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Theorem nmotri 24247
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 2732 . 2 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
3 eqid 2732 . 2 (normβ€˜π‘†) = (normβ€˜π‘†)
4 eqid 2732 . 2 (normβ€˜π‘‡) = (normβ€˜π‘‡)
5 eqid 2732 . 2 (0gβ€˜π‘†) = (0gβ€˜π‘†)
6 nghmrcl1 24240 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) β†’ 𝑆 ∈ NrmGrp)
763ad2ant2 1134 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 𝑆 ∈ NrmGrp)
8 nghmrcl2 24241 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) β†’ 𝑇 ∈ NrmGrp)
983ad2ant2 1134 . 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 24242 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
12 nghmghm 24242 . . 3 (𝐺 ∈ (𝑆 NGHom 𝑇) β†’ 𝐺 ∈ (𝑆 GrpHom 𝑇))
13 nmotri.p . . . 4 + = (+gβ€˜π‘‡)
1413ghmplusg 19708 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
1510, 11, 12, 14syl3an 1160 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑆 GrpHom 𝑇))
161nghmcl 24235 . . . 4 (𝐹 ∈ (𝑆 NGHom 𝑇) β†’ (π‘β€˜πΉ) ∈ ℝ)
17163ad2ant2 1134 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ (π‘β€˜πΉ) ∈ ℝ)
181nghmcl 24235 . . . 4 (𝐺 ∈ (𝑆 NGHom 𝑇) β†’ (π‘β€˜πΊ) ∈ ℝ)
19183ad2ant3 1135 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ (π‘β€˜πΊ) ∈ ℝ)
2017, 19readdcld 11239 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ ((π‘β€˜πΉ) + (π‘β€˜πΊ)) ∈ ℝ)
21113ad2ant2 1134 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
221nmoge0 24229 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) β†’ 0 ≀ (π‘β€˜πΉ))
237, 9, 21, 22syl3anc 1371 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 0 ≀ (π‘β€˜πΉ))
24123ad2ant3 1135 . . . 4 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 𝐺 ∈ (𝑆 GrpHom 𝑇))
251nmoge0 24229 . . . 4 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ 0 ≀ (π‘β€˜πΊ))
267, 9, 24, 25syl3anc 1371 . . 3 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 0 ≀ (π‘β€˜πΊ))
2717, 19, 23, 26addge0d 11786 . 2 ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) β†’ 0 ≀ ((π‘β€˜πΉ) + (π‘β€˜πΊ)))
289adantr 481 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝑇 ∈ NrmGrp)
29 ngpgrp 24099 . . . . . . 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 481 . . . . . . . 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 2732 . . . . . . . . 9 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
332, 32ghmf 19090 . . . . . . . 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 7084 . . . . . 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 481 . . . . . . . 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 19090 . . . . . . . 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 7084 . . . . . 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 18823 . . . . . 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 1371 . . . . 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 24116 . . . . 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 584 . . . 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 24116 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (πΉβ€˜π‘₯) ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)) ∈ ℝ)
4628, 36, 45syl2anc 584 . . . . 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 24116 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (πΊβ€˜π‘₯) ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜(πΊβ€˜π‘₯)) ∈ ℝ)
4828, 40, 47syl2anc 584 . . . . 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 11239 . . . 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 481 . . . . . 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 483 . . . . . . 7 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†)) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
522, 3nmcl 24116 . . . . . . 7 ((𝑆 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘†)) β†’ ((normβ€˜π‘†)β€˜π‘₯) ∈ ℝ)
537, 51, 52syl2an 596 . . . . . 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 11240 . . . . 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 481 . . . . . 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 11240 . . . . 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 11239 . . . 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 24126 . . . . 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 1371 . . . 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 1192 . . . . . 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 24236 . . . . . 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 584 . . . . 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 1193 . . . . . 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 24236 . . . . . 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 584 . . . . 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 11829 . . . 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 11367 . . 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 6715 . . . . 5 (((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))) β†’ 𝐹 Fn (Baseβ€˜π‘†))
6939ffnd 6715 . . . . 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 6903 . . . . 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 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 6892 . . 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 11238 . . . 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 11238 . . . 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 11238 . . . 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 11235 . . 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 5179 . 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 24226 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  Vcvv 3474   class class class wbr 5147   Fn wfn 6535  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ∘f cof 7664  β„cr 11105  0cc0 11106   + caddc 11109   Β· cmul 11111   ≀ cle 11245  Basecbs 17140  +gcplusg 17193  0gc0g 17381  Grpcgrp 18815   GrpHom cghm 19083  Abelcabl 19643  normcnm 24076  NrmGrpcngp 24077   normOp cnmo 24213   NGHom cnghm 24214
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ico 13326  df-0g 17383  df-topgen 17385  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820  df-ghm 19084  df-cmn 19644  df-abl 19645  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-top 22387  df-topon 22404  df-topsp 22426  df-bases 22440  df-xms 23817  df-ms 23818  df-nm 24082  df-ngp 24083  df-nmo 24216  df-nghm 24217
This theorem is referenced by:  nghmplusg  24248
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