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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lincresunit1 | Structured version Visualization version GIF version | ||
| Description: Property 1 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.) |
| Ref | Expression |
|---|---|
| lincresunit.b | β’ π΅ = (Baseβπ) |
| lincresunit.r | β’ π = (Scalarβπ) |
| lincresunit.e | β’ πΈ = (Baseβπ ) |
| lincresunit.u | β’ π = (Unitβπ ) |
| lincresunit.0 | β’ 0 = (0gβπ ) |
| lincresunit.z | β’ π = (0gβπ) |
| lincresunit.n | β’ π = (invgβπ ) |
| lincresunit.i | β’ πΌ = (invrβπ ) |
| lincresunit.t | β’ Β· = (.rβπ ) |
| lincresunit.g | β’ πΊ = (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) |
| Ref | Expression |
|---|---|
| lincresunit1 | β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π)) β πΊ β (πΈ βm (π β {π}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lincresunit.g | . 2 β’ πΊ = (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) | |
| 2 | eldifi 4126 | . . . . 5 β’ (π β (π β {π}) β π β π) | |
| 3 | lincresunit.b | . . . . . 6 β’ π΅ = (Baseβπ) | |
| 4 | lincresunit.r | . . . . . 6 β’ π = (Scalarβπ) | |
| 5 | lincresunit.e | . . . . . 6 β’ πΈ = (Baseβπ ) | |
| 6 | lincresunit.u | . . . . . 6 β’ π = (Unitβπ ) | |
| 7 | lincresunit.0 | . . . . . 6 β’ 0 = (0gβπ ) | |
| 8 | lincresunit.z | . . . . . 6 β’ π = (0gβπ) | |
| 9 | lincresunit.n | . . . . . 6 β’ π = (invgβπ ) | |
| 10 | lincresunit.i | . . . . . 6 β’ πΌ = (invrβπ ) | |
| 11 | lincresunit.t | . . . . . 6 β’ Β· = (.rβπ ) | |
| 12 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 1 | lincresunitlem2 47147 | . . . . 5 β’ ((((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π)) β§ π β π) β ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ )) β πΈ) |
| 13 | 2, 12 | sylan2 593 | . . . 4 β’ ((((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π)) β§ π β (π β {π})) β ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ )) β πΈ) |
| 14 | 13 | fmpttd 7114 | . . 3 β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π)) β (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))):(π β {π})βΆπΈ) |
| 15 | 5 | fvexi 6905 | . . . 4 β’ πΈ β V |
| 16 | difexg 5327 | . . . . . 6 β’ (π β π« π΅ β (π β {π}) β V) | |
| 17 | 16 | 3ad2ant1 1133 | . . . . 5 β’ ((π β π« π΅ β§ π β LMod β§ π β π) β (π β {π}) β V) |
| 18 | 17 | adantr 481 | . . . 4 β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π)) β (π β {π}) β V) |
| 19 | elmapg 8832 | . . . 4 β’ ((πΈ β V β§ (π β {π}) β V) β ((π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) β (πΈ βm (π β {π})) β (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))):(π β {π})βΆπΈ)) | |
| 20 | 15, 18, 19 | sylancr 587 | . . 3 β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π)) β ((π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) β (πΈ βm (π β {π})) β (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))):(π β {π})βΆπΈ)) |
| 21 | 14, 20 | mpbird 256 | . 2 β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π)) β (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) β (πΈ βm (π β {π}))) |
| 22 | 1, 21 | eqeltrid 2837 | 1 β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π)) β πΊ β (πΈ βm (π β {π}))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 Vcvv 3474 β cdif 3945 π« cpw 4602 {csn 4628 β¦ cmpt 5231 βΆwf 6539 βcfv 6543 (class class class)co 7408 βm cmap 8819 Basecbs 17143 .rcmulr 17197 Scalarcsca 17199 0gc0g 17384 invgcminusg 18819 Unitcui 20168 invrcinvr 20200 LModclmod 20470 |
| 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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
| 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 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-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-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-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-minusg 18822 df-mgp 19987 df-ur 20004 df-ring 20057 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-lmod 20472 |
| This theorem is referenced by: lincresunit3lem2 47151 lincresunit3 47152 isldepslvec2 47156 |
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