Theorem nmotri 22353

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 2610	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2610	. 2 ⊢ (norm‘𝑆) = (norm‘𝑆)
4		eqid 2610	. 2 ⊢ (norm‘𝑇) = (norm‘𝑇)
5		eqid 2610	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		nghmrcl1 22346	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
7	6	3ad2ant2 1076	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8		nghmrcl2 22347	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
9	8	3ad2ant2 1076	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑇 ∈ NrmGrp)
10		id 22	. . 3 ⊢ (𝑇 ∈ Abel → 𝑇 ∈ Abel)
11		nghmghm 22348	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
12		nghmghm 22348	. . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
13		nmotri.p	. . . 4 ⊢ + = (+g‘𝑇)
14	13	ghmplusg 18072	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘𝑓 + 𝐺) ∈ (𝑆 GrpHom 𝑇))
15	10, 11, 12, 14	syl3an 1360	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘𝑓 + 𝐺) ∈ (𝑆 GrpHom 𝑇))
16	1	nghmcl 22341	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐹) ∈ ℝ)
17	16	3ad2ant2 1076	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ)
18	1	nghmcl 22341	. . . 4 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐺) ∈ ℝ)
19	18	3ad2ant3 1077	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘𝐺) ∈ ℝ)
20	17, 19	readdcld 9948	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝑁‘𝐹) + (𝑁‘𝐺)) ∈ ℝ)
21	11	3ad2ant2 1076	. . . 4 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
22	1	nmoge0 22335	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
23	7, 9, 21, 22	syl3anc 1318	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
24	12	3ad2ant3 1077	. . . 4 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
25	1	nmoge0 22335	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐺))
26	7, 9, 24, 25	syl3anc 1318	. . 3 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁‘𝐺))
27	17, 19, 23, 26	addge0d 10482	. 2 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ ((𝑁‘𝐹) + (𝑁‘𝐺)))
28	9	adantr 480	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmGrp)
29		ngpgrp 22213	. . . . . . 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 480	. . . . . . . 8 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
32		eqid 2610	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	2, 32	ghmf 17487	. . . . . . . 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 790	. . . . . . 7 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
36	34, 35	ffvelrnd 6268	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝐹‘𝑥) ∈ (Base‘𝑇))
37	24	adantr 480	. . . . . . . 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 17487	. . . . . . . 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 6268	. . . . . 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 17253	. . . . . 6 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇) ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((𝐹‘𝑥) + (𝐺‘𝑥)) ∈ (Base‘𝑇))
42	30, 36, 40, 41	syl3anc 1318	. . . . 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 22230	. . . . 5 ⊢ ((𝑇 ∈ NrmGrp ∧ ((𝐹‘𝑥) + (𝐺‘𝑥)) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ ℝ)
44	28, 42, 43	syl2anc 691	. . . 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 22230	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹‘𝑥)) ∈ ℝ)
46	28, 36, 45	syl2anc 691	. . . . 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 22230	. . . . . 6 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐺‘𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ∈ ℝ)
48	28, 40, 47	syl2anc 691	. . . . 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 9948	. . . 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 480	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐹) ∈ ℝ)
51		simpl 472	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
52	2, 3	nmcl 22230	. . . . . . 7 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
53	7, 51, 52	syl2an 493	. . . . . 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 9949	. . . . 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 480	. . . . . 6 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐺) ∈ ℝ)
56	55, 53	remulcld 9949	. . . . 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 9948	. . . 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 22240	. . . . 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 1318	. . . 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 1058	. . . . . 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 22342	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹‘𝑥)) ≤ ((𝑁‘𝐹) · ((norm‘𝑆)‘𝑥)))
62	60, 35, 61	syl2anc 691	. . . . 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 1059	. . . . . 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 22342	. . . . . 6 ⊢ ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺‘𝑥)) ≤ ((𝑁‘𝐺) · ((norm‘𝑆)‘𝑥)))
65	63, 35, 64	syl2anc 691	. . . . 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 10525	. . . 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 10073	. . 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		ffn 5958	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
69	34, 68	syl 17	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐹 Fn (Base‘𝑆))
70		ffn 5958	. . . . . 6 ⊢ (𝐺:(Base‘𝑆)⟶(Base‘𝑇) → 𝐺 Fn (Base‘𝑆))
71	39, 70	syl 17	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → 𝐺 Fn (Base‘𝑆))
72		fvex 6113	. . . . . 6 ⊢ (Base‘𝑆) ∈ V
73	72	a1i 11	. . . . 5 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (Base‘𝑆) ∈ V)
74		fnfvof 6809	. . . . 5 ⊢ (((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) ∧ ((Base‘𝑆) ∈ V ∧ 𝑥 ∈ (Base‘𝑆))) → ((𝐹 ∘𝑓 + 𝐺)‘𝑥) = ((𝐹‘𝑥) + (𝐺‘𝑥)))
75	69, 71, 73, 35, 74	syl22anc 1319	. . . 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)))
76	75	fveq2d 6107	. . 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))))
77	50	recnd 9947	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐹) ∈ ℂ)
78	55	recnd 9947	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → (𝑁‘𝐺) ∈ ℂ)
79	53	recnd 9947	. . . 4 ⊢ (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
80	77, 78, 79	adddird 9944	. . 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))))
81	67, 76, 80	3brtr4d 4615	. 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)))
82	1, 2, 3, 4, 5, 7, 9, 15, 20, 27, 81	nmolb2d 22332	1 ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘𝑓 + 𝐺)) ≤ ((𝑁‘𝐹) + (𝑁‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   class class class wbr 4583   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  ℝcr 9814  0cc0 9815   + caddc 9818   · cmul 9820   ≤ cle 9954  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923  Grpcgrp 17245   GrpHom cghm 17480  Abelcabl 18017  normcnm 22191  NrmGrpcngp 22192   normOp cnmo 22319   NGHom cnghm 22320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ico 12052  df-0g 15925  df-topgen 15927  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-ghm 17481  df-cmn 18018  df-abl 18019  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-xms 21935  df-ms 21936  df-nm 22197  df-ngp 22198  df-nmo 22322  df-nghm 22323

This theorem is referenced by:  nghmplusg  22354