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Mirrors > Home > MPE Home > Th. List > dipcl | Structured version Visualization version GIF version |
Description: An inner product is a complex number. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ipcl.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
ipcl.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
Ref | Expression |
---|---|
dipcl | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipcl.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | eqid 2821 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
3 | eqid 2821 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | eqid 2821 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
5 | ipcl.7 | . . 3 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
6 | 1, 2, 3, 4, 5 | ipval 28474 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝐵)))↑2)) / 4)) |
7 | fzfid 13335 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1...4) ∈ Fin) | |
8 | ax-icn 10590 | . . . . . . 7 ⊢ i ∈ ℂ | |
9 | elfznn 12930 | . . . . . . . 8 ⊢ (𝑘 ∈ (1...4) → 𝑘 ∈ ℕ) | |
10 | 9 | nnnn0d 11949 | . . . . . . 7 ⊢ (𝑘 ∈ (1...4) → 𝑘 ∈ ℕ0) |
11 | expcl 13441 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (i↑𝑘) ∈ ℂ) | |
12 | 8, 10, 11 | sylancr 589 | . . . . . 6 ⊢ (𝑘 ∈ (1...4) → (i↑𝑘) ∈ ℂ) |
13 | 12 | adantl 484 | . . . . 5 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑘 ∈ (1...4)) → (i↑𝑘) ∈ ℂ) |
14 | 1, 2, 3, 4, 5 | ipval2lem4 28477 | . . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (i↑𝑘) ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ) |
15 | 12, 14 | sylan2 594 | . . . . 5 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑘 ∈ (1...4)) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ) |
16 | 13, 15 | mulcld 10655 | . . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑘 ∈ (1...4)) → ((i↑𝑘) · (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℂ) |
17 | 7, 16 | fsumcl 15084 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℂ) |
18 | 4cn 11716 | . . . 4 ⊢ 4 ∈ ℂ | |
19 | 4ne0 11739 | . . . 4 ⊢ 4 ≠ 0 | |
20 | divcl 11298 | . . . 4 ⊢ ((Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℂ ∧ 4 ∈ ℂ ∧ 4 ≠ 0) → (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝐵)))↑2)) / 4) ∈ ℂ) | |
21 | 18, 19, 20 | mp3an23 1449 | . . 3 ⊢ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℂ → (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝐵)))↑2)) / 4) ∈ ℂ) |
22 | 17, 21 | syl 17 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝐵)))↑2)) / 4) ∈ ℂ) |
23 | 6, 22 | eqeltrd 2913 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 ici 10533 · cmul 10536 / cdiv 11291 2c2 11686 4c4 11688 ℕ0cn0 11891 ...cfz 12886 ↑cexp 13423 Σcsu 15036 NrmCVeccnv 28355 +𝑣 cpv 28356 BaseSetcba 28357 ·𝑠OLD cns 28358 normCVcnmcv 28361 ·𝑖OLDcdip 28471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-grpo 28264 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 df-dip 28472 |
This theorem is referenced by: ipf 28484 ipipcj 28486 ip1ilem 28597 ip2i 28599 ipasslem1 28602 ipasslem2 28603 ipasslem4 28605 ipasslem5 28606 ipasslem7 28607 ipasslem8 28608 ipasslem9 28609 ipasslem10 28610 ipasslem11 28611 dipdi 28614 ip2dii 28615 dipassr 28617 dipsubdir 28619 dipsubdi 28620 pythi 28621 siilem1 28622 siilem2 28623 siii 28624 ipblnfi 28626 ip2eqi 28627 htthlem 28688 |
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