Nearby theorems
|
|
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
| 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 20755 | . . . . 5 β’ (π β LMod β π β Ring) |
| 3 | 2 | 3ad2ant2 1131 | . . . 4 β’ ((π β π« π΅ β§ π β LMod β§ π β π) β π β Ring) |
| 4 | 3 | adantr 479 | . . 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 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 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 47655 | . . 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 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 π) β (πΌβ(πβ(πΉβπ))) β πΈ) |
| 17 | elmapi 8866 | . . . . 5 β’ (πΉ β (πΈ βm π) β πΉ:πβΆπΈ) | |
| 18 | ffvelcdm 7088 | . . . . . 6 β’ ((πΉ:πβΆπΈ β§ π β π) β (πΉβπ) β πΈ) | |
| 19 | 18 | ex 411 | . . . . 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 405 | . 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 20194 | . 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 1368 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β cdif 3942 π« cpw 4603 {csn 4629 β¦ cmpt 5231 βΆwf 6543 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-1st 7992 df-2nd 7993 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8845 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: lincresunit1 47657 lincresunit2 47658 lincresunit3lem1 47659 lincresunit3 47661 |
| Copyright terms: Public domain | W3C validator |