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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 | β’ π = (BaseSetβπ) |
| ip2dii.2 | β’ πΊ = ( +π£ βπ) |
| ip2dii.7 | β’ π = (Β·πOLDβπ) |
| ip2dii.u | β’ π β CPreHilOLD |
| ip2dii.a | β’ π΄ β π |
| ip2dii.b | β’ π΅ β π |
| ip2dii.c | β’ πΆ β π |
| ip2dii.d | β’ π· β π |
| Ref | Expression |
|---|---|
| ip2dii | β’ ((π΄πΊπ΅)π(πΆπΊπ·)) = (((π΄ππΆ) + (π΅ππ·)) + ((π΄ππ·) + (π΅ππΆ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ip2dii.u | . . . 4 β’ π β CPreHilOLD | |
| 2 | ip2dii.a | . . . . 5 β’ π΄ β π | |
| 3 | ip2dii.c | . . . . 5 β’ πΆ β π | |
| 4 | ip2dii.d | . . . . 5 β’ π· β π | |
| 5 | 2, 3, 4 | 3pm3.2i 1338 | . . . 4 β’ (π΄ β π β§ πΆ β π β§ π· β π) |
| 6 | ip2dii.1 | . . . . 5 β’ π = (BaseSetβπ) | |
| 7 | ip2dii.2 | . . . . 5 β’ πΊ = ( +π£ βπ) | |
| 8 | ip2dii.7 | . . . . 5 β’ π = (Β·πOLDβπ) | |
| 9 | 6, 7, 8 | dipdi 30364 | . . . 4 β’ ((π β CPreHilOLD β§ (π΄ β π β§ πΆ β π β§ π· β π)) β (π΄π(πΆπΊπ·)) = ((π΄ππΆ) + (π΄ππ·))) |
| 10 | 1, 5, 9 | mp2an 689 | . . 3 β’ (π΄π(πΆπΊπ·)) = ((π΄ππΆ) + (π΄ππ·)) |
| 11 | ip2dii.b | . . . . 5 β’ π΅ β π | |
| 12 | 11, 3, 4 | 3pm3.2i 1338 | . . . 4 β’ (π΅ β π β§ πΆ β π β§ π· β π) |
| 13 | 6, 7, 8 | dipdi 30364 | . . . 4 β’ ((π β CPreHilOLD β§ (π΅ β π β§ πΆ β π β§ π· β π)) β (π΅π(πΆπΊπ·)) = ((π΅ππΆ) + (π΅ππ·))) |
| 14 | 1, 12, 13 | mp2an 689 | . . 3 β’ (π΅π(πΆπΊπ·)) = ((π΅ππΆ) + (π΅ππ·)) |
| 15 | 10, 14 | oveq12i 7424 | . 2 β’ ((π΄π(πΆπΊπ·)) + (π΅π(πΆπΊπ·))) = (((π΄ππΆ) + (π΄ππ·)) + ((π΅ππΆ) + (π΅ππ·))) |
| 16 | 1 | phnvi 30337 | . . . . 5 β’ π β NrmCVec |
| 17 | 6, 7 | nvgcl 30141 | . . . . 5 β’ ((π β NrmCVec β§ πΆ β π β§ π· β π) β (πΆπΊπ·) β π) |
| 18 | 16, 3, 4, 17 | mp3an 1460 | . . . 4 β’ (πΆπΊπ·) β π |
| 19 | 2, 11, 18 | 3pm3.2i 1338 | . . 3 β’ (π΄ β π β§ π΅ β π β§ (πΆπΊπ·) β π) |
| 20 | 6, 7, 8 | dipdir 30363 | . . 3 β’ ((π β CPreHilOLD β§ (π΄ β π β§ π΅ β π β§ (πΆπΊπ·) β π)) β ((π΄πΊπ΅)π(πΆπΊπ·)) = ((π΄π(πΆπΊπ·)) + (π΅π(πΆπΊπ·)))) |
| 21 | 1, 19, 20 | mp2an 689 | . 2 β’ ((π΄πΊπ΅)π(πΆπΊπ·)) = ((π΄π(πΆπΊπ·)) + (π΅π(πΆπΊπ·))) |
| 22 | 6, 8 | dipcl 30233 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ πΆ β π) β (π΄ππΆ) β β) |
| 23 | 16, 2, 3, 22 | mp3an 1460 | . . 3 β’ (π΄ππΆ) β β |
| 24 | 6, 8 | dipcl 30233 | . . . 4 β’ ((π β NrmCVec β§ π΅ β π β§ π· β π) β (π΅ππ·) β β) |
| 25 | 16, 11, 4, 24 | mp3an 1460 | . . 3 β’ (π΅ππ·) β β |
| 26 | 6, 8 | dipcl 30233 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ π· β π) β (π΄ππ·) β β) |
| 27 | 16, 2, 4, 26 | mp3an 1460 | . . 3 β’ (π΄ππ·) β β |
| 28 | 6, 8 | dipcl 30233 | . . . 4 β’ ((π β NrmCVec β§ π΅ β π β§ πΆ β π) β (π΅ππΆ) β β) |
| 29 | 16, 11, 3, 28 | mp3an 1460 | . . 3 β’ (π΅ππΆ) β β |
| 30 | 23, 25, 27, 29 | add42i 11444 | . 2 β’ (((π΄ππΆ) + (π΅ππ·)) + ((π΄ππ·) + (π΅ππΆ))) = (((π΄ππΆ) + (π΄ππ·)) + ((π΅ππΆ) + (π΅ππ·))) |
| 31 | 15, 21, 30 | 3eqtr4i 2769 | 1 β’ ((π΄πΊπ΅)π(πΆπΊπ·)) = (((π΄ππΆ) + (π΅ππ·)) + ((π΄ππ·) + (π΅ππΆ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β§ w3a 1086 = wceq 1540 β wcel 2105 βcfv 6543 (class class class)co 7412 βcc 11111 + caddc 11116 NrmCVeccnv 30105 +π£ cpv 30106 BaseSetcba 30107 Β·πOLDcdip 30221 CPreHilOLDccphlo 30333 |
| 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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 ax-mulf 11193 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-grpo 30014 df-gid 30015 df-ginv 30016 df-ablo 30066 df-vc 30080 df-nv 30113 df-va 30116 df-ba 30117 df-sm 30118 df-0v 30119 df-nmcv 30121 df-dip 30222 df-ph 30334 |
| This theorem is referenced by: pythi 30371 |
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