MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmotri Structured version   Visualization version   GIF version
Theorem nmotri 24674
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 + = (+gβ€˜π‘‡)
Assertion
Ref Expression
nmotri ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ (π‘β€˜(𝐹 ∘f + 𝐺)) ≀ ((π‘β€˜πΉ) + (π‘β€˜πΊ)))
Proof of Theorem nmotri
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 nmotri.1 . 2 𝑁 = (𝑆 normOp 𝑇)
2 eqid 2727 . 2 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
3 eqid 2727 . 2 (normβ€˜π‘†) = (normβ€˜π‘†)
4 eqid 2727 . 2 (normβ€˜π‘‡) = (normβ€˜π‘‡)
5 eqid 2727 . 2 (0gβ€˜π‘†) = (0gβ€˜π‘†)
6 nghmrcl1 24667 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) β†’ 𝑆 ∈ NrmGrp)
763ad2ant2 1131 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 𝑆 ∈ NrmGrp)
8 nghmrcl2 24668 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) β†’ 𝑇 ∈ NrmGrp)
983ad2ant2 1131 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 𝑇 ∈ NrmGrp)
10 id 22 . . 3 (𝑇 ∈ Abel β†’ 𝑇 ∈ Abel)
11 nghmghm 24669 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
12 nghmghm 24669 . . 3 (𝐺 ∈ (𝑆 NGHom 𝑇) β†’ 𝐺 ∈ (𝑆 GrpHom 𝑇))
13 nmotri.p . . . 4 + = (+gβ€˜π‘‡)
1413ghmplusg 19806 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
1510, 11, 12, 14syl3an 1157 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
161nghmcl 24662 . . . 4 (𝐹 ∈ (𝑆 NGHom 𝑇) β†’ (π‘β€˜πΉ) ∈ ℝ)
17163ad2ant2 1131 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ (π‘β€˜πΉ) ∈ ℝ)
181nghmcl 24662 . . . 4 (𝐺 ∈ (𝑆 NGHom 𝑇) β†’ (π‘β€˜πΊ) ∈ ℝ)
19183ad2ant3 1132 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ (π‘β€˜πΊ) ∈ ℝ)
2017, 19readdcld 11279 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ ((π‘β€˜πΉ) + (π‘β€˜πΊ)) ∈ ℝ)
21113ad2ant2 1131 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
221nmoge0 24656 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) β†’ 0 ≀ (π‘β€˜πΉ))
237, 9, 21, 22syl3anc 1368 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 0 ≀ (π‘β€˜πΉ))
24123ad2ant3 1132 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 𝐺 ∈ (𝑆 GrpHom 𝑇))
251nmoge0 24656 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ 0 ≀ (π‘β€˜πΊ))
267, 9, 24, 25syl3anc 1368 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 0 ≀ (π‘β€˜πΊ))
2717, 19, 23, 26addge0d 11826 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 0 ≀ ((π‘β€˜πΉ) + (π‘β€˜πΊ)))
289adantr 479 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝑇 ∈ NrmGrp)
29 ngpgrp 24526 . . . . . . 7 (𝑇 ∈ NrmGrp β†’ 𝑇 ∈ Grp)
3028, 29syl 17 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝑇 ∈ Grp)
3121adantr 479 . . . . . . . 8 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
32 eqid 2727 . . . . . . . . 9 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
332, 32ghmf 19179 . . . . . . . 8 (𝐹 ∈ (𝑆 GrpHom 𝑇) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
3431, 33syl 17 . . . . . . 7 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
35 simprl 769 . . . . . . 7 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
3634, 35ffvelcdmd 7098 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (πΉβ€˜π‘₯) ∈ (Baseβ€˜π‘‡))
3724adantr 479 . . . . . . . 8 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝐺 ∈ (𝑆 GrpHom 𝑇))
382, 32ghmf 19179 . . . . . . . 8 (𝐺 ∈ (𝑆 GrpHom 𝑇) β†’ 𝐺:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
3937, 38syl 17 . . . . . . 7 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝐺:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
4039, 35ffvelcdmd 7098 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (πΊβ€˜π‘₯) ∈ (Baseβ€˜π‘‡))
4132, 13grpcl 18903 . . . . . 6 ((𝑇 ∈ Grp ∧ (πΉβ€˜π‘₯) ∈ (Baseβ€˜π‘‡) ∧ (πΊβ€˜π‘₯) ∈ (Baseβ€˜π‘‡)) β†’ ((πΉβ€˜π‘₯) + (πΊβ€˜π‘₯)) ∈ (Baseβ€˜π‘‡))
4230, 36, 40, 41syl3anc 1368 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((πΉβ€˜π‘₯) + (πΊβ€˜π‘₯)) ∈ (Baseβ€˜π‘‡))
4332, 4nmcl 24543 . . . . 5 ((𝑇 ∈ NrmGrp ∧ ((πΉβ€˜π‘₯) + (πΊβ€˜π‘₯)) ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜((πΉβ€˜π‘₯) + (πΊβ€˜π‘₯))) ∈ ℝ)
4428, 42, 43syl2anc 582 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((normβ€˜π‘‡)β€˜((πΉβ€˜π‘₯) + (πΊβ€˜π‘₯))) ∈ ℝ)
4532, 4nmcl 24543 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (πΉβ€˜π‘₯) ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)) ∈ ℝ)
4628, 36, 45syl2anc 582 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((normβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)) ∈ ℝ)
4732, 4nmcl 24543 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (πΊβ€˜π‘₯) ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜(πΊβ€˜π‘₯)) ∈ ℝ)
4828, 40, 47syl2anc 582 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((normβ€˜π‘‡)β€˜(πΊβ€˜π‘₯)) ∈ ℝ)
4946, 48readdcld 11279 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (((normβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)) + ((normβ€˜π‘‡)β€˜(πΊβ€˜π‘₯))) ∈ ℝ)
5017adantr 479 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (π‘β€˜πΉ) ∈ ℝ)
51 simpl 481 . . . . . . 7 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†)) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
522, 3nmcl 24543 . . . . . . 7 ((𝑆 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘†)) β†’ ((normβ€˜π‘†)β€˜π‘₯) ∈ ℝ)
537, 51, 52syl2an 594 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((normβ€˜π‘†)β€˜π‘₯) ∈ ℝ)
5450, 53remulcld 11280 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((π‘β€˜πΉ) Β· ((normβ€˜π‘†)β€˜π‘₯)) ∈ ℝ)
5519adantr 479 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (π‘β€˜πΊ) ∈ ℝ)
5655, 53remulcld 11280 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((π‘β€˜πΊ) Β· ((normβ€˜π‘†)β€˜π‘₯)) ∈ ℝ)
5754, 56readdcld 11279 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (((π‘β€˜πΉ) Β· ((normβ€˜π‘†)β€˜π‘₯)) + ((π‘β€˜πΊ) Β· ((normβ€˜π‘†)β€˜π‘₯))) ∈ ℝ)
5832, 4, 13nmtri 24553 . . . . 5 ((𝑇 ∈ NrmGrp ∧ (πΉβ€˜π‘₯) ∈ (Baseβ€˜π‘‡) ∧ (πΊβ€˜π‘₯) ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜((πΉβ€˜π‘₯) + (πΊβ€˜π‘₯))) ≀ (((normβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)) + ((normβ€˜π‘‡)β€˜(πΊβ€˜π‘₯))))
5928, 36, 40, 58syl3anc 1368 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((normβ€˜π‘‡)β€˜((πΉβ€˜π‘₯) + (πΊβ€˜π‘₯))) ≀ (((normβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)) + ((normβ€˜π‘‡)β€˜(πΊβ€˜π‘₯))))
60 simpl2 1189 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝐹 ∈ (𝑆 NGHom 𝑇))
611, 2, 3, 4nmoi 24663 . . . . . 6 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ π‘₯ ∈ (Baseβ€˜π‘†)) β†’ ((normβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)) ≀ ((π‘β€˜πΉ) Β· ((normβ€˜π‘†)β€˜π‘₯)))
6260, 35, 61syl2anc 582 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((normβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)) ≀ ((π‘β€˜πΉ) Β· ((normβ€˜π‘†)β€˜π‘₯)))
63 simpl3 1190 . . . . . 6 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝐺 ∈ (𝑆 NGHom 𝑇))
641, 2, 3, 4nmoi 24663 . . . . . 6 ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ π‘₯ ∈ (Baseβ€˜π‘†)) β†’ ((normβ€˜π‘‡)β€˜(πΊβ€˜π‘₯)) ≀ ((π‘β€˜πΊ) Β· ((normβ€˜π‘†)β€˜π‘₯)))
6563, 35, 64syl2anc 582 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((normβ€˜π‘‡)β€˜(πΊβ€˜π‘₯)) ≀ ((π‘β€˜πΊ) Β· ((normβ€˜π‘†)β€˜π‘₯)))
6646, 48, 54, 56, 62, 65le2addd 11869 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (((normβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)) + ((normβ€˜π‘‡)β€˜(πΊβ€˜π‘₯))) ≀ (((π‘β€˜πΉ) Β· ((normβ€˜π‘†)β€˜π‘₯)) + ((π‘β€˜πΊ) Β· ((normβ€˜π‘†)β€˜π‘₯))))
6744, 49, 57, 59, 66letrd 11407 . . 3 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((normβ€˜π‘‡)β€˜((πΉβ€˜π‘₯) + (πΊβ€˜π‘₯))) ≀ (((π‘β€˜πΉ) Β· ((normβ€˜π‘†)β€˜π‘₯)) + ((π‘β€˜πΊ) Β· ((normβ€˜π‘†)β€˜π‘₯))))
6834ffnd 6726 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝐹 Fn (Baseβ€˜π‘†))
6939ffnd 6726 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝐺 Fn (Baseβ€˜π‘†))
70 fvexd 6915 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (Baseβ€˜π‘†) ∈ V)
71 fnfvof 7706 . . . . 5 (((𝐹 Fn (Baseβ€˜π‘†) ∧ 𝐺 Fn (Baseβ€˜π‘†)) ∧ ((Baseβ€˜π‘†) ∈ V ∧ π‘₯ ∈ (Baseβ€˜π‘†))) β†’ ((𝐹 ∘f + 𝐺)β€˜π‘₯) = ((πΉβ€˜π‘₯) + (πΊβ€˜π‘₯)))
7268, 69, 70, 35, 71syl22anc 837 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((𝐹 ∘f + 𝐺)β€˜π‘₯) = ((πΉβ€˜π‘₯) + (πΊβ€˜π‘₯)))
7372fveq2d 6904 . . 3 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((normβ€˜π‘‡)β€˜((𝐹 ∘f + 𝐺)β€˜π‘₯)) = ((normβ€˜π‘‡)β€˜((πΉβ€˜π‘₯) + (πΊβ€˜π‘₯))))
7450recnd 11278 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (π‘β€˜πΉ) ∈ β„‚)
7555recnd 11278 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (π‘β€˜πΊ) ∈ β„‚)
7653recnd 11278 . . . 4 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((normβ€˜π‘†)β€˜π‘₯) ∈ β„‚)
7774, 75, 76adddird 11275 . . 3 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (((π‘β€˜πΉ) + (π‘β€˜πΊ)) Β· ((normβ€˜π‘†)β€˜π‘₯)) = (((π‘β€˜πΉ) Β· ((normβ€˜π‘†)β€˜π‘₯)) + ((π‘β€˜πΊ) Β· ((normβ€˜π‘†)β€˜π‘₯))))
7867, 73, 773brtr4d 5182 . 2 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ ((normβ€˜π‘‡)β€˜((𝐹 ∘f + 𝐺)β€˜π‘₯)) ≀ (((π‘β€˜πΉ) + (π‘β€˜πΊ)) Β· ((normβ€˜π‘†)β€˜π‘₯)))
791, 2, 3, 4, 5, 7, 9, 15, 20, 27, 78nmolb2d 24653 1 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ (π‘β€˜(𝐹 ∘f + 𝐺)) ≀ ((π‘β€˜πΉ) + (π‘β€˜πΊ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2936  Vcvv 3471   class class class wbr 5150   Fn wfn 6546  βŸΆwf 6547  β€˜cfv 6551  (class class class)co 7424   ∘f cof 7687  β„cr 11143  0cc0 11144   + caddc 11147   Β· cmul 11149   ≀ cle 11285  Basecbs 17185  +gcplusg 17238  0gc0g 17426  Grpcgrp 18895   GrpHom cghm 19172  Abelcabl 19741  normcnm 24503  NrmGrpcngp 24504   normOp cnmo 24640   NGHom cnghm 24641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221  ax-pre-sup 11222
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7689  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-er 8729  df-map 8851  df-en 8969  df-dom 8970  df-sdom 8971  df-sup 9471  df-inf 9472  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-div 11908  df-nn 12249  df-2 12311  df-n0 12509  df-z 12595  df-uz 12859  df-q 12969  df-rp 13013  df-xneg 13130  df-xadd 13131  df-xmul 13132  df-ico 13368  df-0g 17428  df-topgen 17430  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-grp 18898  df-minusg 18899  df-sbg 18900  df-ghm 19173  df-cmn 19742  df-abl 19743  df-psmet 21276  df-xmet 21277  df-met 21278  df-bl 21279  df-mopn 21280  df-top 22814  df-topon 22831  df-topsp 22853  df-bases 22867  df-xms 24244  df-ms 24245  df-nm 24509  df-ngp 24510  df-nmo 24643  df-nghm 24644
This theorem is referenced by:  nghmplusg  24675
  Copyright terms: Public domain W3C validator