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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 29800 | . . . . 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 29604 | . . . . . 6 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (π΄πΊπ΄) β π) |
7 | 2, 3, 3, 6 | mp3an 1462 | . . . . 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 29696 | . . . . 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 1462 | . . . 4 β’ ((π΄πΊπ΄)ππ΅) β β |
12 | 11 | addid1i 11349 | . . 3 β’ (((π΄πΊπ΄)ππ΅) + 0) = ((π΄πΊπ΄)ππ΅) |
13 | ip1i.4 | . . . . . . . 8 β’ π = ( Β·π OLD βπ) | |
14 | eqid 2737 | . . . . . . . 8 β’ (0vecβπ) = (0vecβπ) | |
15 | 4, 5, 13, 14 | nvrinv 29635 | . . . . . . 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 7372 | . . . . 5 β’ ((π΄πΊ(-1ππ΄))ππ΅) = ((0vecβπ)ππ΅) |
18 | 4, 14, 9 | dip0l 29702 | . . . . . 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 2765 | . . . 4 β’ ((π΄πΊ(-1ππ΄))ππ΅) = 0 |
21 | 20 | oveq2i 7373 | . . 3 β’ (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) = (((π΄πΊπ΄)ππ΅) + 0) |
22 | df-2 12223 | . . . . . 6 β’ 2 = (1 + 1) | |
23 | 22 | oveq1i 7372 | . . . . 5 β’ (2ππ΄) = ((1 + 1)ππ΄) |
24 | ax-1cn 11116 | . . . . . . . 8 β’ 1 β β | |
25 | 24, 24, 3 | 3pm3.2i 1340 | . . . . . . 7 β’ (1 β β β§ 1 β β β§ π΄ β π) |
26 | 4, 5, 13 | nvdir 29615 | . . . . . . 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 29611 | . . . . . . . 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 7374 | . . . . . 6 β’ ((1ππ΄)πΊ(1ππ΄)) = (π΄πΊπ΄) |
31 | 27, 30 | eqtri 2765 | . . . . 5 β’ ((1 + 1)ππ΄) = (π΄πΊπ΄) |
32 | 23, 31 | eqtri 2765 | . . . 4 β’ (2ππ΄) = (π΄πΊπ΄) |
33 | 32 | oveq1i 7372 | . . 3 β’ ((2ππ΄)ππ΅) = ((π΄πΊπ΄)ππ΅) |
34 | 12, 21, 33 | 3eqtr4ri 2776 | . 2 β’ ((2ππ΄)ππ΅) = (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) |
35 | 4, 5, 13, 9, 1, 3, 3, 8 | ip1i 29811 | . 2 β’ (((π΄πΊπ΄)ππ΅) + ((π΄πΊ(-1ππ΄))ππ΅)) = (2 Β· (π΄ππ΅)) |
36 | 34, 35 | eqtri 2765 | 1 β’ ((2ππ΄)ππ΅) = (2 Β· (π΄ππ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6501 (class class class)co 7362 βcc 11056 0cc0 11058 1c1 11059 + caddc 11061 Β· cmul 11063 -cneg 11393 2c2 12215 NrmCVeccnv 29568 +π£ cpv 29569 BaseSetcba 29570 Β·π OLD cns 29571 0veccn0v 29572 Β·πOLDcdip 29684 CPreHilOLDccphlo 29796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 df-grpo 29477 df-gid 29478 df-ginv 29479 df-ablo 29529 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-nmcv 29584 df-dip 29685 df-ph 29797 |
This theorem is referenced by: ipdirilem 29813 |
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