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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 24916 | . . 3 β’ (π β βMod β π β LMod) | |
| 2 | 1 | adantr 480 | . 2 β’ ((π β βMod β§ (i β πΎ β§ π΅ β π β§ π β β)) β π β LMod) |
| 3 | simp1 1133 | . . . 4 β’ ((i β πΎ β§ π΅ β π β§ π β β) β i β πΎ) | |
| 4 | 3 | anim2i 616 | . . 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 1193 | . . 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 24970 | . . 3 β’ (((π β βMod β§ i β πΎ) β§ π β β) β (iβπ) β πΎ) |
| 9 | 4, 5, 8 | syl2anc 583 | . 2 β’ ((π β βMod β§ (i β πΎ β§ π΅ β π β§ π β β)) β (iβπ) β πΎ) |
| 10 | simpr2 1192 | . 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 20714 | . 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 1368 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6533 (class class class)co 7401 ici 11108 βcn 12209 βcexp 14024 Basecbs 17143 Scalarcsca 17199 Β·π cvsca 17200 LModclmod 20696 βModcclm 24911 |
| 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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 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 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-seq 13964 df-exp 14025 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-grp 18856 df-mulg 18986 df-cmn 19692 df-mgp 20030 df-ur 20077 df-ring 20130 df-cring 20131 df-subrg 20461 df-lmod 20698 df-cnfld 21229 df-clm 24912 |
| This theorem is referenced by: (None) |
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