 24742	. . . . 5 ⊢ ((𝑇 ∈ NrmGrp ∧ ((𝐹‘𝑥) + (𝐺‘𝑥)) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ ℝ)
44	28, 42, 43	syl2anc 595	. . . 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 24742	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹‘𝑥)) ∈ ℝ)
46	28, 36, 45	syl2anc 595	. . . . 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 24742	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
48	28, 40, 47	syl2anc 595	. . . . 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 11238	. . . 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 485	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐹) ∈ ℝ)
51		simpl 487	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
52	2, 3	nmcl 24742	. . . . . . 7 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
53	7, 51, 52	syl2an 607	. . . . . 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 11239	. . . . 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 485	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐺) ∈ ℝ)
56	55, 53	remulcld 11239	. . . . 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 11238	. . . 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 24752	. . . . 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 1396	. . . 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 1209	. . . . . 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 24854	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹‘𝑥)) ≤ ((𝑁‘𝐹) · ((norm‘𝑆)‘𝑥)))
62	60, 35, 61	syl2anc 595	. . . . 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 1210	. . . . . 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 24854	. . . . . 6 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ≤ ((𝑁‘𝐺) · ((norm‘𝑆)‘𝑥)))
65	63, 35, 64	syl2anc 595	. . . . 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 11833	. . . 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 11367	. . 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 6707	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 Fn (Base‘𝑆))
69	39	ffnd 6707	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺 Fn (Base‘𝑆))
70		fvexd 6897	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (Base‘𝑆) ∈ V)
71		fnfvof 7692	. . . . 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 851	. . . 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 6886	. . 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 11237	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐹) ∈ ℂ)
75	55	recnd 11237	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐺) ∈ ℂ)
76	53	recnd 11237	. . . 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 11234	. . 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 5147	. 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 24844	1 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘f + 𝐺)) ≤ ((𝑁‘𝐹) + (𝑁‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463   class class class wbr 5113   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∘_f cof 7673  ℝcr 11099  0cc0 11100   + caddc 11103   · cmul 11105   ≤ cle 11244  Basecbs 17269  +_gcplusg 17310  0_gc0g 17492  Grpcgrp 19000   GrpHom cghm 19283  Abelcabl 19851  normcnm 24702  NrmGrpcngp 24703   normOp cnmo 24831   NGHom cnghm 24832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-sup 9402  df-inf 9403  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-n0 12505  df-z 12592  df-uz 12863  df-q 12973  df-rp 13017  df-xneg 13137  df-xadd 13138  df-xmul 13139  df-ico 13378  df-0g 17494  df-topgen 17496  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-grp 19003  df-minusg 19004  df-sbg 19005  df-ghm 19284  df-cmn 19852  df-abl 19853  df-psmet 21483  df-xmet 21484  df-met 21485  df-bl 21486  df-mopn 21487  df-top 23020  df-topon 23037  df-topsp 23059  df-bases 23072  df-xms 24446  df-ms 24447  df-nm 24708  df-ngp 24709  df-nmo 24834  df-nghm 24835

This theorem is referenced by:  nghmplusg  24866