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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 1338 | . . . 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 29199 | . . . 4 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃𝐶) + (𝐴𝑃𝐷))) |
10 | 1, 5, 9 | mp2an 689 | . . 3 ⊢ (𝐴𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃𝐶) + (𝐴𝑃𝐷)) |
11 | ip2dii.b | . . . . 5 ⊢ 𝐵 ∈ 𝑋 | |
12 | 11, 3, 4 | 3pm3.2i 1338 | . . . 4 ⊢ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) |
13 | 6, 7, 8 | dipdi 29199 | . . . 4 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐵𝑃(𝐶𝐺𝐷)) = ((𝐵𝑃𝐶) + (𝐵𝑃𝐷))) |
14 | 1, 12, 13 | mp2an 689 | . . 3 ⊢ (𝐵𝑃(𝐶𝐺𝐷)) = ((𝐵𝑃𝐶) + (𝐵𝑃𝐷)) |
15 | 10, 14 | oveq12i 7281 | . 2 ⊢ ((𝐴𝑃(𝐶𝐺𝐷)) + (𝐵𝑃(𝐶𝐺𝐷))) = (((𝐴𝑃𝐶) + (𝐴𝑃𝐷)) + ((𝐵𝑃𝐶) + (𝐵𝑃𝐷))) |
16 | 1 | phnvi 29172 | . . . . 5 ⊢ 𝑈 ∈ NrmCVec |
17 | 6, 7 | nvgcl 28976 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐶𝐺𝐷) ∈ 𝑋) |
18 | 16, 3, 4, 17 | mp3an 1460 | . . . 4 ⊢ (𝐶𝐺𝐷) ∈ 𝑋 |
19 | 2, 11, 18 | 3pm3.2i 1338 | . . 3 ⊢ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋) |
20 | 6, 7, 8 | dipdir 29198 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃(𝐶𝐺𝐷)) + (𝐵𝑃(𝐶𝐺𝐷)))) |
21 | 1, 19, 20 | mp2an 689 | . 2 ⊢ ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃(𝐶𝐺𝐷)) + (𝐵𝑃(𝐶𝐺𝐷))) |
22 | 6, 8 | dipcl 29068 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑃𝐶) ∈ ℂ) |
23 | 16, 2, 3, 22 | mp3an 1460 | . . 3 ⊢ (𝐴𝑃𝐶) ∈ ℂ |
24 | 6, 8 | dipcl 29068 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐵𝑃𝐷) ∈ ℂ) |
25 | 16, 11, 4, 24 | mp3an 1460 | . . 3 ⊢ (𝐵𝑃𝐷) ∈ ℂ |
26 | 6, 8 | dipcl 29068 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐴𝑃𝐷) ∈ ℂ) |
27 | 16, 2, 4, 26 | mp3an 1460 | . . 3 ⊢ (𝐴𝑃𝐷) ∈ ℂ |
28 | 6, 8 | dipcl 29068 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝑃𝐶) ∈ ℂ) |
29 | 16, 11, 3, 28 | mp3an 1460 | . . 3 ⊢ (𝐵𝑃𝐶) ∈ ℂ |
30 | 23, 25, 27, 29 | add42i 11198 | . 2 ⊢ (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶))) = (((𝐴𝑃𝐶) + (𝐴𝑃𝐷)) + ((𝐵𝑃𝐶) + (𝐵𝑃𝐷))) |
31 | 15, 21, 30 | 3eqtr4i 2778 | 1 ⊢ ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ‘cfv 6431 (class class class)co 7269 ℂcc 10868 + caddc 10873 NrmCVeccnv 28940 +𝑣 cpv 28941 BaseSetcba 28942 ·𝑖OLDcdip 29056 CPreHilOLDccphlo 29168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9375 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 ax-addf 10949 ax-mulf 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-sup 9177 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-n0 12232 df-z 12318 df-uz 12580 df-rp 12728 df-fz 13237 df-fzo 13380 df-seq 13718 df-exp 13779 df-hash 14041 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-clim 15193 df-sum 15394 df-grpo 28849 df-gid 28850 df-ginv 28851 df-ablo 28901 df-vc 28915 df-nv 28948 df-va 28951 df-ba 28952 df-sm 28953 df-0v 28954 df-nmcv 28956 df-dip 29057 df-ph 29169 |
This theorem is referenced by: pythi 29206 |
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