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Mirrors > Home > MPE Home > Th. List > ip2dii | Structured version Visualization version GIF version |
Description: Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ip2dii.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
ip2dii.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
ip2dii.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
ip2dii.u | ⊢ 𝑈 ∈ CPreHilOLD |
ip2dii.a | ⊢ 𝐴 ∈ 𝑋 |
ip2dii.b | ⊢ 𝐵 ∈ 𝑋 |
ip2dii.c | ⊢ 𝐶 ∈ 𝑋 |
ip2dii.d | ⊢ 𝐷 ∈ 𝑋 |
Ref | Expression |
---|---|
ip2dii | ⊢ ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ip2dii.u | . . . 4 ⊢ 𝑈 ∈ CPreHilOLD | |
2 | ip2dii.a | . . . . 5 ⊢ 𝐴 ∈ 𝑋 | |
3 | ip2dii.c | . . . . 5 ⊢ 𝐶 ∈ 𝑋 | |
4 | ip2dii.d | . . . . 5 ⊢ 𝐷 ∈ 𝑋 | |
5 | 2, 3, 4 | 3pm3.2i 1336 | . . . 4 ⊢ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) |
6 | ip2dii.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
7 | ip2dii.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
8 | ip2dii.7 | . . . . 5 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
9 | 6, 7, 8 | dipdi 28626 | . . . 4 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃𝐶) + (𝐴𝑃𝐷))) |
10 | 1, 5, 9 | mp2an 691 | . . 3 ⊢ (𝐴𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃𝐶) + (𝐴𝑃𝐷)) |
11 | ip2dii.b | . . . . 5 ⊢ 𝐵 ∈ 𝑋 | |
12 | 11, 3, 4 | 3pm3.2i 1336 | . . . 4 ⊢ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) |
13 | 6, 7, 8 | dipdi 28626 | . . . 4 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐵𝑃(𝐶𝐺𝐷)) = ((𝐵𝑃𝐶) + (𝐵𝑃𝐷))) |
14 | 1, 12, 13 | mp2an 691 | . . 3 ⊢ (𝐵𝑃(𝐶𝐺𝐷)) = ((𝐵𝑃𝐶) + (𝐵𝑃𝐷)) |
15 | 10, 14 | oveq12i 7147 | . 2 ⊢ ((𝐴𝑃(𝐶𝐺𝐷)) + (𝐵𝑃(𝐶𝐺𝐷))) = (((𝐴𝑃𝐶) + (𝐴𝑃𝐷)) + ((𝐵𝑃𝐶) + (𝐵𝑃𝐷))) |
16 | 1 | phnvi 28599 | . . . . 5 ⊢ 𝑈 ∈ NrmCVec |
17 | 6, 7 | nvgcl 28403 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐶𝐺𝐷) ∈ 𝑋) |
18 | 16, 3, 4, 17 | mp3an 1458 | . . . 4 ⊢ (𝐶𝐺𝐷) ∈ 𝑋 |
19 | 2, 11, 18 | 3pm3.2i 1336 | . . 3 ⊢ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋) |
20 | 6, 7, 8 | dipdir 28625 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃(𝐶𝐺𝐷)) + (𝐵𝑃(𝐶𝐺𝐷)))) |
21 | 1, 19, 20 | mp2an 691 | . 2 ⊢ ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃(𝐶𝐺𝐷)) + (𝐵𝑃(𝐶𝐺𝐷))) |
22 | 6, 8 | dipcl 28495 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑃𝐶) ∈ ℂ) |
23 | 16, 2, 3, 22 | mp3an 1458 | . . 3 ⊢ (𝐴𝑃𝐶) ∈ ℂ |
24 | 6, 8 | dipcl 28495 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐵𝑃𝐷) ∈ ℂ) |
25 | 16, 11, 4, 24 | mp3an 1458 | . . 3 ⊢ (𝐵𝑃𝐷) ∈ ℂ |
26 | 6, 8 | dipcl 28495 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐴𝑃𝐷) ∈ ℂ) |
27 | 16, 2, 4, 26 | mp3an 1458 | . . 3 ⊢ (𝐴𝑃𝐷) ∈ ℂ |
28 | 6, 8 | dipcl 28495 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝑃𝐶) ∈ ℂ) |
29 | 16, 11, 3, 28 | mp3an 1458 | . . 3 ⊢ (𝐵𝑃𝐶) ∈ ℂ |
30 | 23, 25, 27, 29 | add42i 10854 | . 2 ⊢ (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶))) = (((𝐴𝑃𝐶) + (𝐴𝑃𝐷)) + ((𝐵𝑃𝐶) + (𝐵𝑃𝐷))) |
31 | 15, 21, 30 | 3eqtr4i 2831 | 1 ⊢ ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 + caddc 10529 NrmCVeccnv 28367 +𝑣 cpv 28368 BaseSetcba 28369 ·𝑖OLDcdip 28483 CPreHilOLDccphlo 28595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-grpo 28276 df-gid 28277 df-ginv 28278 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-nmcv 28383 df-dip 28484 df-ph 28596 |
This theorem is referenced by: pythi 28633 |
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