MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmotri Structured version   Visualization version   GIF version

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 + = (+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 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 (0g𝑆) = (0g𝑆)
6 nghmrcl1 24096 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
763ad2ant2 1134 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8 nghmrcl2 24097 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
983ad2ant2 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 + = (+g𝑇)
1413ghmplusg 19624 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
1510, 11, 12, 14syl3an 1160 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
161nghmcl 24091 . . . 4 (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁𝐹) ∈ ℝ)
17163ad2ant2 1134 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁𝐹) ∈ ℝ)
181nghmcl 24091 . . . 4 (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑁𝐺) ∈ ℝ)
19183ad2ant3 1135 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁𝐺) ∈ ℝ)
2017, 19readdcld 11184 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝑁𝐹) + (𝑁𝐺)) ∈ ℝ)
21113ad2ant2 1134 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
221nmoge0 24085 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁𝐹))
237, 9, 21, 22syl3anc 1371 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁𝐹))
24123ad2ant3 1135 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
251nmoge0 24085 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁𝐺))
267, 9, 24, 25syl3anc 1371 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁𝐺))
2717, 19, 23, 26addge0d 11731 . 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 23955 . . . . . . 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 2736 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
332, 32ghmf 19012 . . . . . . . 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 769 . . . . . . 7 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝑥 ∈ (Base‘𝑆))
3634, 35ffvelcdmd 7036 . . . . . 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 19012 . . . . . . . 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, 35ffvelcdmd 7036 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝐺𝑥) ∈ (Base‘𝑇))
4132, 13grpcl 18756 . . . . . 6 ((𝑇 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((𝐹𝑥) + (𝐺𝑥)) ∈ (Base‘𝑇))
4230, 36, 40, 41syl3anc 1371 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐹𝑥) + (𝐺𝑥)) ∈ (Base‘𝑇))
4332, 4nmcl 23972 . . . . 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 23972 . . . . . 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 23972 . . . . . 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 11184 . . . 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 23972 . . . . . . 7 ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
537, 51, 52syl2an 596 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
5450, 53remulcld 11185 . . . . 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 11185 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝑁𝐺) · ((norm‘𝑆)‘𝑥)) ∈ ℝ)
5754, 56readdcld 11184 . . . 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 23982 . . . . 5 ((𝑇 ∈ NrmGrp ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ≤ (((norm‘𝑇)‘(𝐹𝑥)) + ((norm‘𝑇)‘(𝐺𝑥))))
5928, 36, 40, 58syl3anc 1371 . . . 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 1192 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹 ∈ (𝑆 NGHom 𝑇))
611, 2, 3, 4nmoi 24092 . . . . . 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 1193 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺 ∈ (𝑆 NGHom 𝑇))
641, 2, 3, 4nmoi 24092 . . . . . 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 11774 . . . 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 11312 . . 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 6669 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹 Fn (Base‘𝑆))
6939ffnd 6669 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺 Fn (Base‘𝑆))
70 fvexd 6857 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (Base‘𝑆) ∈ V)
71 fnfvof 7634 . . . . 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 837 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐹f + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
7372fveq2d 6846 . . 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 11183 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐹) ∈ ℂ)
7555recnd 11183 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐺) ∈ ℂ)
7653recnd 11183 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
7774, 75, 76adddird 11180 . . 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 5137 . 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 24082 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 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  cr 11050  0cc0 11051   + caddc 11054   · cmul 11056  cle 11190  Basecbs 17083  +gcplusg 17133  0gc0g 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
  Copyright terms: Public domain W3C validator