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Theorem branmfn 31089
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 6848 . . 3 (𝐴 = 0β„Ž β†’ (normfnβ€˜(braβ€˜π΄)) = (normfnβ€˜(braβ€˜0β„Ž)))
2 fveq2 6843 . . 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 30931 . . . . 5 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄): β„‹βŸΆβ„‚)
5 nmfnval 30860 . . . . 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 30861 . . . . . . . 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 11204 . . . . . . 7 ℝ βŠ† ℝ*
119, 10sstrdi 3957 . . . . . 6 (𝐴 ∈ β„‹ β†’ {π‘₯ ∣ βˆƒπ‘¦ ∈ β„‹ ((normβ„Žβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (absβ€˜((braβ€˜π΄)β€˜π‘¦)))} βŠ† ℝ*)
12 normcl 30109 . . . . . . 7 (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
1312rexrd 11210 . . . . . 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 3448 . . . . . . . 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 3172 . . . . . . . 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 3631 . . . . . . 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 30928 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π‘¦) = (𝑦 Β·ih 𝐴))
2322fveq2d 6847 . . . . . . . . . . . . . 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 30166 . . . . . . . . . . . . . . 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 5128 . . . . . . . . . . 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 3147 . . . . . . . 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 3140 . . . . 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 11188 . . . . . . . . . 10 (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) ∈ β„‚)
3938adantr 482 . . . . . . . . 9 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ β„‚)
40 normne0 30114 . . . . . . . . . . . . . 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 11929 . . . . . . . . . . . 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 29997 . . . . . . . . . . . 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 30233 . . . . . . . . . . . 12 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = 1)
47 1le1 11788 . . . . . . . . . . . 12 1 ≀ 1
4846, 47eqbrtrdi 5145 . . . . . . . . . . 11 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1)
49 ax-his3 30068 . . . . . . . . . . . . 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 11987 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (1 / (normβ„Žβ€˜π΄)) ∈ ℝ)
53 hiidrcl 30079 . . . . . . . . . . . . . . . 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 11190 . . . . . . . . . . . . . 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 30111 . . . . . . . . . . . . . . . . . 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 12094 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 < (1 / (normβ„Žβ€˜π΄)))
60 0re 11162 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ
61 ltle 11248 . . . . . . . . . . . . . . . . 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 30082 . . . . . . . . . . . . . . . 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 11737 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 ≀ ((1 / (normβ„Žβ€˜π΄)) Β· (𝐴 Β·ih 𝐴)))
6766, 50breqtrrd 5134 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 ≀ (((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) Β·ih 𝐴))
6856, 67absidd 15313 . . . . . . . . . . . 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 11932 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((1 / (normβ„Žβ€˜π΄)) Β· (normβ„Žβ€˜π΄)) = 1)
7069oveq2d 7374 . . . . . . . . . . . . 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 11369 . . . . . . . . . . . . . 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 14054 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄)↑2) = ((normβ„Žβ€˜π΄) Β· (normβ„Žβ€˜π΄)))
73 normsq 30118 . . . . . . . . . . . . . . . . 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 7374 . . . . . . . . . . . . . 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)))
7838mulid1d 11177 . . . . . . . . . . . . . 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 6843 . . . . . . . . . . . . . 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 5116 . . . . . . . . . . . . 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 7381 . . . . . . . . . . . . . 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 3580 . . . . . . . . . . 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 3170 . . . . . . . . . . 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 3172 . . . . . . . . 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 3634 . . . . . . . 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 5110 . . . . . . . . 9 (𝑀 = (normβ„Žβ€˜π΄) β†’ (𝑧 < 𝑀 ↔ 𝑧 < (normβ„Žβ€˜π΄)))
9998rspcev 3580 . . . . . . . 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 3140 . . . 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 13239 . . . 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 30971 . . . 4 (normfnβ€˜( β„‹ Γ— {0})) = 0
108 bra0 30934 . . . . 5 (braβ€˜0β„Ž) = ( β„‹ Γ— {0})
109108fveq2i 6846 . . . 4 (normfnβ€˜(braβ€˜0β„Ž)) = (normfnβ€˜( β„‹ Γ— {0}))
110 norm0 30112 . . . 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 3027 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 2940  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3911  {csn 4587   class class class wbr 5106   Γ— cxp 5632  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  supcsup 9381  β„‚cc 11054  β„cr 11055  0cc0 11056  1c1 11057   Β· cmul 11061  β„*cxr 11193   < clt 11194   ≀ cle 11195   / cdiv 11817  2c2 12213  β†‘cexp 13973  abscabs 15125   β„‹chba 29903   Β·β„Ž csm 29905   Β·ih csp 29906  normβ„Žcno 29907  0β„Žc0v 29908  normfncnmf 29935  bracbr 29940
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134  ax-addf 11135  ax-mulf 11136  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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9309  df-fi 9352  df-sup 9383  df-inf 9384  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-q 12879  df-rp 12921  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13274  df-icc 13277  df-fz 13431  df-fzo 13574  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-sum 15577  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-starv 17153  df-sca 17154  df-vsca 17155  df-ip 17156  df-tset 17157  df-ple 17158  df-ds 17160  df-unif 17161  df-hom 17162  df-cco 17163  df-rest 17309  df-topn 17310  df-0g 17328  df-gsum 17329  df-topgen 17330  df-pt 17331  df-prds 17334  df-xrs 17389  df-qtop 17394  df-imas 17395  df-xps 17397  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-submnd 18607  df-mulg 18878  df-cntz 19102  df-cmn 19569  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-nmfn 30829  df-lnfn 30832  df-bra 30834
This theorem is referenced by:  brabn  31090
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