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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lincresunitlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for properties 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 |
|---|---|
| lincresunitlem2 | 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) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lincresunit.r | . . . . . 6 β’ π = (Scalarβπ) | |
| 2 | 1 | lmodring 20471 | . . . . 5 β’ (π β LMod β π β Ring) |
| 3 | 2 | 3ad2ant2 1134 | . . . 4 β’ ((π β π« π΅ β§ π β LMod β§ π β π) β π β Ring) |
| 4 | 3 | adantr 481 | . . 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 Ring) |
| 5 | 4 | adantr 481 | . 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 Ring) |
| 6 | lincresunit.b | . . . 4 β’ π΅ = (Baseβπ) | |
| 7 | lincresunit.e | . . . 4 β’ πΈ = (Baseβπ ) | |
| 8 | lincresunit.u | . . . 4 β’ π = (Unitβπ ) | |
| 9 | lincresunit.0 | . . . 4 β’ 0 = (0gβπ ) | |
| 10 | lincresunit.z | . . . 4 β’ π = (0gβπ) | |
| 11 | lincresunit.n | . . . 4 β’ π = (invgβπ ) | |
| 12 | lincresunit.i | . . . 4 β’ πΌ = (invrβπ ) | |
| 13 | lincresunit.t | . . . 4 β’ Β· = (.rβπ ) | |
| 14 | lincresunit.g | . . . 4 β’ πΊ = (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) | |
| 15 | 6, 1, 7, 8, 9, 10, 11, 12, 13, 14 | lincresunitlem1 47109 | . . 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 (πΌβ(πβ(πΉβπ))) β πΈ) |
| 16 | 15 | adantr 481 | . 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 (πΌβ(πβ(πΉβπ))) β πΈ) |
| 17 | elmapi 8839 | . . . . 5 β’ (πΉ β (πΈ βm π) β πΉ:πβΆπΈ) | |
| 18 | ffvelcdm 7080 | . . . . . 6 β’ ((πΉ:πβΆπΈ β§ π β π) β (πΉβπ) β πΈ) | |
| 19 | 18 | ex 413 | . . . . 5 β’ (πΉ:πβΆπΈ β (π β π β (πΉβπ) β πΈ)) |
| 20 | 17, 19 | syl 17 | . . . 4 β’ (πΉ β (πΈ βm π) β (π β π β (πΉβπ) β πΈ)) |
| 21 | 20 | ad2antrl 726 | . . 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 (πΉβπ) β πΈ)) |
| 22 | 21 | imp 407 | . 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 (πΉβπ) β πΈ) |
| 23 | 7, 13 | ringcl 20066 | . 2 β’ ((π β Ring β§ (πΌβ(πβ(πΉβπ))) β πΈ β§ (πΉβπ) β πΈ) β ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ)) β πΈ) |
| 24 | 5, 16, 22, 23 | syl3anc 1371 | 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§ π β π) β ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ)) β πΈ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β cdif 3944 π« cpw 4601 {csn 4627 β¦ cmpt 5230 βΆwf 6536 βcfv 6540 (class class class)co 7405 βm cmap 8816 Basecbs 17140 .rcmulr 17194 Scalarcsca 17196 0gc0g 17381 invgcminusg 18816 Ringcrg 20049 Unitcui 20161 invrcinvr 20193 LModclmod 20463 |
| 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-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 |
| 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-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-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-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-lmod 20465 |
| This theorem is referenced by: lincresunit1 47111 lincresunit2 47112 lincresunit3lem1 47113 lincresunit3 47115 |
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