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Theorem cnlnadjlem2 31052
Description: Lemma for cnlnadji 31060. 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 30953 . . . . . . 7 𝑇: β„‹βŸΆ β„‹
32ffvelcdmi 7039 . . . . . 6 (𝑔 ∈ β„‹ β†’ (π‘‡β€˜π‘”) ∈ β„‹)
4 hicl 30064 . . . . . 6 (((π‘‡β€˜π‘”) ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘”) Β·ih 𝑦) ∈ β„‚)
53, 4sylan 581 . . . . 5 ((𝑔 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘”) Β·ih 𝑦) ∈ β„‚)
65ancoms 460 . . . 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 7067 . . 3 (𝑦 ∈ β„‹ β†’ 𝐺: β„‹βŸΆβ„‚)
9 hvmulcl 29997 . . . . . . 7 ((π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹) β†’ (π‘₯ Β·β„Ž 𝑀) ∈ β„‹)
101lnopaddi 30955 . . . . . . . . . . . 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 1133 . . . . . . . . . . 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 7377 . . . . . . . . . 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 7039 . . . . . . . . . . 11 ((π‘₯ Β·β„Ž 𝑀) ∈ β„‹ β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) ∈ β„‹)
142ffvelcdmi 7039 . . . . . . . . . . 11 (𝑧 ∈ β„‹ β†’ (π‘‡β€˜π‘§) ∈ β„‹)
15 id 22 . . . . . . . . . . 11 (𝑦 ∈ β„‹ β†’ 𝑦 ∈ β„‹)
16 ax-his2 30067 . . . . . . . . . . 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 1161 . . . . . . . . . 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 2777 . . . . . . . . 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 1126 . . . . . . . 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 1119 . . . . . . 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 680 . . . . . 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 29996 . . . . . . . . 9 (((π‘₯ Β·β„Ž 𝑀) ∈ β„‹ ∧ 𝑧 ∈ β„‹) β†’ ((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧) ∈ β„‹)
239, 22sylan 581 . . . . . . . 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 31051 . . . . . . . 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 713 . . . . . 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 7039 . . . . . . . . . . 11 (𝑀 ∈ β„‹ β†’ (π‘‡β€˜π‘€) ∈ β„‹)
29 ax-his3 30068 . . . . . . . . . . 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 1165 . . . . . . . . . 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 1126 . . . . . . . . 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 1121 . . . . . . . 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 30956 . . . . . . . . . 10 ((π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹) β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) = (π‘₯ Β·β„Ž (π‘‡β€˜π‘€)))
3433oveq1d 7377 . . . . . . . . 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 483 . . . . . . . 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 31051 . . . . . . . . . 10 (𝑀 ∈ β„‹ β†’ (πΊβ€˜π‘€) = ((π‘‡β€˜π‘€) Β·ih 𝑦))
3736oveq2d 7378 . . . . . . . . 9 (𝑀 ∈ β„‹ β†’ (π‘₯ Β· (πΊβ€˜π‘€)) = (π‘₯ Β· ((π‘‡β€˜π‘€) Β·ih 𝑦)))
3837ad2antll 728 . . . . . . . 8 ((𝑦 ∈ β„‹ ∧ (π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹)) β†’ (π‘₯ Β· (πΊβ€˜π‘€)) = (π‘₯ Β· ((π‘‡β€˜π‘€) Β·ih 𝑦)))
3932, 35, 383eqtr4rd 2788 . . . . . . 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 31051 . . . . . . 7 (𝑧 ∈ β„‹ β†’ (πΊβ€˜π‘§) = ((π‘‡β€˜π‘§) Β·ih 𝑦))
4139, 40oveqan12d 7381 . . . . . 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 2787 . . . . 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 3144 . . . 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 3198 . . 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 30867 . . 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 584 . 2 (𝑦 ∈ β„‹ β†’ 𝐺 ∈ LinFn)
471, 24nmcopexi 31011 . . . . 5 (normopβ€˜π‘‡) ∈ ℝ
48 normcl 30109 . . . . 5 (𝑦 ∈ β„‹ β†’ (normβ„Žβ€˜π‘¦) ∈ ℝ)
49 remulcl 11143 . . . . 5 (((normopβ€˜π‘‡) ∈ ℝ ∧ (normβ„Žβ€˜π‘¦) ∈ ℝ) β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
5047, 48, 49sylancr 588 . . . 4 (𝑦 ∈ β„‹ β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
5140adantr 482 . . . . . . . . . 10 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (πΊβ€˜π‘§) = ((π‘‡β€˜π‘§) Β·ih 𝑦))
52 hicl 30064 . . . . . . . . . . 11 (((π‘‡β€˜π‘§) ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘§) Β·ih 𝑦) ∈ β„‚)
5314, 52sylan 581 . . . . . . . . . 10 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘§) Β·ih 𝑦) ∈ β„‚)
5451, 53eqeltrd 2838 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (πΊβ€˜π‘§) ∈ β„‚)
5554abscld 15328 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ∈ ℝ)
56 normcl 30109 . . . . . . . . . 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 11143 . . . . . . . . 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 597 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
60 normcl 30109 . . . . . . . . . 10 (𝑧 ∈ β„‹ β†’ (normβ„Žβ€˜π‘§) ∈ ℝ)
61 remulcl 11143 . . . . . . . . . 10 (((normopβ€˜π‘‡) ∈ ℝ ∧ (normβ„Žβ€˜π‘§) ∈ ℝ) β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) ∈ ℝ)
6247, 60, 61sylancr 588 . . . . . . . . 9 (𝑧 ∈ β„‹ β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) ∈ ℝ)
63 remulcl 11143 . . . . . . . . 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 597 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
6551fveq2d 6851 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) = (absβ€˜((π‘‡β€˜π‘§) Β·ih 𝑦)))
66 bcs 30165 . . . . . . . . . 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 581 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜((π‘‡β€˜π‘§) Β·ih 𝑦)) ≀ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)))
6865, 67eqbrtrd 5132 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ≀ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)))
6957adantr 482 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (normβ„Žβ€˜(π‘‡β€˜π‘§)) ∈ ℝ)
7062adantr 482 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) ∈ ℝ)
71 normge0 30110 . . . . . . . . . . 11 (𝑦 ∈ β„‹ β†’ 0 ≀ (normβ„Žβ€˜π‘¦))
7248, 71jca 513 . . . . . . . . . 10 (𝑦 ∈ β„‹ β†’ ((normβ„Žβ€˜π‘¦) ∈ ℝ ∧ 0 ≀ (normβ„Žβ€˜π‘¦)))
7372adantl 483 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((normβ„Žβ€˜π‘¦) ∈ ℝ ∧ 0 ≀ (normβ„Žβ€˜π‘¦)))
741, 24nmcoplbi 31012 . . . . . . . . . 10 (𝑧 ∈ β„‹ β†’ (normβ„Žβ€˜(π‘‡β€˜π‘§)) ≀ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)))
7574adantr 482 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (normβ„Žβ€˜(π‘‡β€˜π‘§)) ≀ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)))
76 lemul1a 12016 . . . . . . . . 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 1374 . . . . . . . 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 11319 . . . . . . 7 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)))
7960recnd 11190 . . . . . . . 8 (𝑧 ∈ β„‹ β†’ (normβ„Žβ€˜π‘§) ∈ β„‚)
8048recnd 11190 . . . . . . . 8 (𝑦 ∈ β„‹ β†’ (normβ„Žβ€˜π‘¦) ∈ β„‚)
8147recni 11176 . . . . . . . . 9 (normopβ€˜π‘‡) ∈ β„‚
82 mul32 11328 . . . . . . . . 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 1449 . . . . . . . 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 597 . . . . . . 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 5136 . . . . . 6 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§)))
8685ancoms 460 . . . . 5 ((𝑦 ∈ β„‹ ∧ 𝑧 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§)))
8786ralrimiva 3144 . . . 4 (𝑦 ∈ β„‹ β†’ βˆ€π‘§ ∈ β„‹ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§)))
88 oveq1 7369 . . . . . . 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 5122 . . . . . 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 3175 . . . . 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 3584 . . . 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 585 . . 3 (𝑦 ∈ β„‹ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ β„‹ (absβ€˜(πΊβ€˜π‘§)) ≀ (π‘₯ Β· (normβ„Žβ€˜π‘§)))
93 lnfncon 31040 . . . 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 513 1 (𝑦 ∈ β„‹ β†’ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074   class class class wbr 5110   ↦ cmpt 5193  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056  β„cr 11057  0cc0 11058   + caddc 11061   Β· cmul 11063   ≀ cle 11197  abscabs 15126   β„‹chba 29903   +β„Ž cva 29904   Β·β„Ž csm 29905   Β·ih csp 29906  normβ„Žcno 29907  normopcnop 29929  ContOpccop 29930  LinOpclo 29931  ContFnccnfn 29937  LinFnclf 29938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136  ax-addf 11137  ax-mulf 11138  ax-hilex 29983  ax-hfvadd 29984  ax-hvcom 29985  ax-hvass 29986  ax-hv0cl 29987  ax-hvaddid 29988  ax-hfvmul 29989  ax-hvmulid 29990  ax-hvmulass 29991  ax-hvdistr1 29992  ax-hvdistr2 29993  ax-hvmul0 29994  ax-hfi 30063  ax-his1 30066  ax-his2 30067  ax-his3 30068  ax-his4 30069
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-map 8774  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9313  df-fi 9354  df-sup 9385  df-inf 9386  df-oi 9453  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-q 12881  df-rp 12923  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-ioo 13275  df-icc 13278  df-fz 13432  df-fzo 13575  df-seq 13914  df-exp 13975  df-hash 14238  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-clim 15377  df-sum 15578  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-starv 17155  df-sca 17156  df-vsca 17157  df-ip 17158  df-tset 17159  df-ple 17160  df-ds 17162  df-unif 17163  df-hom 17164  df-cco 17165  df-rest 17311  df-topn 17312  df-0g 17330  df-gsum 17331  df-topgen 17332  df-pt 17333  df-prds 17336  df-xrs 17391  df-qtop 17396  df-imas 17397  df-xps 17399  df-mre 17473  df-mrc 17474  df-acs 17476  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-mulg 18880  df-cntz 19104  df-cmn 19571  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-cnfld 20813  df-top 22259  df-topon 22276  df-topsp 22298  df-bases 22312  df-cld 22386  df-ntr 22387  df-cls 22388  df-cn 22594  df-cnp 22595  df-t1 22681  df-haus 22682  df-tx 22929  df-hmeo 23122  df-xms 23689  df-ms 23690  df-tms 23691  df-grpo 29477  df-gid 29478  df-ginv 29479  df-gdiv 29480  df-ablo 29529  df-vc 29543  df-nv 29576  df-va 29579  df-ba 29580  df-sm 29581  df-0v 29582  df-vs 29583  df-nmcv 29584  df-ims 29585  df-dip 29685  df-ph 29797  df-hnorm 29952  df-hba 29953  df-hvsub 29955  df-nmop 30823  df-cnop 30824  df-lnop 30825  df-nmfn 30829  df-cnfn 30831  df-lnfn 30832
This theorem is referenced by:  cnlnadjlem3  31053  cnlnadjlem5  31055
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