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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lincresunitlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 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 |
|---|---|
| lincresunitlem1 | 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) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lincresunit.r | . . . . 5 β’ π = (Scalarβπ) | |
| 2 | 1 | lmodring 20755 | . . . 4 β’ (π β LMod β π β Ring) |
| 3 | 2 | 3ad2ant2 1131 | . . 3 β’ ((π β π« π΅ β§ π β LMod β§ π β π) β π β Ring) |
| 4 | 3 | adantr 479 | . 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 Ring) |
| 5 | simpr 483 | . . 3 β’ ((πΉ β (πΈ βm π) β§ (πΉβπ) β π) β (πΉβπ) β π) | |
| 6 | lincresunit.u | . . . 4 β’ π = (Unitβπ ) | |
| 7 | lincresunit.n | . . . 4 β’ π = (invgβπ ) | |
| 8 | 6, 7 | unitnegcl 20340 | . . 3 β’ ((π β Ring β§ (πΉβπ) β π) β (πβ(πΉβπ)) β π) |
| 9 | 3, 5, 8 | syl2an 594 | . 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 (πβ(πΉβπ)) β π) |
| 10 | lincresunit.i | . . 3 β’ πΌ = (invrβπ ) | |
| 11 | lincresunit.e | . . 3 β’ πΈ = (Baseβπ ) | |
| 12 | 6, 10, 11 | ringinvcl 20335 | . 2 β’ ((π β Ring β§ (πβ(πΉβπ)) β π) β (πΌβ(πβ(πΉβπ))) β πΈ) |
| 13 | 4, 9, 12 | syl2anc 582 | 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 (πΌβ(πβ(πΉβπ))) β πΈ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β cdif 3942 π« cpw 4603 {csn 4629 β¦ cmpt 5231 βcfv 6547 (class class class)co 7417 βm cmap 8843 Basecbs 17179 .rcmulr 17233 Scalarcsca 17235 0gc0g 17420 invgcminusg 18895 Ringcrg 20177 Unitcui 20298 invrcinvr 20330 LModclmod 20747 |
| 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 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7870 df-2nd 7993 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-lmod 20749 |
| This theorem is referenced by: lincresunitlem2 47656 lincresunit2 47658 |
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