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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 20740 | . . . 4 β’ (π β LMod β π β Ring) |
| 3 | 2 | 3ad2ant2 1132 | . . 3 β’ ((π β π« π΅ β§ π β LMod β§ π β π) β π β Ring) |
| 4 | 3 | adantr 480 | . 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 484 | . . 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 20325 | . . 3 β’ ((π β Ring β§ (πΉβπ) β π) β (πβ(πΉβπ)) β π) |
| 9 | 3, 5, 8 | syl2an 595 | . 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 20320 | . 2 β’ ((π β Ring β§ (πβ(πΉβπ)) β π) β (πΌβ(πβ(πΉβπ))) β πΈ) |
| 13 | 4, 9, 12 | syl2anc 583 | 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 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β cdif 3941 π« cpw 4598 {csn 4624 β¦ cmpt 5225 βcfv 6542 (class class class)co 7414 βm cmap 8836 Basecbs 17171 .rcmulr 17225 Scalarcsca 17227 0gc0g 17412 invgcminusg 18882 Ringcrg 20164 Unitcui 20283 invrcinvr 20315 LModclmod 20732 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-lmod 20734 |
| This theorem is referenced by: lincresunitlem2 47467 lincresunit2 47469 |
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