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| Mirrors > Home > MPE Home > Th. List > ip2i | Structured version Visualization version GIF version | ||
| Description: Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| ip1i.1 | β’ π = (BaseSetβπ) |
| ip1i.2 | β’ πΊ = ( +π£ βπ) |
| ip1i.4 | β’ π = ( Β·π OLD βπ) |
| ip1i.7 | β’ π = (Β·πOLDβπ) |
| ip1i.9 | β’ π β CPreHilOLD |
| ip2i.8 | β’ π΄ β π |
| ip2i.9 | β’ π΅ β π |
| Ref | Expression |
|---|---|
| ip2i | β’ ((2ππ΄)ππ΅) = (2 Β· (π΄ππ΅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ip1i.9 | . . . . . 6 β’ π β CPreHilOLD | |
| 2 | 1 | phnvi 30056 | . . . . 5 β’ π β NrmCVec |
| 3 | ip2i.8 | . . . . . 6 β’ π΄ β π | |
| 4 | ip1i.1 | . . . . . . 7 β’ π = (BaseSetβπ) | |
| 5 | ip1i.2 | . . . . . . 7 β’ πΊ = ( +π£ βπ) | |
| 6 | 4, 5 | nvgcl 29860 | . . . . . 6 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (π΄πΊπ΄) β π) |
| 7 | 2, 3, 3, 6 | mp3an 1461 | . . . . 5 β’ (π΄πΊπ΄) β π |
| 8 | ip2i.9 | . . . . 5 β’ π΅ β π | |
| 9 | ip1i.7 | . . . . . 6 β’ π = (Β·πOLDβπ) | |
| 10 | 4, 9 | dipcl 29952 | . . . . 5 β’ ((π β NrmCVec β§ (π΄πΊπ΄) β π β§ π΅ β π) β ((π΄πΊπ΄)ππ΅) β β) |
| 11 | 2, 7, 8, 10 | mp3an 1461 | . . . 4 β’ ((π΄πΊπ΄)ππ΅) β β |
| 12 | 11 | addridi 11397 | . . 3 β’ (((π΄πΊπ΄)ππ΅) + 0) = ((π΄πΊπ΄)ππ΅) |
| 13 | ip1i.4 | . . . . . . . 8 β’ π = ( Β·π OLD βπ) | |
| 14 | eqid 2732 | . . . . . . . 8 β’ (0vecβπ) = (0vecβπ) | |
| 15 | 4, 5, 13, 14 | nvrinv 29891 | . . . . . . 7 β’ ((π β NrmCVec β§ π΄ β π) β (π΄πΊ(-1ππ΄)) = (0vecβπ)) |
| 16 | 2, 3, 15 | mp2an 690 | . . . . . 6 β’ (π΄πΊ(-1ππ΄)) = (0vecβπ) |
| 17 | 16 | oveq1i 7415 | . . . . 5 β’ ((π΄πΊ(-1ππ΄))ππ΅) = ((0vecβπ)ππ΅) |
| 18 | 4, 14, 9 | dip0l 29958 | . . . . . 6 β’ ((π β NrmCVec β§ π΅ β π) β ((0vecβπ)ππ΅) = 0) |
| 19 | 2, 8, 18 | mp2an 690 | . . . . 5 β’ ((0vecβπ)ππ΅) = 0 |
| 20 | 17, 19 | eqtri 2760 | . . . 4 β’ ((π΄πΊ(-1ππ΄))ππ΅) = 0 |
| 21 | 20 | oveq2i 7416 | . . 3 β’ (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) = (((π΄πΊπ΄)ππ΅) + 0) |
| 22 | df-2 12271 | . . . . . 6 β’ 2 = (1 + 1) | |
| 23 | 22 | oveq1i 7415 | . . . . 5 β’ (2ππ΄) = ((1 + 1)ππ΄) |
| 24 | ax-1cn 11164 | . . . . . . . 8 β’ 1 β β | |
| 25 | 24, 24, 3 | 3pm3.2i 1339 | . . . . . . 7 β’ (1 β β β§ 1 β β β§ π΄ β π) |
| 26 | 4, 5, 13 | nvdir 29871 | . . . . . . 7 β’ ((π β NrmCVec β§ (1 β β β§ 1 β β β§ π΄ β π)) β ((1 + 1)ππ΄) = ((1ππ΄)πΊ(1ππ΄))) |
| 27 | 2, 25, 26 | mp2an 690 | . . . . . 6 β’ ((1 + 1)ππ΄) = ((1ππ΄)πΊ(1ππ΄)) |
| 28 | 4, 13 | nvsid 29867 | . . . . . . . 8 β’ ((π β NrmCVec β§ π΄ β π) β (1ππ΄) = π΄) |
| 29 | 2, 3, 28 | mp2an 690 | . . . . . . 7 β’ (1ππ΄) = π΄ |
| 30 | 29, 29 | oveq12i 7417 | . . . . . 6 β’ ((1ππ΄)πΊ(1ππ΄)) = (π΄πΊπ΄) |
| 31 | 27, 30 | eqtri 2760 | . . . . 5 β’ ((1 + 1)ππ΄) = (π΄πΊπ΄) |
| 32 | 23, 31 | eqtri 2760 | . . . 4 β’ (2ππ΄) = (π΄πΊπ΄) |
| 33 | 32 | oveq1i 7415 | . . 3 β’ ((2ππ΄)ππ΅) = ((π΄πΊπ΄)ππ΅) |
| 34 | 12, 21, 33 | 3eqtr4ri 2771 | . 2 β’ ((2ππ΄)ππ΅) = (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) |
| 35 | 4, 5, 13, 9, 1, 3, 3, 8 | ip1i 30067 | . 2 β’ (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) = (2 Β· (π΄ππ΅)) |
| 36 | 34, 35 | eqtri 2760 | 1 β’ ((2ππ΄)ππ΅) = (2 Β· (π΄ππ΅)) |
| Colors of variables: wff setvar class |
| Syntax hints: β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6540 (class class class)co 7405 βcc 11104 0cc0 11106 1c1 11107 + caddc 11109 Β· cmul 11111 -cneg 11441 2c2 12263 NrmCVeccnv 29824 +π£ cpv 29825 BaseSetcba 29826 Β·π OLD cns 29827 0veccn0v 29828 Β·πOLDcdip 29940 CPreHilOLDccphlo 30052 |
| 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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-grpo 29733 df-gid 29734 df-ginv 29735 df-ablo 29785 df-vc 29799 df-nv 29832 df-va 29835 df-ba 29836 df-sm 29837 df-0v 29838 df-nmcv 29840 df-dip 29941 df-ph 30053 |
| This theorem is referenced by: ipdirilem 30069 |
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