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Theorem cnlnadjlem2 31830
Description: Lemma for cnlnadji 31838. 𝐺 is a continuous linear functional. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
cnlnadjlem.1 𝑇 ∈ LinOp
cnlnadjlem.2 𝑇 ∈ ContOp
cnlnadjlem.3 𝐺 = (𝑔 ∈ β„‹ ↦ ((π‘‡β€˜π‘”) Β·ih 𝑦))
Assertion
Ref Expression
cnlnadjlem2 (𝑦 ∈ β„‹ β†’ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
Distinct variable group:   𝑦,𝑔,𝑇
Allowed substitution hints:   𝐺(𝑦,𝑔)
Proof of Theorem cnlnadjlem2
Dummy variables 𝑀 𝑧 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnlnadjlem.1 . . . . . . . 8 𝑇 ∈ LinOp
21lnopfi 31731 . . . . . . 7 𝑇: β„‹βŸΆ β„‹
32ffvelcdmi 7079 . . . . . 6 (𝑔 ∈ β„‹ β†’ (π‘‡β€˜π‘”) ∈ β„‹)
4 hicl 30842 . . . . . 6 (((π‘‡β€˜π‘”) ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘”) Β·ih 𝑦) ∈ β„‚)
53, 4sylan 579 . . . . 5 ((𝑔 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘”) Β·ih 𝑦) ∈ β„‚)
65ancoms 458 . . . 4 ((𝑦 ∈ β„‹ ∧ 𝑔 ∈ β„‹) β†’ ((π‘‡β€˜π‘”) Β·ih 𝑦) ∈ β„‚)
7 cnlnadjlem.3 . . . 4 𝐺 = (𝑔 ∈ β„‹ ↦ ((π‘‡β€˜π‘”) Β·ih 𝑦))
86, 7fmptd 7109 . . 3 (𝑦 ∈ β„‹ β†’ 𝐺: β„‹βŸΆβ„‚)
9 hvmulcl 30775 . . . . . . 7 ((π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹) β†’ (π‘₯ Β·β„Ž 𝑀) ∈ β„‹)
101lnopaddi 31733 . . . . . . . . . . . 12 (((π‘₯ Β·β„Ž 𝑀) ∈ β„‹ ∧ 𝑧 ∈ β„‹) β†’ (π‘‡β€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) = ((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) +β„Ž (π‘‡β€˜π‘§)))
11103adant3 1129 . . . . . . . . . . 11 (((π‘₯ Β·β„Ž 𝑀) ∈ β„‹ ∧ 𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (π‘‡β€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) = ((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) +β„Ž (π‘‡β€˜π‘§)))
1211oveq1d 7420 . . . . . . . . . 10 (((π‘₯ Β·β„Ž 𝑀) ∈ β„‹ ∧ 𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) Β·ih 𝑦) = (((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) +β„Ž (π‘‡β€˜π‘§)) Β·ih 𝑦))
132ffvelcdmi 7079 . . . . . . . . . . 11 ((π‘₯ Β·β„Ž 𝑀) ∈ β„‹ β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) ∈ β„‹)
142ffvelcdmi 7079 . . . . . . . . . . 11 (𝑧 ∈ β„‹ β†’ (π‘‡β€˜π‘§) ∈ β„‹)
15 id 22 . . . . . . . . . . 11 (𝑦 ∈ β„‹ β†’ 𝑦 ∈ β„‹)
16 ax-his2 30845 . . . . . . . . . . 11 (((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) ∈ β„‹ ∧ (π‘‡β€˜π‘§) ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) +β„Ž (π‘‡β€˜π‘§)) Β·ih 𝑦) = (((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) Β·ih 𝑦) + ((π‘‡β€˜π‘§) Β·ih 𝑦)))
1713, 14, 15, 16syl3an 1157 . . . . . . . . . 10 (((π‘₯ Β·β„Ž 𝑀) ∈ β„‹ ∧ 𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) +β„Ž (π‘‡β€˜π‘§)) Β·ih 𝑦) = (((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) Β·ih 𝑦) + ((π‘‡β€˜π‘§) Β·ih 𝑦)))
1812, 17eqtrd 2766 . . . . . . . . 9 (((π‘₯ Β·β„Ž 𝑀) ∈ β„‹ ∧ 𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) Β·ih 𝑦) = (((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) Β·ih 𝑦) + ((π‘‡β€˜π‘§) Β·ih 𝑦)))
19183comr 1122 . . . . . . . 8 ((𝑦 ∈ β„‹ ∧ (π‘₯ Β·β„Ž 𝑀) ∈ β„‹ ∧ 𝑧 ∈ β„‹) β†’ ((π‘‡β€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) Β·ih 𝑦) = (((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) Β·ih 𝑦) + ((π‘‡β€˜π‘§) Β·ih 𝑦)))
20193expa 1115 . . . . . . 7 (((𝑦 ∈ β„‹ ∧ (π‘₯ Β·β„Ž 𝑀) ∈ β„‹) ∧ 𝑧 ∈ β„‹) β†’ ((π‘‡β€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) Β·ih 𝑦) = (((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) Β·ih 𝑦) + ((π‘‡β€˜π‘§) Β·ih 𝑦)))
219, 20sylanl2 678 . . . . . 6 (((𝑦 ∈ β„‹ ∧ (π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹)) ∧ 𝑧 ∈ β„‹) β†’ ((π‘‡β€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) Β·ih 𝑦) = (((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) Β·ih 𝑦) + ((π‘‡β€˜π‘§) Β·ih 𝑦)))
22 hvaddcl 30774 . . . . . . . . 9 (((π‘₯ Β·β„Ž 𝑀) ∈ β„‹ ∧ 𝑧 ∈ β„‹) β†’ ((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧) ∈ β„‹)
239, 22sylan 579 . . . . . . . 8 (((π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹) ∧ 𝑧 ∈ β„‹) β†’ ((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧) ∈ β„‹)
24 cnlnadjlem.2 . . . . . . . . 9 𝑇 ∈ ContOp
251, 24, 7cnlnadjlem1 31829 . . . . . . . 8 (((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧) ∈ β„‹ β†’ (πΊβ€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) = ((π‘‡β€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) Β·ih 𝑦))
2623, 25syl 17 . . . . . . 7 (((π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹) ∧ 𝑧 ∈ β„‹) β†’ (πΊβ€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) = ((π‘‡β€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) Β·ih 𝑦))
2726adantll 711 . . . . . 6 (((𝑦 ∈ β„‹ ∧ (π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹)) ∧ 𝑧 ∈ β„‹) β†’ (πΊβ€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) = ((π‘‡β€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) Β·ih 𝑦))
282ffvelcdmi 7079 . . . . . . . . . . 11 (𝑀 ∈ β„‹ β†’ (π‘‡β€˜π‘€) ∈ β„‹)
29 ax-his3 30846 . . . . . . . . . . 11 ((π‘₯ ∈ β„‚ ∧ (π‘‡β€˜π‘€) ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘₯ Β·β„Ž (π‘‡β€˜π‘€)) Β·ih 𝑦) = (π‘₯ Β· ((π‘‡β€˜π‘€) Β·ih 𝑦)))
3028, 29syl3an2 1161 . . . . . . . . . 10 ((π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘₯ Β·β„Ž (π‘‡β€˜π‘€)) Β·ih 𝑦) = (π‘₯ Β· ((π‘‡β€˜π‘€) Β·ih 𝑦)))
31303comr 1122 . . . . . . . . 9 ((𝑦 ∈ β„‹ ∧ π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹) β†’ ((π‘₯ Β·β„Ž (π‘‡β€˜π‘€)) Β·ih 𝑦) = (π‘₯ Β· ((π‘‡β€˜π‘€) Β·ih 𝑦)))
32313expb 1117 . . . . . . . 8 ((𝑦 ∈ β„‹ ∧ (π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹)) β†’ ((π‘₯ Β·β„Ž (π‘‡β€˜π‘€)) Β·ih 𝑦) = (π‘₯ Β· ((π‘‡β€˜π‘€) Β·ih 𝑦)))
331lnopmuli 31734 . . . . . . . . . 10 ((π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹) β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) = (π‘₯ Β·β„Ž (π‘‡β€˜π‘€)))
3433oveq1d 7420 . . . . . . . . 9 ((π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹) β†’ ((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) Β·ih 𝑦) = ((π‘₯ Β·β„Ž (π‘‡β€˜π‘€)) Β·ih 𝑦))
3534adantl 481 . . . . . . . 8 ((𝑦 ∈ β„‹ ∧ (π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹)) β†’ ((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) Β·ih 𝑦) = ((π‘₯ Β·β„Ž (π‘‡β€˜π‘€)) Β·ih 𝑦))
361, 24, 7cnlnadjlem1 31829 . . . . . . . . . 10 (𝑀 ∈ β„‹ β†’ (πΊβ€˜π‘€) = ((π‘‡β€˜π‘€) Β·ih 𝑦))
3736oveq2d 7421 . . . . . . . . 9 (𝑀 ∈ β„‹ β†’ (π‘₯ Β· (πΊβ€˜π‘€)) = (π‘₯ Β· ((π‘‡β€˜π‘€) Β·ih 𝑦)))
3837ad2antll 726 . . . . . . . 8 ((𝑦 ∈ β„‹ ∧ (π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹)) β†’ (π‘₯ Β· (πΊβ€˜π‘€)) = (π‘₯ Β· ((π‘‡β€˜π‘€) Β·ih 𝑦)))
3932, 35, 383eqtr4rd 2777 . . . . . . 7 ((𝑦 ∈ β„‹ ∧ (π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹)) β†’ (π‘₯ Β· (πΊβ€˜π‘€)) = ((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) Β·ih 𝑦))
401, 24, 7cnlnadjlem1 31829 . . . . . . 7 (𝑧 ∈ β„‹ β†’ (πΊβ€˜π‘§) = ((π‘‡β€˜π‘§) Β·ih 𝑦))
4139, 40oveqan12d 7424 . . . . . 6 (((𝑦 ∈ β„‹ ∧ (π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹)) ∧ 𝑧 ∈ β„‹) β†’ ((π‘₯ Β· (πΊβ€˜π‘€)) + (πΊβ€˜π‘§)) = (((π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) Β·ih 𝑦) + ((π‘‡β€˜π‘§) Β·ih 𝑦)))
4221, 27, 413eqtr4d 2776 . . . . 5 (((𝑦 ∈ β„‹ ∧ (π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹)) ∧ 𝑧 ∈ β„‹) β†’ (πΊβ€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) = ((π‘₯ Β· (πΊβ€˜π‘€)) + (πΊβ€˜π‘§)))
4342ralrimiva 3140 . . . 4 ((𝑦 ∈ β„‹ ∧ (π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹)) β†’ βˆ€π‘§ ∈ β„‹ (πΊβ€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) = ((π‘₯ Β· (πΊβ€˜π‘€)) + (πΊβ€˜π‘§)))
4443ralrimivva 3194 . . 3 (𝑦 ∈ β„‹ β†’ βˆ€π‘₯ ∈ β„‚ βˆ€π‘€ ∈ β„‹ βˆ€π‘§ ∈ β„‹ (πΊβ€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) = ((π‘₯ Β· (πΊβ€˜π‘€)) + (πΊβ€˜π‘§)))
45 ellnfn 31645 . . 3 (𝐺 ∈ LinFn ↔ (𝐺: β„‹βŸΆβ„‚ ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘€ ∈ β„‹ βˆ€π‘§ ∈ β„‹ (πΊβ€˜((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧)) = ((π‘₯ Β· (πΊβ€˜π‘€)) + (πΊβ€˜π‘§))))
468, 44, 45sylanbrc 582 . 2 (𝑦 ∈ β„‹ β†’ 𝐺 ∈ LinFn)
471, 24nmcopexi 31789 . . . . 5 (normopβ€˜π‘‡) ∈ ℝ
48 normcl 30887 . . . . 5 (𝑦 ∈ β„‹ β†’ (normβ„Žβ€˜π‘¦) ∈ ℝ)
49 remulcl 11197 . . . . 5 (((normopβ€˜π‘‡) ∈ ℝ ∧ (normβ„Žβ€˜π‘¦) ∈ ℝ) β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
5047, 48, 49sylancr 586 . . . 4 (𝑦 ∈ β„‹ β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
5140adantr 480 . . . . . . . . . 10 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (πΊβ€˜π‘§) = ((π‘‡β€˜π‘§) Β·ih 𝑦))
52 hicl 30842 . . . . . . . . . . 11 (((π‘‡β€˜π‘§) ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘§) Β·ih 𝑦) ∈ β„‚)
5314, 52sylan 579 . . . . . . . . . 10 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘§) Β·ih 𝑦) ∈ β„‚)
5451, 53eqeltrd 2827 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (πΊβ€˜π‘§) ∈ β„‚)
5554abscld 15389 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ∈ ℝ)
56 normcl 30887 . . . . . . . . . 10 ((π‘‡β€˜π‘§) ∈ β„‹ β†’ (normβ„Žβ€˜(π‘‡β€˜π‘§)) ∈ ℝ)
5714, 56syl 17 . . . . . . . . 9 (𝑧 ∈ β„‹ β†’ (normβ„Žβ€˜(π‘‡β€˜π‘§)) ∈ ℝ)
58 remulcl 11197 . . . . . . . . 9 (((normβ„Žβ€˜(π‘‡β€˜π‘§)) ∈ ℝ ∧ (normβ„Žβ€˜π‘¦) ∈ ℝ) β†’ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
5957, 48, 58syl2an 595 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
60 normcl 30887 . . . . . . . . . 10 (𝑧 ∈ β„‹ β†’ (normβ„Žβ€˜π‘§) ∈ ℝ)
61 remulcl 11197 . . . . . . . . . 10 (((normopβ€˜π‘‡) ∈ ℝ ∧ (normβ„Žβ€˜π‘§) ∈ ℝ) β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) ∈ ℝ)
6247, 60, 61sylancr 586 . . . . . . . . 9 (𝑧 ∈ β„‹ β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) ∈ ℝ)
63 remulcl 11197 . . . . . . . . 9 ((((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) ∈ ℝ ∧ (normβ„Žβ€˜π‘¦) ∈ ℝ) β†’ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
6462, 48, 63syl2an 595 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
6551fveq2d 6889 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) = (absβ€˜((π‘‡β€˜π‘§) Β·ih 𝑦)))
66 bcs 30943 . . . . . . . . . 10 (((π‘‡β€˜π‘§) ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜((π‘‡β€˜π‘§) Β·ih 𝑦)) ≀ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)))
6714, 66sylan 579 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜((π‘‡β€˜π‘§) Β·ih 𝑦)) ≀ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)))
6865, 67eqbrtrd 5163 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ≀ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)))
6957adantr 480 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (normβ„Žβ€˜(π‘‡β€˜π‘§)) ∈ ℝ)
7062adantr 480 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) ∈ ℝ)
71 normge0 30888 . . . . . . . . . . 11 (𝑦 ∈ β„‹ β†’ 0 ≀ (normβ„Žβ€˜π‘¦))
7248, 71jca 511 . . . . . . . . . 10 (𝑦 ∈ β„‹ β†’ ((normβ„Žβ€˜π‘¦) ∈ ℝ ∧ 0 ≀ (normβ„Žβ€˜π‘¦)))
7372adantl 481 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((normβ„Žβ€˜π‘¦) ∈ ℝ ∧ 0 ≀ (normβ„Žβ€˜π‘¦)))
741, 24nmcoplbi 31790 . . . . . . . . . 10 (𝑧 ∈ β„‹ β†’ (normβ„Žβ€˜(π‘‡β€˜π‘§)) ≀ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)))
7574adantr 480 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (normβ„Žβ€˜(π‘‡β€˜π‘§)) ≀ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)))
76 lemul1a 12072 . . . . . . . . 9 ((((normβ„Žβ€˜(π‘‡β€˜π‘§)) ∈ ℝ ∧ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) ∈ ℝ ∧ ((normβ„Žβ€˜π‘¦) ∈ ℝ ∧ 0 ≀ (normβ„Žβ€˜π‘¦))) ∧ (normβ„Žβ€˜(π‘‡β€˜π‘§)) ≀ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§))) β†’ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)))
7769, 70, 73, 75, 76syl31anc 1370 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)))
7855, 59, 64, 68, 77letrd 11375 . . . . . . 7 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)))
7960recnd 11246 . . . . . . . 8 (𝑧 ∈ β„‹ β†’ (normβ„Žβ€˜π‘§) ∈ β„‚)
8048recnd 11246 . . . . . . . 8 (𝑦 ∈ β„‹ β†’ (normβ„Žβ€˜π‘¦) ∈ β„‚)
8147recni 11232 . . . . . . . . 9 (normopβ€˜π‘‡) ∈ β„‚
82 mul32 11384 . . . . . . . . 9 (((normopβ€˜π‘‡) ∈ β„‚ ∧ (normβ„Žβ€˜π‘§) ∈ β„‚ ∧ (normβ„Žβ€˜π‘¦) ∈ β„‚) β†’ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) = (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§)))
8381, 82mp3an1 1444 . . . . . . . 8 (((normβ„Žβ€˜π‘§) ∈ β„‚ ∧ (normβ„Žβ€˜π‘¦) ∈ β„‚) β†’ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) = (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§)))
8479, 80, 83syl2an 595 . . . . . . 7 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) = (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§)))
8578, 84breqtrd 5167 . . . . . 6 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§)))
8685ancoms 458 . . . . 5 ((𝑦 ∈ β„‹ ∧ 𝑧 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§)))
8786ralrimiva 3140 . . . 4 (𝑦 ∈ β„‹ β†’ βˆ€π‘§ ∈ β„‹ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§)))
88 oveq1 7412 . . . . . . 7 (π‘₯ = ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) β†’ (π‘₯ Β· (normβ„Žβ€˜π‘§)) = (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§)))
8988breq2d 5153 . . . . . 6 (π‘₯ = ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) β†’ ((absβ€˜(πΊβ€˜π‘§)) ≀ (π‘₯ Β· (normβ„Žβ€˜π‘§)) ↔ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§))))
9089ralbidv 3171 . . . . 5 (π‘₯ = ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) β†’ (βˆ€π‘§ ∈ β„‹ (absβ€˜(πΊβ€˜π‘§)) ≀ (π‘₯ Β· (normβ„Žβ€˜π‘§)) ↔ βˆ€π‘§ ∈ β„‹ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§))))
9190rspcev 3606 . . . 4 ((((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ ∧ βˆ€π‘§ ∈ β„‹ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§))) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ β„‹ (absβ€˜(πΊβ€˜π‘§)) ≀ (π‘₯ Β· (normβ„Žβ€˜π‘§)))
9250, 87, 91syl2anc 583 . . 3 (𝑦 ∈ β„‹ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ β„‹ (absβ€˜(πΊβ€˜π‘§)) ≀ (π‘₯ Β· (normβ„Žβ€˜π‘§)))
93 lnfncon 31818 . . . 4 (𝐺 ∈ LinFn β†’ (𝐺 ∈ ContFn ↔ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ β„‹ (absβ€˜(πΊβ€˜π‘§)) ≀ (π‘₯ Β· (normβ„Žβ€˜π‘§))))
9446, 93syl 17 . . 3 (𝑦 ∈ β„‹ β†’ (𝐺 ∈ ContFn ↔ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ β„‹ (absβ€˜(πΊβ€˜π‘§)) ≀ (π‘₯ Β· (normβ„Žβ€˜π‘§))))
9592, 94mpbird 257 . 2 (𝑦 ∈ β„‹ β†’ 𝐺 ∈ ContFn)
9646, 95jca 511 1 (𝑦 ∈ β„‹ β†’ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064   class class class wbr 5141   ↦ cmpt 5224  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  β„‚cc 11110  β„cr 11111  0cc0 11112   + caddc 11115   Β· cmul 11117   ≀ cle 11253  abscabs 15187   β„‹chba 30681   +β„Ž cva 30682   Β·β„Ž csm 30683   Β·ih csp 30684  normβ„Žcno 30685  normopcnop 30707  ContOpccop 30708  LinOpclo 30709  ContFnccnfn 30715  LinFnclf 30716
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 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192  ax-hilex 30761  ax-hfvadd 30762  ax-hvcom 30763  ax-hvass 30764  ax-hv0cl 30765  ax-hvaddid 30766  ax-hfvmul 30767  ax-hvmulid 30768  ax-hvmulass 30769  ax-hvdistr1 30770  ax-hvdistr2 30771  ax-hvmul0 30772  ax-hfi 30841  ax-his1 30844  ax-his2 30845  ax-his3 30846  ax-his4 30847
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-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  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-se 5625  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 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-ioo 13334  df-icc 13337  df-fz 13491  df-fzo 13634  df-seq 13973  df-exp 14033  df-hash 14296  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438  df-sum 15639  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-starv 17221  df-sca 17222  df-vsca 17223  df-ip 17224  df-tset 17225  df-ple 17226  df-ds 17228  df-unif 17229  df-hom 17230  df-cco 17231  df-rest 17377  df-topn 17378  df-0g 17396  df-gsum 17397  df-topgen 17398  df-pt 17399  df-prds 17402  df-xrs 17457  df-qtop 17462  df-imas 17463  df-xps 17465  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-mulg 18996  df-cntz 19233  df-cmn 19702  df-psmet 21232  df-xmet 21233  df-met 21234  df-bl 21235  df-mopn 21236  df-cnfld 21241  df-top 22751  df-topon 22768  df-topsp 22790  df-bases 22804  df-cld 22878  df-ntr 22879  df-cls 22880  df-cn 23086  df-cnp 23087  df-t1 23173  df-haus 23174  df-tx 23421  df-hmeo 23614  df-xms 24181  df-ms 24182  df-tms 24183  df-grpo 30255  df-gid 30256  df-ginv 30257  df-gdiv 30258  df-ablo 30307  df-vc 30321  df-nv 30354  df-va 30357  df-ba 30358  df-sm 30359  df-0v 30360  df-vs 30361  df-nmcv 30362  df-ims 30363  df-dip 30463  df-ph 30575  df-hnorm 30730  df-hba 30731  df-hvsub 30733  df-nmop 31601  df-cnop 31602  df-lnop 31603  df-nmfn 31607  df-cnfn 31609  df-lnfn 31610
This theorem is referenced by:  cnlnadjlem3  31831  cnlnadjlem5  31833
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