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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 1335 | . . . 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 28620 | . . . 4 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃𝐶) + (𝐴𝑃𝐷))) |
10 | 1, 5, 9 | mp2an 690 | . . 3 ⊢ (𝐴𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃𝐶) + (𝐴𝑃𝐷)) |
11 | ip2dii.b | . . . . 5 ⊢ 𝐵 ∈ 𝑋 | |
12 | 11, 3, 4 | 3pm3.2i 1335 | . . . 4 ⊢ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) |
13 | 6, 7, 8 | dipdi 28620 | . . . 4 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐵𝑃(𝐶𝐺𝐷)) = ((𝐵𝑃𝐶) + (𝐵𝑃𝐷))) |
14 | 1, 12, 13 | mp2an 690 | . . 3 ⊢ (𝐵𝑃(𝐶𝐺𝐷)) = ((𝐵𝑃𝐶) + (𝐵𝑃𝐷)) |
15 | 10, 14 | oveq12i 7168 | . 2 ⊢ ((𝐴𝑃(𝐶𝐺𝐷)) + (𝐵𝑃(𝐶𝐺𝐷))) = (((𝐴𝑃𝐶) + (𝐴𝑃𝐷)) + ((𝐵𝑃𝐶) + (𝐵𝑃𝐷))) |
16 | 1 | phnvi 28593 | . . . . 5 ⊢ 𝑈 ∈ NrmCVec |
17 | 6, 7 | nvgcl 28397 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐶𝐺𝐷) ∈ 𝑋) |
18 | 16, 3, 4, 17 | mp3an 1457 | . . . 4 ⊢ (𝐶𝐺𝐷) ∈ 𝑋 |
19 | 2, 11, 18 | 3pm3.2i 1335 | . . 3 ⊢ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋) |
20 | 6, 7, 8 | dipdir 28619 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃(𝐶𝐺𝐷)) + (𝐵𝑃(𝐶𝐺𝐷)))) |
21 | 1, 19, 20 | mp2an 690 | . 2 ⊢ ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = ((𝐴𝑃(𝐶𝐺𝐷)) + (𝐵𝑃(𝐶𝐺𝐷))) |
22 | 6, 8 | dipcl 28489 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑃𝐶) ∈ ℂ) |
23 | 16, 2, 3, 22 | mp3an 1457 | . . 3 ⊢ (𝐴𝑃𝐶) ∈ ℂ |
24 | 6, 8 | dipcl 28489 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐵𝑃𝐷) ∈ ℂ) |
25 | 16, 11, 4, 24 | mp3an 1457 | . . 3 ⊢ (𝐵𝑃𝐷) ∈ ℂ |
26 | 6, 8 | dipcl 28489 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐴𝑃𝐷) ∈ ℂ) |
27 | 16, 2, 4, 26 | mp3an 1457 | . . 3 ⊢ (𝐴𝑃𝐷) ∈ ℂ |
28 | 6, 8 | dipcl 28489 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝑃𝐶) ∈ ℂ) |
29 | 16, 11, 3, 28 | mp3an 1457 | . . 3 ⊢ (𝐵𝑃𝐶) ∈ ℂ |
30 | 23, 25, 27, 29 | add42i 10865 | . 2 ⊢ (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶))) = (((𝐴𝑃𝐶) + (𝐴𝑃𝐷)) + ((𝐵𝑃𝐶) + (𝐵𝑃𝐷))) |
31 | 15, 21, 30 | 3eqtr4i 2854 | 1 ⊢ ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 + caddc 10540 NrmCVeccnv 28361 +𝑣 cpv 28362 BaseSetcba 28363 ·𝑖OLDcdip 28477 CPreHilOLDccphlo 28589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-grpo 28270 df-gid 28271 df-ginv 28272 df-ablo 28322 df-vc 28336 df-nv 28369 df-va 28372 df-ba 28373 df-sm 28374 df-0v 28375 df-nmcv 28377 df-dip 28478 df-ph 28590 |
This theorem is referenced by: pythi 28627 |
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