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Theorem nmotri 22590
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 + = (+g𝑇)
Assertion
Ref Expression
nmotri ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹𝑓 + 𝐺)) ≤ ((𝑁𝐹) + (𝑁𝐺)))

Proof of Theorem nmotri
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nmotri.1 . 2 𝑁 = (𝑆 normOp 𝑇)
2 eqid 2651 . 2 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2651 . 2 (norm‘𝑆) = (norm‘𝑆)
4 eqid 2651 . 2 (norm‘𝑇) = (norm‘𝑇)
5 eqid 2651 . 2 (0g𝑆) = (0g𝑆)
6 nghmrcl1 22583 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
763ad2ant2 1103 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8 nghmrcl2 22584 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
983ad2ant2 1103 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑇 ∈ NrmGrp)
10 id 22 . . 3 (𝑇 ∈ Abel → 𝑇 ∈ Abel)
11 nghmghm 22585 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
12 nghmghm 22585 . . 3 (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
13 nmotri.p . . . 4 + = (+g𝑇)
1413ghmplusg 18295 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝑓 + 𝐺) ∈ (𝑆 GrpHom 𝑇))
1510, 11, 12, 14syl3an 1408 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹𝑓 + 𝐺) ∈ (𝑆 GrpHom 𝑇))
161nghmcl 22578 . . . 4 (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁𝐹) ∈ ℝ)
17163ad2ant2 1103 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁𝐹) ∈ ℝ)
181nghmcl 22578 . . . 4 (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑁𝐺) ∈ ℝ)
19183ad2ant3 1104 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁𝐺) ∈ ℝ)
2017, 19readdcld 10107 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝑁𝐹) + (𝑁𝐺)) ∈ ℝ)
21113ad2ant2 1103 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
221nmoge0 22572 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁𝐹))
237, 9, 21, 22syl3anc 1366 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁𝐹))
24123ad2ant3 1104 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
251nmoge0 22572 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁𝐺))
267, 9, 24, 25syl3anc 1366 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁𝐺))
2717, 19, 23, 26addge0d 10641 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ ((𝑁𝐹) + (𝑁𝐺)))
289adantr 480 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝑇 ∈ NrmGrp)
29 ngpgrp 22450 . . . . . . 7 (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
3028, 29syl 17 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝑇 ∈ Grp)
3121adantr 480 . . . . . . . 8 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
32 eqid 2651 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
332, 32ghmf 17711 . . . . . . . 8 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
3431, 33syl 17 . . . . . . 7 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
35 simprl 809 . . . . . . 7 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝑥 ∈ (Base‘𝑆))
3634, 35ffvelrnd 6400 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝐹𝑥) ∈ (Base‘𝑇))
3724adantr 480 . . . . . . . 8 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
382, 32ghmf 17711 . . . . . . . 8 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
3937, 38syl 17 . . . . . . 7 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
4039, 35ffvelrnd 6400 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝐺𝑥) ∈ (Base‘𝑇))
4132, 13grpcl 17477 . . . . . 6 ((𝑇 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((𝐹𝑥) + (𝐺𝑥)) ∈ (Base‘𝑇))
4230, 36, 40, 41syl3anc 1366 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐹𝑥) + (𝐺𝑥)) ∈ (Base‘𝑇))
4332, 4nmcl 22467 . . . . 5 ((𝑇 ∈ NrmGrp ∧ ((𝐹𝑥) + (𝐺𝑥)) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ∈ ℝ)
4428, 42, 43syl2anc 694 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ∈ ℝ)
4532, 4nmcl 22467 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (𝐹𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹𝑥)) ∈ ℝ)
4628, 36, 45syl2anc 694 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘(𝐹𝑥)) ∈ ℝ)
4732, 4nmcl 22467 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺𝑥)) ∈ ℝ)
4828, 40, 47syl2anc 694 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘(𝐺𝑥)) ∈ ℝ)
4946, 48readdcld 10107 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (((norm‘𝑇)‘(𝐹𝑥)) + ((norm‘𝑇)‘(𝐺𝑥))) ∈ ℝ)
5017adantr 480 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐹) ∈ ℝ)
51 simpl 472 . . . . . . 7 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆)) → 𝑥 ∈ (Base‘𝑆))
522, 3nmcl 22467 . . . . . . 7 ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
537, 51, 52syl2an 493 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
5450, 53remulcld 10108 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝑁𝐹) · ((norm‘𝑆)‘𝑥)) ∈ ℝ)
5519adantr 480 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐺) ∈ ℝ)
5655, 53remulcld 10108 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝑁𝐺) · ((norm‘𝑆)‘𝑥)) ∈ ℝ)
5754, 56readdcld 10107 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (((𝑁𝐹) · ((norm‘𝑆)‘𝑥)) + ((𝑁𝐺) · ((norm‘𝑆)‘𝑥))) ∈ ℝ)
5832, 4, 13nmtri 22477 . . . . 5 ((𝑇 ∈ NrmGrp ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ≤ (((norm‘𝑇)‘(𝐹𝑥)) + ((norm‘𝑇)‘(𝐺𝑥))))
5928, 36, 40, 58syl3anc 1366 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ≤ (((norm‘𝑇)‘(𝐹𝑥)) + ((norm‘𝑇)‘(𝐺𝑥))))
60 simpl2 1085 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹 ∈ (𝑆 NGHom 𝑇))
611, 2, 3, 4nmoi 22579 . . . . . 6 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹𝑥)) ≤ ((𝑁𝐹) · ((norm‘𝑆)‘𝑥)))
6260, 35, 61syl2anc 694 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘(𝐹𝑥)) ≤ ((𝑁𝐹) · ((norm‘𝑆)‘𝑥)))
63 simpl3 1086 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺 ∈ (𝑆 NGHom 𝑇))
641, 2, 3, 4nmoi 22579 . . . . . 6 ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺𝑥)) ≤ ((𝑁𝐺) · ((norm‘𝑆)‘𝑥)))
6563, 35, 64syl2anc 694 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘(𝐺𝑥)) ≤ ((𝑁𝐺) · ((norm‘𝑆)‘𝑥)))
6646, 48, 54, 56, 62, 65le2addd 10684 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (((norm‘𝑇)‘(𝐹𝑥)) + ((norm‘𝑇)‘(𝐺𝑥))) ≤ (((𝑁𝐹) · ((norm‘𝑆)‘𝑥)) + ((𝑁𝐺) · ((norm‘𝑆)‘𝑥))))
6744, 49, 57, 59, 66letrd 10232 . . 3 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ≤ (((𝑁𝐹) · ((norm‘𝑆)‘𝑥)) + ((𝑁𝐺) · ((norm‘𝑆)‘𝑥))))
68 ffn 6083 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
6934, 68syl 17 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹 Fn (Base‘𝑆))
70 ffn 6083 . . . . . 6 (𝐺:(Base‘𝑆)⟶(Base‘𝑇) → 𝐺 Fn (Base‘𝑆))
7139, 70syl 17 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺 Fn (Base‘𝑆))
72 fvexd 6241 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (Base‘𝑆) ∈ V)
73 fnfvof 6953 . . . . 5 (((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) ∧ ((Base‘𝑆) ∈ V ∧ 𝑥 ∈ (Base‘𝑆))) → ((𝐹𝑓 + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
7469, 71, 72, 35, 73syl22anc 1367 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐹𝑓 + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
7574fveq2d 6233 . . 3 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘((𝐹𝑓 + 𝐺)‘𝑥)) = ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))))
7650recnd 10106 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐹) ∈ ℂ)
7755recnd 10106 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐺) ∈ ℂ)
7853recnd 10106 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
7976, 77, 78adddird 10103 . . 3 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (((𝑁𝐹) + (𝑁𝐺)) · ((norm‘𝑆)‘𝑥)) = (((𝑁𝐹) · ((norm‘𝑆)‘𝑥)) + ((𝑁𝐺) · ((norm‘𝑆)‘𝑥))))
8067, 75, 793brtr4d 4717 . 2 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘((𝐹𝑓 + 𝐺)‘𝑥)) ≤ (((𝑁𝐹) + (𝑁𝐺)) · ((norm‘𝑆)‘𝑥)))
811, 2, 3, 4, 5, 7, 9, 15, 20, 27, 80nmolb2d 22569 1 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹𝑓 + 𝐺)) ≤ ((𝑁𝐹) + (𝑁𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  Vcvv 3231   class class class wbr 4685   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937  cr 9973  0cc0 9974   + caddc 9977   · cmul 9979  cle 10113  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Grpcgrp 17469   GrpHom cghm 17704  Abelcabl 18240  normcnm 22428  NrmGrpcngp 22429   normOp cnmo 22556   NGHom cnghm 22557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ico 12219  df-0g 16149  df-topgen 16151  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-ghm 17705  df-cmn 18241  df-abl 18242  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-xms 22172  df-ms 22173  df-nm 22434  df-ngp 22435  df-nmo 22559  df-nghm 22560
This theorem is referenced by:  nghmplusg  22591
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