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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 30541 | . . . . 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 30345 | . . . . . 6 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (π΄πΊπ΄) β π) |
| 7 | 2, 3, 3, 6 | mp3an 1457 | . . . . 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 30437 | . . . . 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 1457 | . . . 4 β’ ((π΄πΊπ΄)ππ΅) β β |
| 12 | 11 | addridi 11399 | . . 3 β’ (((π΄πΊπ΄)ππ΅) + 0) = ((π΄πΊπ΄)ππ΅) |
| 13 | ip1i.4 | . . . . . . . 8 β’ π = ( Β·π OLD βπ) | |
| 14 | eqid 2724 | . . . . . . . 8 β’ (0vecβπ) = (0vecβπ) | |
| 15 | 4, 5, 13, 14 | nvrinv 30376 | . . . . . . 7 β’ ((π β NrmCVec β§ π΄ β π) β (π΄πΊ(-1ππ΄)) = (0vecβπ)) |
| 16 | 2, 3, 15 | mp2an 689 | . . . . . 6 β’ (π΄πΊ(-1ππ΄)) = (0vecβπ) |
| 17 | 16 | oveq1i 7412 | . . . . 5 β’ ((π΄πΊ(-1ππ΄))ππ΅) = ((0vecβπ)ππ΅) |
| 18 | 4, 14, 9 | dip0l 30443 | . . . . . 6 β’ ((π β NrmCVec β§ π΅ β π) β ((0vecβπ)ππ΅) = 0) |
| 19 | 2, 8, 18 | mp2an 689 | . . . . 5 β’ ((0vecβπ)ππ΅) = 0 |
| 20 | 17, 19 | eqtri 2752 | . . . 4 β’ ((π΄πΊ(-1ππ΄))ππ΅) = 0 |
| 21 | 20 | oveq2i 7413 | . . 3 β’ (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) = (((π΄πΊπ΄)ππ΅) + 0) |
| 22 | df-2 12273 | . . . . . 6 β’ 2 = (1 + 1) | |
| 23 | 22 | oveq1i 7412 | . . . . 5 β’ (2ππ΄) = ((1 + 1)ππ΄) |
| 24 | ax-1cn 11165 | . . . . . . . 8 β’ 1 β β | |
| 25 | 24, 24, 3 | 3pm3.2i 1336 | . . . . . . 7 β’ (1 β β β§ 1 β β β§ π΄ β π) |
| 26 | 4, 5, 13 | nvdir 30356 | . . . . . . 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 689 | . . . . . 6 β’ ((1 + 1)ππ΄) = ((1ππ΄)πΊ(1ππ΄)) |
| 28 | 4, 13 | nvsid 30352 | . . . . . . . 8 β’ ((π β NrmCVec β§ π΄ β π) β (1ππ΄) = π΄) |
| 29 | 2, 3, 28 | mp2an 689 | . . . . . . 7 β’ (1ππ΄) = π΄ |
| 30 | 29, 29 | oveq12i 7414 | . . . . . 6 β’ ((1ππ΄)πΊ(1ππ΄)) = (π΄πΊπ΄) |
| 31 | 27, 30 | eqtri 2752 | . . . . 5 β’ ((1 + 1)ππ΄) = (π΄πΊπ΄) |
| 32 | 23, 31 | eqtri 2752 | . . . 4 β’ (2ππ΄) = (π΄πΊπ΄) |
| 33 | 32 | oveq1i 7412 | . . 3 β’ ((2ππ΄)ππ΅) = ((π΄πΊπ΄)ππ΅) |
| 34 | 12, 21, 33 | 3eqtr4ri 2763 | . 2 β’ ((2ππ΄)ππ΅) = (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) |
| 35 | 4, 5, 13, 9, 1, 3, 3, 8 | ip1i 30552 | . 2 β’ (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) = (2 Β· (π΄ππ΅)) |
| 36 | 34, 35 | eqtri 2752 | 1 β’ ((2ππ΄)ππ΅) = (2 Β· (π΄ππ΅)) |
| Colors of variables: wff setvar class |
| Syntax hints: β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 βcc 11105 0cc0 11107 1c1 11108 + caddc 11110 Β· cmul 11112 -cneg 11443 2c2 12265 NrmCVeccnv 30309 +π£ cpv 30310 BaseSetcba 30311 Β·π OLD cns 30312 0veccn0v 30313 Β·πOLDcdip 30425 CPreHilOLDccphlo 30537 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-n0 12471 df-z 12557 df-uz 12821 df-rp 12973 df-fz 13483 df-fzo 13626 df-seq 13965 df-exp 14026 df-hash 14289 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-clim 15430 df-sum 15631 df-grpo 30218 df-gid 30219 df-ginv 30220 df-ablo 30270 df-vc 30284 df-nv 30317 df-va 30320 df-ba 30321 df-sm 30322 df-0v 30323 df-nmcv 30325 df-dip 30426 df-ph 30538 |
| This theorem is referenced by: ipdirilem 30554 |
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