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Theorem nmotri 24607
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 2726 . 2 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
3 eqid 2726 . 2 (normβ€˜π‘†) = (normβ€˜π‘†)
4 eqid 2726 . 2 (normβ€˜π‘‡) = (normβ€˜π‘‡)
5 eqid 2726 . 2 (0gβ€˜π‘†) = (0gβ€˜π‘†)
6 nghmrcl1 24600 . . 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 24601 . . 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 24602 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
12 nghmghm 24602 . . 3 (𝐺 ∈ (𝑆 NGHom 𝑇) β†’ 𝐺 ∈ (𝑆 GrpHom 𝑇))
13 nmotri.p . . . 4 + = (+gβ€˜π‘‡)
1413ghmplusg 19764 . . 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 24595 . . . 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 24595 . . . 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 11244 . 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 24589 . . . 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 24589 . . . 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 11791 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 0 ≀ ((π‘β€˜πΉ) + (π‘β€˜πΊ)))
289adantr 480 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝑇 ∈ NrmGrp)
29 ngpgrp 24459 . . . . . . 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 480 . . . . . . . 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 2726 . . . . . . . . 9 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
332, 32ghmf 19143 . . . . . . . 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 768 . . . . . . 7 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
3634, 35ffvelcdmd 7080 . . . . . 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 480 . . . . . . . 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 19143 . . . . . . . 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 7080 . . . . . 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 18869 . . . . . 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 24476 . . . . 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 583 . . . 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 24476 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (πΉβ€˜π‘₯) ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)) ∈ ℝ)
4628, 36, 45syl2anc 583 . . . . 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 24476 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (πΊβ€˜π‘₯) ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜(πΊβ€˜π‘₯)) ∈ ℝ)
4828, 40, 47syl2anc 583 . . . . 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 11244 . . . 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 480 . . . . . 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 482 . . . . . . 7 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†)) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
522, 3nmcl 24476 . . . . . . 7 ((𝑆 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘†)) β†’ ((normβ€˜π‘†)β€˜π‘₯) ∈ ℝ)
537, 51, 52syl2an 595 . . . . . 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 11245 . . . . 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 480 . . . . . 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 11245 . . . . 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 11244 . . . 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 24486 . . . . 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 24596 . . . . . 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 583 . . . . 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 24596 . . . . . 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 583 . . . . 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 11834 . . . 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 11372 . . 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 6711 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝐹 Fn (Baseβ€˜π‘†))
6939ffnd 6711 . . . . 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 6899 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ (Baseβ€˜π‘†) ∈ V)
71 fnfvof 7683 . . . . 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 836 . . . 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 6888 . . 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 11243 . . . 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 11243 . . . 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 11243 . . . 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 11240 . . 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 5173 . 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 24586 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  Vcvv 3468   class class class wbr 5141   Fn wfn 6531  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404   ∘f cof 7664  β„cr 11108  0cc0 11109   + caddc 11112   Β· cmul 11114   ≀ cle 11250  Basecbs 17151  +gcplusg 17204  0gc0g 17392  Grpcgrp 18861   GrpHom cghm 19136  Abelcabl 19699  normcnm 24436  NrmGrpcngp 24437   normOp cnmo 24573   NGHom cnghm 24574
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-inf 9437  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-div 11873  df-nn 12214  df-2 12276  df-n0 12474  df-z 12560  df-uz 12824  df-q 12934  df-rp 12978  df-xneg 13095  df-xadd 13096  df-xmul 13097  df-ico 13333  df-0g 17394  df-topgen 17396  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-minusg 18865  df-sbg 18866  df-ghm 19137  df-cmn 19700  df-abl 19701  df-psmet 21228  df-xmet 21229  df-met 21230  df-bl 21231  df-mopn 21232  df-top 22747  df-topon 22764  df-topsp 22786  df-bases 22800  df-xms 24177  df-ms 24178  df-nm 24442  df-ngp 24443  df-nmo 24576  df-nghm 24577
This theorem is referenced by:  nghmplusg  24608
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