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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 20740 | . . . . 5 β’ (π β LMod β π β Ring) |
| 3 | 2 | 3ad2ant2 1132 | . . . 4 β’ ((π β π« π΅ β§ π β LMod β§ π β π) β π β Ring) |
| 4 | 3 | adantr 480 | . . 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 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 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 47466 | . . 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 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 π) β (πΌβ(πβ(πΉβπ))) β πΈ) |
| 17 | elmapi 8859 | . . . . 5 β’ (πΉ β (πΈ βm π) β πΉ:πβΆπΈ) | |
| 18 | ffvelcdm 7085 | . . . . . 6 β’ ((πΉ:πβΆπΈ β§ π β π) β (πΉβπ) β πΈ) | |
| 19 | 18 | ex 412 | . . . . 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 727 | . . 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 406 | . 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 20181 | . 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 1369 | 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 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β cdif 3941 π« cpw 4598 {csn 4624 β¦ cmpt 5225 βΆwf 6538 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-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8838 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: lincresunit1 47468 lincresunit2 47469 lincresunit3lem1 47470 lincresunit3 47472 |
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