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| Mirrors > Home > MPE Home > Th. List > cmodscmulexp | Structured version Visualization version GIF version | ||
| Description: The scalar product of a vector with powers of i belongs to the base set of a subcomplex module if the scalar subring of th subcomplex module contains i. (Contributed by AV, 18-Oct-2021.) |
| Ref | Expression |
|---|---|
| cmodscexp.f | β’ πΉ = (Scalarβπ) |
| cmodscexp.k | β’ πΎ = (BaseβπΉ) |
| cmodscmulexp.x | β’ π = (Baseβπ) |
| cmodscmulexp.s | β’ Β· = ( Β·π βπ) |
| Ref | Expression |
|---|---|
| cmodscmulexp | β’ ((π β βMod β§ (i β πΎ β§ π΅ β π β§ π β β)) β ((iβπ) Β· π΅) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | clmlmod 24507 | . . 3 β’ (π β βMod β π β LMod) | |
| 2 | 1 | adantr 481 | . 2 β’ ((π β βMod β§ (i β πΎ β§ π΅ β π β§ π β β)) β π β LMod) |
| 3 | simp1 1136 | . . . 4 β’ ((i β πΎ β§ π΅ β π β§ π β β) β i β πΎ) | |
| 4 | 3 | anim2i 617 | . . 3 β’ ((π β βMod β§ (i β πΎ β§ π΅ β π β§ π β β)) β (π β βMod β§ i β πΎ)) |
| 5 | simpr3 1196 | . . 3 β’ ((π β βMod β§ (i β πΎ β§ π΅ β π β§ π β β)) β π β β) | |
| 6 | cmodscexp.f | . . . 4 β’ πΉ = (Scalarβπ) | |
| 7 | cmodscexp.k | . . . 4 β’ πΎ = (BaseβπΉ) | |
| 8 | 6, 7 | cmodscexp 24561 | . . 3 β’ (((π β βMod β§ i β πΎ) β§ π β β) β (iβπ) β πΎ) |
| 9 | 4, 5, 8 | syl2anc 584 | . 2 β’ ((π β βMod β§ (i β πΎ β§ π΅ β π β§ π β β)) β (iβπ) β πΎ) |
| 10 | simpr2 1195 | . 2 β’ ((π β βMod β§ (i β πΎ β§ π΅ β π β§ π β β)) β π΅ β π) | |
| 11 | cmodscmulexp.x | . . 3 β’ π = (Baseβπ) | |
| 12 | cmodscmulexp.s | . . 3 β’ Β· = ( Β·π βπ) | |
| 13 | 11, 6, 12, 7 | lmodvscl 20433 | . 2 β’ ((π β LMod β§ (iβπ) β πΎ β§ π΅ β π) β ((iβπ) Β· π΅) β π) |
| 14 | 2, 9, 10, 13 | syl3anc 1371 | 1 β’ ((π β βMod β§ (i β πΎ β§ π΅ β π β§ π β β)) β ((iβπ) Β· π΅) β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6529 (class class class)co 7390 ici 11091 βcn 12191 βcexp 14006 Basecbs 17123 Scalarcsca 17179 Β·π cvsca 17180 LModclmod 20415 βModcclm 24502 |
| 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 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 ax-addf 11168 ax-mulf 11169 |
| 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-tp 4624 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7836 df-1st 7954 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-1o 8445 df-er 8683 df-en 8920 df-dom 8921 df-sdom 8922 df-fin 8923 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-5 12257 df-6 12258 df-7 12259 df-8 12260 df-9 12261 df-n0 12452 df-z 12538 df-dec 12657 df-uz 12802 df-fz 13464 df-seq 13946 df-exp 14007 df-struct 17059 df-sets 17076 df-slot 17094 df-ndx 17106 df-base 17124 df-ress 17153 df-plusg 17189 df-mulr 17190 df-starv 17191 df-tset 17195 df-ple 17196 df-ds 17198 df-unif 17199 df-0g 17366 df-mgm 18540 df-sgrp 18589 df-mnd 18600 df-submnd 18645 df-grp 18794 df-mulg 18920 df-cmn 19611 df-mgp 19944 df-ur 19961 df-ring 20013 df-cring 20014 df-subrg 20305 df-lmod 20417 df-cnfld 20874 df-clm 24503 |
| This theorem is referenced by: (None) |
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