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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 + = (+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 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 (0g𝑆) = (0g𝑆)
6 nghmrcl1 22346 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
763ad2ant2 1076 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8 nghmrcl2 22347 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
983ad2ant2 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 + = (+g𝑇)
1413ghmplusg 18072 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝑓 + 𝐺) ∈ (𝑆 GrpHom 𝑇))
1510, 11, 12, 14syl3an 1360 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹𝑓 + 𝐺) ∈ (𝑆 GrpHom 𝑇))
161nghmcl 22341 . . . 4 (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁𝐹) ∈ ℝ)
17163ad2ant2 1076 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁𝐹) ∈ ℝ)
181nghmcl 22341 . . . 4 (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑁𝐺) ∈ ℝ)
19183ad2ant3 1077 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁𝐺) ∈ ℝ)
2017, 19readdcld 9948 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝑁𝐹) + (𝑁𝐺)) ∈ ℝ)
21113ad2ant2 1076 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
221nmoge0 22335 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁𝐹))
237, 9, 21, 22syl3anc 1318 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁𝐹))
24123ad2ant3 1077 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
251nmoge0 22335 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁𝐺))
267, 9, 24, 25syl3anc 1318 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ (𝑁𝐺))
2717, 19, 23, 26addge0d 10482 . 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 22213 . . . . . . 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 2610 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
332, 32ghmf 17487 . . . . . . . 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 790 . . . . . . 7 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝑥 ∈ (Base‘𝑆))
3634, 35ffvelrnd 6268 . . . . . 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 17487 . . . . . . . 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 6268 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝐺𝑥) ∈ (Base‘𝑇))
4132, 13grpcl 17253 . . . . . 6 ((𝑇 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((𝐹𝑥) + (𝐺𝑥)) ∈ (Base‘𝑇))
4230, 36, 40, 41syl3anc 1318 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐹𝑥) + (𝐺𝑥)) ∈ (Base‘𝑇))
4332, 4nmcl 22230 . . . . 5 ((𝑇 ∈ NrmGrp ∧ ((𝐹𝑥) + (𝐺𝑥)) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ∈ ℝ)
4428, 42, 43syl2anc 691 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ∈ ℝ)
4532, 4nmcl 22230 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (𝐹𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐹𝑥)) ∈ ℝ)
4628, 36, 45syl2anc 691 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘(𝐹𝑥)) ∈ ℝ)
4732, 4nmcl 22230 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺𝑥)) ∈ ℝ)
4828, 40, 47syl2anc 691 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘(𝐺𝑥)) ∈ ℝ)
4946, 48readdcld 9948 . . . 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 22230 . . . . . . 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 9949 . . . . 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 9949 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝑁𝐺) · ((norm‘𝑆)‘𝑥)) ∈ ℝ)
5754, 56readdcld 9948 . . . 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 22240 . . . . 5 ((𝑇 ∈ NrmGrp ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((𝐹𝑥) + (𝐺𝑥))) ≤ (((norm‘𝑇)‘(𝐹𝑥)) + ((norm‘𝑇)‘(𝐺𝑥))))
5928, 36, 40, 58syl3anc 1318 . . . 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 1058 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹 ∈ (𝑆 NGHom 𝑇))
611, 2, 3, 4nmoi 22342 . . . . . 6 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐹𝑥)) ≤ ((𝑁𝐹) · ((norm‘𝑆)‘𝑥)))
6260, 35, 61syl2anc 691 . . . . 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 1059 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺 ∈ (𝑆 NGHom 𝑇))
641, 2, 3, 4nmoi 22342 . . . . . 6 ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺𝑥)) ≤ ((𝑁𝐺) · ((norm‘𝑆)‘𝑥)))
6563, 35, 64syl2anc 691 . . . . 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 10525 . . . 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 10073 . . 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 5958 . . . . . 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 5958 . . . . . 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 fvex 6113 . . . . . 6 (Base‘𝑆) ∈ V
7372a1i 11 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (Base‘𝑆) ∈ V)
74 fnfvof 6809 . . . . 5 (((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) ∧ ((Base‘𝑆) ∈ V ∧ 𝑥 ∈ (Base‘𝑆))) → ((𝐹𝑓 + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
7569, 71, 73, 35, 74syl22anc 1319 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐹𝑓 + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
7675fveq2d 6107 . . 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))))
7750recnd 9947 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐹) ∈ ℂ)
7855recnd 9947 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑁𝐺) ∈ ℂ)
7953recnd 9947 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
8077, 78, 79adddird 9944 . . 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))))
8167, 76, 803brtr4d 4615 . 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)))
821, 2, 3, 4, 5, 7, 9, 15, 20, 27, 81nmolb2d 22332 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 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  0gc0g 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
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