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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 30619 | . . . . 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 30423 | . . . . . 6 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (π΄πΊπ΄) β π) |
| 7 | 2, 3, 3, 6 | mp3an 1458 | . . . . 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 30515 | . . . . 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 1458 | . . . 4 β’ ((π΄πΊπ΄)ππ΅) β β |
| 12 | 11 | addridi 11425 | . . 3 β’ (((π΄πΊπ΄)ππ΅) + 0) = ((π΄πΊπ΄)ππ΅) |
| 13 | ip1i.4 | . . . . . . . 8 β’ π = ( Β·π OLD βπ) | |
| 14 | eqid 2728 | . . . . . . . 8 β’ (0vecβπ) = (0vecβπ) | |
| 15 | 4, 5, 13, 14 | nvrinv 30454 | . . . . . . 7 β’ ((π β NrmCVec β§ π΄ β π) β (π΄πΊ(-1ππ΄)) = (0vecβπ)) |
| 16 | 2, 3, 15 | mp2an 691 | . . . . . 6 β’ (π΄πΊ(-1ππ΄)) = (0vecβπ) |
| 17 | 16 | oveq1i 7424 | . . . . 5 β’ ((π΄πΊ(-1ππ΄))ππ΅) = ((0vecβπ)ππ΅) |
| 18 | 4, 14, 9 | dip0l 30521 | . . . . . 6 β’ ((π β NrmCVec β§ π΅ β π) β ((0vecβπ)ππ΅) = 0) |
| 19 | 2, 8, 18 | mp2an 691 | . . . . 5 β’ ((0vecβπ)ππ΅) = 0 |
| 20 | 17, 19 | eqtri 2756 | . . . 4 β’ ((π΄πΊ(-1ππ΄))ππ΅) = 0 |
| 21 | 20 | oveq2i 7425 | . . 3 β’ (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) = (((π΄πΊπ΄)ππ΅) + 0) |
| 22 | df-2 12299 | . . . . . 6 β’ 2 = (1 + 1) | |
| 23 | 22 | oveq1i 7424 | . . . . 5 β’ (2ππ΄) = ((1 + 1)ππ΄) |
| 24 | ax-1cn 11190 | . . . . . . . 8 β’ 1 β β | |
| 25 | 24, 24, 3 | 3pm3.2i 1337 | . . . . . . 7 β’ (1 β β β§ 1 β β β§ π΄ β π) |
| 26 | 4, 5, 13 | nvdir 30434 | . . . . . . 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 691 | . . . . . 6 β’ ((1 + 1)ππ΄) = ((1ππ΄)πΊ(1ππ΄)) |
| 28 | 4, 13 | nvsid 30430 | . . . . . . . 8 β’ ((π β NrmCVec β§ π΄ β π) β (1ππ΄) = π΄) |
| 29 | 2, 3, 28 | mp2an 691 | . . . . . . 7 β’ (1ππ΄) = π΄ |
| 30 | 29, 29 | oveq12i 7426 | . . . . . 6 β’ ((1ππ΄)πΊ(1ππ΄)) = (π΄πΊπ΄) |
| 31 | 27, 30 | eqtri 2756 | . . . . 5 β’ ((1 + 1)ππ΄) = (π΄πΊπ΄) |
| 32 | 23, 31 | eqtri 2756 | . . . 4 β’ (2ππ΄) = (π΄πΊπ΄) |
| 33 | 32 | oveq1i 7424 | . . 3 β’ ((2ππ΄)ππ΅) = ((π΄πΊπ΄)ππ΅) |
| 34 | 12, 21, 33 | 3eqtr4ri 2767 | . 2 β’ ((2ππ΄)ππ΅) = (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) |
| 35 | 4, 5, 13, 9, 1, 3, 3, 8 | ip1i 30630 | . 2 β’ (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) = (2 Β· (π΄ππ΅)) |
| 36 | 34, 35 | eqtri 2756 | 1 β’ ((2ππ΄)ππ΅) = (2 Β· (π΄ππ΅)) |
| Colors of variables: wff setvar class |
| Syntax hints: β§ w3a 1085 = wceq 1534 β wcel 2099 βcfv 6542 (class class class)co 7414 βcc 11130 0cc0 11132 1c1 11133 + caddc 11135 Β· cmul 11137 -cneg 11469 2c2 12291 NrmCVeccnv 30387 +π£ cpv 30388 BaseSetcba 30389 Β·π OLD cns 30390 0veccn0v 30391 Β·πOLDcdip 30503 CPreHilOLDccphlo 30615 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-sum 15659 df-grpo 30296 df-gid 30297 df-ginv 30298 df-ablo 30348 df-vc 30362 df-nv 30395 df-va 30398 df-ba 30399 df-sm 30400 df-0v 30401 df-nmcv 30403 df-dip 30504 df-ph 30616 |
| This theorem is referenced by: ipdirilem 30632 |
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