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Theorem cnlnadjlem2 31588
Description: Lemma for cnlnadji 31596. 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 31489 . . . . . . 7 𝑇: β„‹βŸΆ β„‹
32ffvelcdmi 7084 . . . . . 6 (𝑔 ∈ β„‹ β†’ (π‘‡β€˜π‘”) ∈ β„‹)
4 hicl 30600 . . . . . 6 (((π‘‡β€˜π‘”) ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘”) Β·ih 𝑦) ∈ β„‚)
53, 4sylan 578 . . . . 5 ((𝑔 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘”) Β·ih 𝑦) ∈ β„‚)
65ancoms 457 . . . 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 7114 . . 3 (𝑦 ∈ β„‹ β†’ 𝐺: β„‹βŸΆβ„‚)
9 hvmulcl 30533 . . . . . . 7 ((π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹) β†’ (π‘₯ Β·β„Ž 𝑀) ∈ β„‹)
101lnopaddi 31491 . . . . . . . . . . . 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 1130 . . . . . . . . . . 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 7426 . . . . . . . . . 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 7084 . . . . . . . . . . 11 ((π‘₯ Β·β„Ž 𝑀) ∈ β„‹ β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) ∈ β„‹)
142ffvelcdmi 7084 . . . . . . . . . . 11 (𝑧 ∈ β„‹ β†’ (π‘‡β€˜π‘§) ∈ β„‹)
15 id 22 . . . . . . . . . . 11 (𝑦 ∈ β„‹ β†’ 𝑦 ∈ β„‹)
16 ax-his2 30603 . . . . . . . . . . 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 1158 . . . . . . . . . 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 2770 . . . . . . . . 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 1123 . . . . . . . 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 1116 . . . . . . 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 677 . . . . . 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 30532 . . . . . . . . 9 (((π‘₯ Β·β„Ž 𝑀) ∈ β„‹ ∧ 𝑧 ∈ β„‹) β†’ ((π‘₯ Β·β„Ž 𝑀) +β„Ž 𝑧) ∈ β„‹)
239, 22sylan 578 . . . . . . . 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 31587 . . . . . . . 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 710 . . . . . 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 7084 . . . . . . . . . . 11 (𝑀 ∈ β„‹ β†’ (π‘‡β€˜π‘€) ∈ β„‹)
29 ax-his3 30604 . . . . . . . . . . 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 1162 . . . . . . . . . 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 1123 . . . . . . . . 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 1118 . . . . . . . 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 31492 . . . . . . . . . 10 ((π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹) β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑀)) = (π‘₯ Β·β„Ž (π‘‡β€˜π‘€)))
3433oveq1d 7426 . . . . . . . . 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 480 . . . . . . . 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 31587 . . . . . . . . . 10 (𝑀 ∈ β„‹ β†’ (πΊβ€˜π‘€) = ((π‘‡β€˜π‘€) Β·ih 𝑦))
3736oveq2d 7427 . . . . . . . . 9 (𝑀 ∈ β„‹ β†’ (π‘₯ Β· (πΊβ€˜π‘€)) = (π‘₯ Β· ((π‘‡β€˜π‘€) Β·ih 𝑦)))
3837ad2antll 725 . . . . . . . 8 ((𝑦 ∈ β„‹ ∧ (π‘₯ ∈ β„‚ ∧ 𝑀 ∈ β„‹)) β†’ (π‘₯ Β· (πΊβ€˜π‘€)) = (π‘₯ Β· ((π‘‡β€˜π‘€) Β·ih 𝑦)))
3932, 35, 383eqtr4rd 2781 . . . . . . 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 31587 . . . . . . 7 (𝑧 ∈ β„‹ β†’ (πΊβ€˜π‘§) = ((π‘‡β€˜π‘§) Β·ih 𝑦))
4139, 40oveqan12d 7430 . . . . . 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 2780 . . . . 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 31403 . . 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 581 . 2 (𝑦 ∈ β„‹ β†’ 𝐺 ∈ LinFn)
471, 24nmcopexi 31547 . . . . 5 (normopβ€˜π‘‡) ∈ ℝ
48 normcl 30645 . . . . 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 585 . . . 4 (𝑦 ∈ β„‹ β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
5140adantr 479 . . . . . . . . . 10 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (πΊβ€˜π‘§) = ((π‘‡β€˜π‘§) Β·ih 𝑦))
52 hicl 30600 . . . . . . . . . . 11 (((π‘‡β€˜π‘§) ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘§) Β·ih 𝑦) ∈ β„‚)
5314, 52sylan 578 . . . . . . . . . 10 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((π‘‡β€˜π‘§) Β·ih 𝑦) ∈ β„‚)
5451, 53eqeltrd 2831 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (πΊβ€˜π‘§) ∈ β„‚)
5554abscld 15387 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ∈ ℝ)
56 normcl 30645 . . . . . . . . . 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 594 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
60 normcl 30645 . . . . . . . . . 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 585 . . . . . . . . 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 594 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)) ∈ ℝ)
6551fveq2d 6894 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) = (absβ€˜((π‘‡β€˜π‘§) Β·ih 𝑦)))
66 bcs 30701 . . . . . . . . . 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 578 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜((π‘‡β€˜π‘§) Β·ih 𝑦)) ≀ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)))
6865, 67eqbrtrd 5169 . . . . . . . 8 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ≀ ((normβ„Žβ€˜(π‘‡β€˜π‘§)) Β· (normβ„Žβ€˜π‘¦)))
6957adantr 479 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (normβ„Žβ€˜(π‘‡β€˜π‘§)) ∈ ℝ)
7062adantr 479 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)) ∈ ℝ)
71 normge0 30646 . . . . . . . . . . 11 (𝑦 ∈ β„‹ β†’ 0 ≀ (normβ„Žβ€˜π‘¦))
7248, 71jca 510 . . . . . . . . . 10 (𝑦 ∈ β„‹ β†’ ((normβ„Žβ€˜π‘¦) ∈ ℝ ∧ 0 ≀ (normβ„Žβ€˜π‘¦)))
7372adantl 480 . . . . . . . . 9 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((normβ„Žβ€˜π‘¦) ∈ ℝ ∧ 0 ≀ (normβ„Žβ€˜π‘¦)))
741, 24nmcoplbi 31548 . . . . . . . . . 10 (𝑧 ∈ β„‹ β†’ (normβ„Žβ€˜(π‘‡β€˜π‘§)) ≀ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘§)))
7574adantr 479 . . . . . . . . 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 1371 . . . . . . . 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 1446 . . . . . . . 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 594 . . . . . . 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 5173 . . . . . 6 ((𝑧 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜(πΊβ€˜π‘§)) ≀ (((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π‘¦)) Β· (normβ„Žβ€˜π‘§)))
8685ancoms 457 . . . . 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 7418 . . . . . . 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 5159 . . . . . 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 3611 . . . 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 582 . . 3 (𝑦 ∈ β„‹ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ β„‹ (absβ€˜(πΊβ€˜π‘§)) ≀ (π‘₯ Β· (normβ„Žβ€˜π‘§)))
93 lnfncon 31576 . . . 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 256 . 2 (𝑦 ∈ β„‹ β†’ 𝐺 ∈ ContFn)
9646, 95jca 510 1 (𝑦 ∈ β„‹ β†’ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068   class class class wbr 5147   ↦ cmpt 5230  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  β„cr 11111  0cc0 11112   + caddc 11115   Β· cmul 11117   ≀ cle 11253  abscabs 15185   β„‹chba 30439   +β„Ž cva 30440   Β·β„Ž csm 30441   Β·ih csp 30442  normβ„Žcno 30443  normopcnop 30465  ContOpccop 30466  LinOpclo 30467  ContFnccnfn 30473  LinFnclf 30474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  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 30519  ax-hfvadd 30520  ax-hvcom 30521  ax-hvass 30522  ax-hv0cl 30523  ax-hvaddid 30524  ax-hfvmul 30525  ax-hvmulid 30526  ax-hvmulass 30527  ax-hvdistr1 30528  ax-hvdistr2 30529  ax-hvmul0 30530  ax-hfi 30599  ax-his1 30602  ax-his2 30603  ax-his3 30604  ax-his4 30605
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  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-se 5631  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  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 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ioo 13332  df-icc 13335  df-fz 13489  df-fzo 13632  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-sum 15637  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-hom 17225  df-cco 17226  df-rest 17372  df-topn 17373  df-0g 17391  df-gsum 17392  df-topgen 17393  df-pt 17394  df-prds 17397  df-xrs 17452  df-qtop 17457  df-imas 17458  df-xps 17460  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-mulg 18987  df-cntz 19222  df-cmn 19691  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-cnfld 21145  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-ntr 22744  df-cls 22745  df-cn 22951  df-cnp 22952  df-t1 23038  df-haus 23039  df-tx 23286  df-hmeo 23479  df-xms 24046  df-ms 24047  df-tms 24048  df-grpo 30013  df-gid 30014  df-ginv 30015  df-gdiv 30016  df-ablo 30065  df-vc 30079  df-nv 30112  df-va 30115  df-ba 30116  df-sm 30117  df-0v 30118  df-vs 30119  df-nmcv 30120  df-ims 30121  df-dip 30221  df-ph 30333  df-hnorm 30488  df-hba 30489  df-hvsub 30491  df-nmop 31359  df-cnop 31360  df-lnop 31361  df-nmfn 31365  df-cnfn 31367  df-lnfn 31368
This theorem is referenced by:  cnlnadjlem3  31589  cnlnadjlem5  31591
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