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Theorem nmotri 23275
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 2818 . 2 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2818 . 2 (norm‘𝑆) = (norm‘𝑆)
4 eqid 2818 . 2 (norm‘𝑇) = (norm‘𝑇)
5 eqid 2818 . 2 (0g𝑆) = (0g𝑆)
6 nghmrcl1 23268 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
763ad2ant2 1126 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8 nghmrcl2 23269 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
983ad2ant2 1126 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑇 ∈ NrmGrp)
10 id 22 . . 3 (𝑇 ∈ Abel → 𝑇 ∈ Abel)
11 nghmghm 23270 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
12 nghmghm 23270 . . 3 (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
13 nmotri.p . . . 4 + = (+g𝑇)
1413ghmplusg 18895 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
1510, 11, 12, 14syl3an 1152 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
161nghmcl 23263 . . . 4 (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁𝐹) ∈ ℝ)
17163ad2ant2 1126 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁𝐹) ∈ ℝ)
181nghmcl 23263 . . . 4 (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑁𝐺) ∈ ℝ)
19183ad2ant3 1127 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁𝐺) ∈ ℝ)
2017, 19readdcld 10658 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝑁𝐹) + (𝑁𝐺)) ∈ ℝ)
21113ad2ant2 1126 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
221nmoge0 23257 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁𝐹))
237, 9, 21, 22syl3anc 1363 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁𝐹))
24123ad2ant3 1127 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
251nmoge0 23257 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁𝐺))
267, 9, 24, 25syl3anc 1363 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁𝐺))
2717, 19, 23, 26addge0d 11204 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ ((𝑁𝐹) + (𝑁𝐺)))
289adantr 481 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝑇 ∈ NrmGrp)
29 ngpgrp 23135 . . . . . . 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 481 . . . . . . . 8 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
32 eqid 2818 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
332, 32ghmf 18300 . . . . . . . 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 767 . . . . . . 7 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝑥 ∈ (Base‘𝑆))
3634, 35ffvelrnd 6844 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝐹𝑥) ∈ (Base‘𝑇))
3724adantr 481 . . . . . . . 8 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
382, 32ghmf 18300 . . . . . . . 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 6844 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝐺𝑥) ∈ (Base‘𝑇))
4132, 13grpcl 18049 . . . . . 6 ((𝑇 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((𝐹𝑥) + (𝐺𝑥)) ∈ (Base‘𝑇))
4230, 36, 40, 41syl3anc 1363 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐹𝑥) + (𝐺𝑥)) ∈ (Base‘𝑇))
4332, 4nmcl 23152 . . . . 5 ((𝑇 ∈ NrmGrp ∧ ((𝐹𝑥) + (𝐺𝑥)) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ∈ ℝ)
4428, 42, 43syl2anc 584 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ∈ ℝ)
4532, 4nmcl 23152 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (𝐹𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹𝑥)) ∈ ℝ)
4628, 36, 45syl2anc 584 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘(𝐹𝑥)) ∈ ℝ)
4732, 4nmcl 23152 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺𝑥)) ∈ ℝ)
4828, 40, 47syl2anc 584 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘(𝐺𝑥)) ∈ ℝ)
4946, 48readdcld 10658 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (((norm‘𝑇)‘(𝐹𝑥)) + ((norm‘𝑇)‘(𝐺𝑥))) ∈ ℝ)
5017adantr 481 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐹) ∈ ℝ)
51 simpl 483 . . . . . . 7 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆)) → 𝑥 ∈ (Base‘𝑆))
522, 3nmcl 23152 . . . . . . 7 ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
537, 51, 52syl2an 595 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
5450, 53remulcld 10659 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝑁𝐹) · ((norm‘𝑆)‘𝑥)) ∈ ℝ)
5519adantr 481 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐺) ∈ ℝ)
5655, 53remulcld 10659 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝑁𝐺) · ((norm‘𝑆)‘𝑥)) ∈ ℝ)
5754, 56readdcld 10658 . . . 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 23162 . . . . 5 ((𝑇 ∈ NrmGrp ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ≤ (((norm‘𝑇)‘(𝐹𝑥)) + ((norm‘𝑇)‘(𝐺𝑥))))
5928, 36, 40, 58syl3anc 1363 . . . 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 1184 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹 ∈ (𝑆 NGHom 𝑇))
611, 2, 3, 4nmoi 23264 . . . . . 6 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹𝑥)) ≤ ((𝑁𝐹) · ((norm‘𝑆)‘𝑥)))
6260, 35, 61syl2anc 584 . . . . 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 1185 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺 ∈ (𝑆 NGHom 𝑇))
641, 2, 3, 4nmoi 23264 . . . . . 6 ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺𝑥)) ≤ ((𝑁𝐺) · ((norm‘𝑆)‘𝑥)))
6563, 35, 64syl2anc 584 . . . . 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 11247 . . . 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 10785 . . 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))))
6834ffnd 6508 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹 Fn (Base‘𝑆))
6939ffnd 6508 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺 Fn (Base‘𝑆))
70 fvexd 6678 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (Base‘𝑆) ∈ V)
71 fnfvof 7412 . . . . 5 (((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) ∧ ((Base‘𝑆) ∈ V ∧ 𝑥 ∈ (Base‘𝑆))) → ((𝐹f + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
7268, 69, 70, 35, 71syl22anc 834 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐹f + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
7372fveq2d 6667 . . 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))))
7450recnd 10657 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐹) ∈ ℂ)
7555recnd 10657 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐺) ∈ ℂ)
7653recnd 10657 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
7774, 75, 76adddird 10654 . . 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))))
7867, 73, 773brtr4d 5089 . 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)))
791, 2, 3, 4, 5, 7, 9, 15, 20, 27, 78nmolb2d 23254 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 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  Vcvv 3492   class class class wbr 5057   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  f cof 7396  cr 10524  0cc0 10525   + caddc 10528   · cmul 10530  cle 10664  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Grpcgrp 18041   GrpHom cghm 18293  Abelcabl 18836  normcnm 23113  NrmGrpcngp 23114   normOp cnmo 23241   NGHom cnghm 23242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ico 12732  df-0g 16703  df-topgen 16705  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-sbg 18046  df-ghm 18294  df-cmn 18837  df-abl 18838  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-xms 22857  df-ms 22858  df-nm 23119  df-ngp 23120  df-nmo 23244  df-nghm 23245
This theorem is referenced by:  nghmplusg  23276
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