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| Mirrors > Home > MPE Home > Th. List > ipval | Structured version Visualization version GIF version | ||
| Description: Value of the inner product. The definition is meaningful for normed complex vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law, although for convenience we define it for any normed complex vector space. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, the norm is 𝑁, and the set of vectors is 𝑋. Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| dipfval.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
| dipfval.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
| dipfval.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
| dipfval.6 | ⊢ 𝑁 = (normCV‘𝑈) |
| dipfval.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
| Ref | Expression |
|---|---|
| ipval | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dipfval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 2 | dipfval.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
| 3 | dipfval.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
| 4 | dipfval.6 | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
| 5 | dipfval.7 | . . . . 5 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
| 6 | 1, 2, 3, 4, 5 | dipfval 30721 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))) |
| 7 | 6 | oveqd 7448 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝐴𝑃𝐵) = (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))𝐵)) |
| 8 | fvoveq1 7454 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦))) = (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))) | |
| 9 | 8 | oveq1d 7446 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) |
| 10 | 9 | oveq2d 7447 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2))) |
| 11 | 10 | sumeq2sdv 15739 | . . . . 5 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) = Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2))) |
| 12 | 11 | oveq1d 7446 | . . . 4 ⊢ (𝑥 = 𝐴 → (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)) |
| 13 | oveq2 7439 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((i↑𝑘)𝑆𝑦) = ((i↑𝑘)𝑆𝐵)) | |
| 14 | 13 | oveq2d 7447 | . . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝐴𝐺((i↑𝑘)𝑆𝑦)) = (𝐴𝐺((i↑𝑘)𝑆𝐵))) |
| 15 | 14 | fveq2d 6910 | . . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦))) = (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))) |
| 16 | 15 | oveq1d 7446 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) |
| 17 | 16 | oveq2d 7447 | . . . . . 6 ⊢ (𝑦 = 𝐵 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2))) |
| 18 | 17 | sumeq2sdv 15739 | . . . . 5 ⊢ (𝑦 = 𝐵 → Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) = Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2))) |
| 19 | 18 | oveq1d 7446 | . . . 4 ⊢ (𝑦 = 𝐵 → (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4)) |
| 20 | eqid 2737 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)) | |
| 21 | ovex 7464 | . . . 4 ⊢ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4) ∈ V | |
| 22 | 12, 19, 20, 21 | ovmpo 7593 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4)) |
| 23 | 7, 22 | sylan9eq 2797 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4)) |
| 24 | 23 | 3impb 1115 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1c1 11156 ici 11157 · cmul 11160 / cdiv 11920 2c2 12321 4c4 12323 ...cfz 13547 ↑cexp 14102 Σcsu 15722 NrmCVeccnv 30603 +𝑣 cpv 30604 BaseSetcba 30605 ·𝑠OLD cns 30606 normCVcnmcv 30609 ·𝑖OLDcdip 30719 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seq 14043 df-sum 15723 df-dip 30720 |
| This theorem is referenced by: ipval2 30726 dipcl 30731 ipf 30732 |
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