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Theorem branmfn 31358
Description: The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
branmfn (𝐴 ∈ β„‹ β†’ (normfnβ€˜(braβ€˜π΄)) = (normβ„Žβ€˜π΄))

Proof of Theorem branmfn
Dummy variables π‘₯ 𝑦 𝑧 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2fveq3 6897 . . 3 (𝐴 = 0β„Ž β†’ (normfnβ€˜(braβ€˜π΄)) = (normfnβ€˜(braβ€˜0β„Ž)))
2 fveq2 6892 . . 3 (𝐴 = 0β„Ž β†’ (normβ„Žβ€˜π΄) = (normβ„Žβ€˜0β„Ž))
31, 2eqeq12d 2749 . 2 (𝐴 = 0β„Ž β†’ ((normfnβ€˜(braβ€˜π΄)) = (normβ„Žβ€˜π΄) ↔ (normfnβ€˜(braβ€˜0β„Ž)) = (normβ„Žβ€˜0β„Ž)))
4 brafn 31200 . . . . 5 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄): β„‹βŸΆβ„‚)
5 nmfnval 31129 . . . . 5 ((braβ€˜π΄): β„‹βŸΆβ„‚ β†’ (normfnβ€˜(braβ€˜π΄)) = sup({π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}, ℝ*, < ))
64, 5syl 17 . . . 4 (𝐴 ∈ β„‹ β†’ (normfnβ€˜(braβ€˜π΄)) = sup({π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}, ℝ*, < ))
76adantr 482 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normfnβ€˜(braβ€˜π΄)) = sup({π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}, ℝ*, < ))
8 nmfnsetre 31130 . . . . . . . 8 ((braβ€˜π΄): β„‹βŸΆβ„‚ β†’ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))} βŠ† ℝ)
94, 8syl 17 . . . . . . 7 (𝐴 ∈ β„‹ β†’ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))} βŠ† ℝ)
10 ressxr 11258 . . . . . . 7 ℝ βŠ† ℝ*
119, 10sstrdi 3995 . . . . . 6 (𝐴 ∈ β„‹ β†’ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))} βŠ† ℝ*)
12 normcl 30378 . . . . . . 7 (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
1312rexrd 11264 . . . . . 6 (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) ∈ ℝ*)
1411, 13jca 513 . . . . 5 (𝐴 ∈ β„‹ β†’ ({π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))} βŠ† ℝ* ∧ (normβ„Žβ€˜π΄) ∈ ℝ*))
1514adantr 482 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ({π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))} βŠ† ℝ* ∧ (normβ„Žβ€˜π΄) ∈ ℝ*))
16 vex 3479 . . . . . . . 8 𝑧 ∈ V
17 eqeq1 2737 . . . . . . . . . 10 (π‘₯ = 𝑧 β†’ (π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)) ↔ 𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦))))
1817anbi2d 630 . . . . . . . . 9 (π‘₯ = 𝑧 β†’ (((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) ↔ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ 𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))))
1918rexbidv 3179 . . . . . . . 8 (π‘₯ = 𝑧 β†’ (βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) ↔ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ 𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))))
2016, 19elab 3669 . . . . . . 7 (𝑧 ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))} ↔ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ 𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦))))
21 id 22 . . . . . . . . . . . . 13 (𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦)) β†’ 𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))
22 braval 31197 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π‘¦) = (𝑦 Β·ih 𝐴))
2322fveq2d 6896 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (absβ€˜((braβ€˜π΄)β€˜π‘¦)) = (absβ€˜(𝑦 Β·ih 𝐴)))
2423adantr 482 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‹ ∧ 𝑦 ∈ β„‹) ∧ (normβ„Žβ€˜π‘¦) ≀ 1) β†’ (absβ€˜((braβ€˜π΄)β€˜π‘¦)) = (absβ€˜(𝑦 Β·ih 𝐴)))
2521, 24sylan9eqr 2795 . . . . . . . . . . . 12 ((((𝐴 ∈ β„‹ ∧ 𝑦 ∈ β„‹) ∧ (normβ„Žβ€˜π‘¦) ≀ 1) ∧ 𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) β†’ 𝑧 = (absβ€˜(𝑦 Β·ih 𝐴)))
26 bcs2 30435 . . . . . . . . . . . . . . 15 ((𝑦 ∈ β„‹ ∧ 𝐴 ∈ β„‹ ∧ (normβ„Žβ€˜π‘¦) ≀ 1) β†’ (absβ€˜(𝑦 Β·ih 𝐴)) ≀ (normβ„Žβ€˜π΄))
27263expa 1119 . . . . . . . . . . . . . 14 (((𝑦 ∈ β„‹ ∧ 𝐴 ∈ β„‹) ∧ (normβ„Žβ€˜π‘¦) ≀ 1) β†’ (absβ€˜(𝑦 Β·ih 𝐴)) ≀ (normβ„Žβ€˜π΄))
2827ancom1s 652 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‹ ∧ 𝑦 ∈ β„‹) ∧ (normβ„Žβ€˜π‘¦) ≀ 1) β†’ (absβ€˜(𝑦 Β·ih 𝐴)) ≀ (normβ„Žβ€˜π΄))
2928adantr 482 . . . . . . . . . . . 12 ((((𝐴 ∈ β„‹ ∧ 𝑦 ∈ β„‹) ∧ (normβ„Žβ€˜π‘¦) ≀ 1) ∧ 𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) β†’ (absβ€˜(𝑦 Β·ih 𝐴)) ≀ (normβ„Žβ€˜π΄))
3025, 29eqbrtrd 5171 . . . . . . . . . . 11 ((((𝐴 ∈ β„‹ ∧ 𝑦 ∈ β„‹) ∧ (normβ„Žβ€˜π‘¦) ≀ 1) ∧ 𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) β†’ 𝑧 ≀ (normβ„Žβ€˜π΄))
3130exp41 436 . . . . . . . . . 10 (𝐴 ∈ β„‹ β†’ (𝑦 ∈ β„‹ β†’ ((normβ„Žβ€˜π‘¦) ≀ 1 β†’ (𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦)) β†’ 𝑧 ≀ (normβ„Žβ€˜π΄)))))
3231imp4a 424 . . . . . . . . 9 (𝐴 ∈ β„‹ β†’ (𝑦 ∈ β„‹ β†’ (((normβ„Žβ€˜π‘¦) ≀ 1 ∧ 𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) β†’ 𝑧 ≀ (normβ„Žβ€˜π΄))))
3332rexlimdv 3154 . . . . . . . 8 (𝐴 ∈ β„‹ β†’ (βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ 𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) β†’ 𝑧 ≀ (normβ„Žβ€˜π΄)))
3433imp 408 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ 𝑧 = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))) β†’ 𝑧 ≀ (normβ„Žβ€˜π΄))
3520, 34sylan2b 595 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝑧 ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}) β†’ 𝑧 ≀ (normβ„Žβ€˜π΄))
3635ralrimiva 3147 . . . . 5 (𝐴 ∈ β„‹ β†’ βˆ€π‘§ ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}𝑧 ≀ (normβ„Žβ€˜π΄))
3736adantr 482 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ βˆ€π‘§ ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}𝑧 ≀ (normβ„Žβ€˜π΄))
3812recnd 11242 . . . . . . . . . 10 (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) ∈ β„‚)
3938adantr 482 . . . . . . . . 9 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ β„‚)
40 normne0 30383 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄) β‰  0 ↔ 𝐴 β‰  0β„Ž))
4140biimpar 479 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) β‰  0)
4239, 41reccld 11983 . . . . . . . . . . . 12 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (1 / (normβ„Žβ€˜π΄)) ∈ β„‚)
43 simpl 484 . . . . . . . . . . . 12 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 𝐴 ∈ β„‹)
44 hvmulcl 30266 . . . . . . . . . . . 12 (((1 / (normβ„Žβ€˜π΄)) ∈ β„‚ ∧ 𝐴 ∈ β„‹) β†’ ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹)
4542, 43, 44syl2anc 585 . . . . . . . . . . 11 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹)
46 norm1 30502 . . . . . . . . . . . 12 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = 1)
47 1le1 11842 . . . . . . . . . . . 12 1 ≀ 1
4846, 47eqbrtrdi 5188 . . . . . . . . . . 11 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1)
49 ax-his3 30337 . . . . . . . . . . . . 13 (((1 / (normβ„Žβ€˜π΄)) ∈ β„‚ ∧ 𝐴 ∈ β„‹ ∧ 𝐴 ∈ β„‹) β†’ (((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴) = ((1 / (normβ„Žβ€˜π΄)) Β· (𝐴 Β·ih 𝐴)))
5042, 43, 43, 49syl3anc 1372 . . . . . . . . . . . 12 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴) = ((1 / (normβ„Žβ€˜π΄)) Β· (𝐴 Β·ih 𝐴)))
5112adantr 482 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
5251, 41rereccld 12041 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (1 / (normβ„Žβ€˜π΄)) ∈ ℝ)
53 hiidrcl 30348 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‹ β†’ (𝐴 Β·ih 𝐴) ∈ ℝ)
5453adantr 482 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (𝐴 Β·ih 𝐴) ∈ ℝ)
5552, 54remulcld 11244 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((1 / (normβ„Žβ€˜π΄)) Β· (𝐴 Β·ih 𝐴)) ∈ ℝ)
5650, 55eqeltrd 2834 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴) ∈ ℝ)
57 normgt0 30380 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ β„‹ β†’ (𝐴 β‰  0β„Ž ↔ 0 < (normβ„Žβ€˜π΄)))
5857biimpa 478 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 < (normβ„Žβ€˜π΄))
5951, 58recgt0d 12148 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 < (1 / (normβ„Žβ€˜π΄)))
60 0re 11216 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ
61 ltle 11302 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℝ ∧ (1 / (normβ„Žβ€˜π΄)) ∈ ℝ) β†’ (0 < (1 / (normβ„Žβ€˜π΄)) β†’ 0 ≀ (1 / (normβ„Žβ€˜π΄))))
6260, 61mpan 689 . . . . . . . . . . . . . . . 16 ((1 / (normβ„Žβ€˜π΄)) ∈ ℝ β†’ (0 < (1 / (normβ„Žβ€˜π΄)) β†’ 0 ≀ (1 / (normβ„Žβ€˜π΄))))
6352, 59, 62sylc 65 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 ≀ (1 / (normβ„Žβ€˜π΄)))
64 hiidge0 30351 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‹ β†’ 0 ≀ (𝐴 Β·ih 𝐴))
6564adantr 482 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 ≀ (𝐴 Β·ih 𝐴))
6652, 54, 63, 65mulge0d 11791 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 ≀ ((1 / (normβ„Žβ€˜π΄)) Β· (𝐴 Β·ih 𝐴)))
6766, 50breqtrrd 5177 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 ≀ (((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴))
6856, 67absidd 15369 . . . . . . . . . . . 12 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜(((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴)) = (((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴))
6939, 41recid2d 11986 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((1 / (normβ„Žβ€˜π΄)) Β· (normβ„Žβ€˜π΄)) = 1)
7069oveq2d 7425 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((normβ„Žβ€˜π΄) Β· ((1 / (normβ„Žβ€˜π΄)) Β· (normβ„Žβ€˜π΄))) = ((normβ„Žβ€˜π΄) Β· 1))
7139, 42, 39mul12d 11423 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((normβ„Žβ€˜π΄) Β· ((1 / (normβ„Žβ€˜π΄)) Β· (normβ„Žβ€˜π΄))) = ((1 / (normβ„Žβ€˜π΄)) Β· ((normβ„Žβ€˜π΄) Β· (normβ„Žβ€˜π΄))))
7238sqvald 14108 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄)↑2) = ((normβ„Žβ€˜π΄) Β· (normβ„Žβ€˜π΄)))
73 normsq 30387 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄)↑2) = (𝐴 Β·ih 𝐴))
7472, 73eqtr3d 2775 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄) Β· (normβ„Žβ€˜π΄)) = (𝐴 Β·ih 𝐴))
7574adantr 482 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((normβ„Žβ€˜π΄) Β· (normβ„Žβ€˜π΄)) = (𝐴 Β·ih 𝐴))
7675oveq2d 7425 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((1 / (normβ„Žβ€˜π΄)) Β· ((normβ„Žβ€˜π΄) Β· (normβ„Žβ€˜π΄))) = ((1 / (normβ„Žβ€˜π΄)) Β· (𝐴 Β·ih 𝐴)))
7771, 76eqtrd 2773 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((normβ„Žβ€˜π΄) Β· ((1 / (normβ„Žβ€˜π΄)) Β· (normβ„Žβ€˜π΄))) = ((1 / (normβ„Žβ€˜π΄)) Β· (𝐴 Β·ih 𝐴)))
7838mulridd 11231 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄) Β· 1) = (normβ„Žβ€˜π΄))
7978adantr 482 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((normβ„Žβ€˜π΄) Β· 1) = (normβ„Žβ€˜π΄))
8070, 77, 793eqtr3rd 2782 . . . . . . . . . . . 12 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) = ((1 / (normβ„Žβ€˜π΄)) Β· (𝐴 Β·ih 𝐴)))
8150, 68, 803eqtr4rd 2784 . . . . . . . . . . 11 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) = (absβ€˜(((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴)))
82 fveq2 6892 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) β†’ (normβ„Žβ€˜π‘¦) = (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)))
8382breq1d 5159 . . . . . . . . . . . . 13 (𝑦 = ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) β†’ ((normβ„Žβ€˜π‘¦) ≀ 1 ↔ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1))
84 fvoveq1 7432 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) β†’ (absβ€˜(𝑦 Β·ih 𝐴)) = (absβ€˜(((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴)))
8584eqeq2d 2744 . . . . . . . . . . . . 13 (𝑦 = ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) β†’ ((normβ„Žβ€˜π΄) = (absβ€˜(𝑦 Β·ih 𝐴)) ↔ (normβ„Žβ€˜π΄) = (absβ€˜(((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴))))
8683, 85anbi12d 632 . . . . . . . . . . . 12 (𝑦 = ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) β†’ (((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜(𝑦 Β·ih 𝐴))) ↔ ((normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜(((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴)))))
8786rspcev 3613 . . . . . . . . . . 11 ((((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹ ∧ ((normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜(((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴)))) β†’ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜(𝑦 Β·ih 𝐴))))
8845, 48, 81, 87syl12anc 836 . . . . . . . . . 10 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜(𝑦 Β·ih 𝐴))))
8923eqeq2d 2744 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((normβ„Žβ€˜π΄) = (absβ€˜((braβ€˜π΄)β€˜π‘¦)) ↔ (normβ„Žβ€˜π΄) = (absβ€˜(𝑦 Β·ih 𝐴))))
9089anbi2d 630 . . . . . . . . . . . 12 ((𝐴 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) ↔ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜(𝑦 Β·ih 𝐴)))))
9190rexbidva 3177 . . . . . . . . . . 11 (𝐴 ∈ β„‹ β†’ (βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) ↔ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜(𝑦 Β·ih 𝐴)))))
9291adantr 482 . . . . . . . . . 10 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) ↔ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜(𝑦 Β·ih 𝐴)))))
9388, 92mpbird 257 . . . . . . . . 9 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜((braβ€˜π΄)β€˜π‘¦))))
94 eqeq1 2737 . . . . . . . . . . 11 (π‘₯ = (normβ„Žβ€˜π΄) β†’ (π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)) ↔ (normβ„Žβ€˜π΄) = (absβ€˜((braβ€˜π΄)β€˜π‘¦))))
9594anbi2d 630 . . . . . . . . . 10 (π‘₯ = (normβ„Žβ€˜π΄) β†’ (((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) ↔ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))))
9695rexbidv 3179 . . . . . . . . 9 (π‘₯ = (normβ„Žβ€˜π΄) β†’ (βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦))) ↔ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ (normβ„Žβ€˜π΄) = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))))
9739, 93, 96elabd 3672 . . . . . . . 8 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))})
98 breq2 5153 . . . . . . . . 9 (𝑀 = (normβ„Žβ€˜π΄) β†’ (𝑧 < 𝑀 ↔ 𝑧 < (normβ„Žβ€˜π΄)))
9998rspcev 3613 . . . . . . . 8 (((normβ„Žβ€˜π΄) ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))} ∧ 𝑧 < (normβ„Žβ€˜π΄)) β†’ βˆƒπ‘€ ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}𝑧 < 𝑀)
10097, 99sylan 581 . . . . . . 7 (((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) ∧ 𝑧 < (normβ„Žβ€˜π΄)) β†’ βˆƒπ‘€ ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}𝑧 < 𝑀)
101100adantlr 714 . . . . . 6 ((((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (normβ„Žβ€˜π΄)) β†’ βˆƒπ‘€ ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}𝑧 < 𝑀)
102101ex 414 . . . . 5 (((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) ∧ 𝑧 ∈ ℝ) β†’ (𝑧 < (normβ„Žβ€˜π΄) β†’ βˆƒπ‘€ ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}𝑧 < 𝑀))
103102ralrimiva 3147 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ βˆ€π‘§ ∈ ℝ (𝑧 < (normβ„Žβ€˜π΄) β†’ βˆƒπ‘€ ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}𝑧 < 𝑀))
104 supxr2 13293 . . . 4 ((({π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))} βŠ† ℝ* ∧ (normβ„Žβ€˜π΄) ∈ ℝ*) ∧ (βˆ€π‘§ ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}𝑧 ≀ (normβ„Žβ€˜π΄) ∧ βˆ€π‘§ ∈ ℝ (𝑧 < (normβ„Žβ€˜π΄) β†’ βˆƒπ‘€ ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}𝑧 < 𝑀))) β†’ sup({π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}, ℝ*, < ) = (normβ„Žβ€˜π΄))
10515, 37, 103, 104syl12anc 836 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ sup({π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))}, ℝ*, < ) = (normβ„Žβ€˜π΄))
1067, 105eqtrd 2773 . 2 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normfnβ€˜(braβ€˜π΄)) = (normβ„Žβ€˜π΄))
107 nmfn0 31240 . . . 4 (normfnβ€˜( β„‹ Γ— {0})) = 0
108 bra0 31203 . . . . 5 (braβ€˜0β„Ž) = ( β„‹ Γ— {0})
109108fveq2i 6895 . . . 4 (normfnβ€˜(braβ€˜0β„Ž)) = (normfnβ€˜( β„‹ Γ— {0}))
110 norm0 30381 . . . 4 (normβ„Žβ€˜0β„Ž) = 0
111107, 109, 1103eqtr4i 2771 . . 3 (normfnβ€˜(braβ€˜0β„Ž)) = (normβ„Žβ€˜0β„Ž)
112111a1i 11 . 2 (𝐴 ∈ β„‹ β†’ (normfnβ€˜(braβ€˜0β„Ž)) = (normβ„Žβ€˜0β„Ž))
1133, 106, 112pm2.61ne 3028 1 (𝐴 ∈ β„‹ β†’ (normfnβ€˜(braβ€˜π΄)) = (normβ„Žβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071   βŠ† wss 3949  {csn 4629   class class class wbr 5149   Γ— cxp 5675  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  supcsup 9435  β„‚cc 11108  β„cr 11109  0cc0 11110  1c1 11111   Β· cmul 11115  β„*cxr 11247   < clt 11248   ≀ cle 11249   / cdiv 11871  2c2 12267  β†‘cexp 14027  abscabs 15181   β„‹chba 30172   Β·β„Ž csm 30174   Β·ih csp 30175  normβ„Žcno 30176  0β„Žc0v 30177  normfncnmf 30204  bracbr 30209
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190  ax-hilex 30252  ax-hfvadd 30253  ax-hvcom 30254  ax-hvass 30255  ax-hv0cl 30256  ax-hvaddid 30257  ax-hfvmul 30258  ax-hvmulid 30259  ax-hvmulass 30260  ax-hvdistr1 30261  ax-hvdistr2 30262  ax-hvmul0 30263  ax-hfi 30332  ax-his1 30335  ax-his2 30336  ax-his3 30337  ax-his4 30338
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-icc 13331  df-fz 13485  df-fzo 13628  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-sum 15633  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-hom 17221  df-cco 17222  df-rest 17368  df-topn 17369  df-0g 17387  df-gsum 17388  df-topgen 17389  df-pt 17390  df-prds 17393  df-xrs 17448  df-qtop 17453  df-imas 17454  df-xps 17456  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-mulg 18951  df-cntz 19181  df-cmn 19650  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-cnfld 20945  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-cn 22731  df-cnp 22732  df-t1 22818  df-haus 22819  df-tx 23066  df-hmeo 23259  df-xms 23826  df-ms 23827  df-tms 23828  df-grpo 29746  df-gid 29747  df-ginv 29748  df-gdiv 29749  df-ablo 29798  df-vc 29812  df-nv 29845  df-va 29848  df-ba 29849  df-sm 29850  df-0v 29851  df-vs 29852  df-nmcv 29853  df-ims 29854  df-dip 29954  df-ph 30066  df-hnorm 30221  df-hba 30222  df-hvsub 30224  df-nmfn 31098  df-lnfn 31101  df-bra 31103
This theorem is referenced by:  brabn  31359
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