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| Mirrors > Home > MPE Home > Th. List > clmpm1dir | Structured version Visualization version GIF version | ||
| Description: Subtractive distributive law for the scalar product of a subcomplex module. (Contributed by NM, 31-Jul-2007.) (Revised by AV, 21-Sep-2021.) |
| Ref | Expression |
|---|---|
| clmpm1dir.v | β’ π = (Baseβπ) |
| clmpm1dir.s | β’ Β· = ( Β·π βπ) |
| clmpm1dir.a | β’ + = (+gβπ) |
| clmpm1dir.k | β’ πΎ = (Baseβ(Scalarβπ)) |
| Ref | Expression |
|---|---|
| clmpm1dir | β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β ((π΄ β π΅) Β· πΆ) = ((π΄ Β· πΆ) + (-1 Β· (π΅ Β· πΆ)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | clmpm1dir.v | . . 3 β’ π = (Baseβπ) | |
| 2 | clmpm1dir.s | . . 3 β’ Β· = ( Β·π βπ) | |
| 3 | eqid 2731 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
| 4 | clmpm1dir.k | . . 3 β’ πΎ = (Baseβ(Scalarβπ)) | |
| 5 | eqid 2731 | . . 3 β’ (-gβπ) = (-gβπ) | |
| 6 | simpl 482 | . . 3 β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β π β βMod) | |
| 7 | simpr1 1193 | . . 3 β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β π΄ β πΎ) | |
| 8 | simpr2 1194 | . . 3 β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β π΅ β πΎ) | |
| 9 | simpr3 1195 | . . 3 β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β πΆ β π) | |
| 10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | clmsubdir 24850 | . 2 β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β ((π΄ β π΅) Β· πΆ) = ((π΄ Β· πΆ)(-gβπ)(π΅ Β· πΆ))) |
| 11 | 1, 3, 2, 4 | clmvscl 24836 | . . . 4 β’ ((π β βMod β§ π΄ β πΎ β§ πΆ β π) β (π΄ Β· πΆ) β π) |
| 12 | 6, 7, 9, 11 | syl3anc 1370 | . . 3 β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β (π΄ Β· πΆ) β π) |
| 13 | 1, 3, 2, 4 | clmvscl 24836 | . . . 4 β’ ((π β βMod β§ π΅ β πΎ β§ πΆ β π) β (π΅ Β· πΆ) β π) |
| 14 | 6, 8, 9, 13 | syl3anc 1370 | . . 3 β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β (π΅ Β· πΆ) β π) |
| 15 | clmpm1dir.a | . . . 4 β’ + = (+gβπ) | |
| 16 | eqid 2731 | . . . 4 β’ (invgβπ) = (invgβπ) | |
| 17 | 1, 15, 16, 5 | grpsubval 18907 | . . 3 β’ (((π΄ Β· πΆ) β π β§ (π΅ Β· πΆ) β π) β ((π΄ Β· πΆ)(-gβπ)(π΅ Β· πΆ)) = ((π΄ Β· πΆ) + ((invgβπ)β(π΅ Β· πΆ)))) |
| 18 | 12, 14, 17 | syl2anc 583 | . 2 β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β ((π΄ Β· πΆ)(-gβπ)(π΅ Β· πΆ)) = ((π΄ Β· πΆ) + ((invgβπ)β(π΅ Β· πΆ)))) |
| 19 | 1, 16, 3, 2 | clmvneg1 24847 | . . . . 5 β’ ((π β βMod β§ (π΅ Β· πΆ) β π) β (-1 Β· (π΅ Β· πΆ)) = ((invgβπ)β(π΅ Β· πΆ))) |
| 20 | 19 | eqcomd 2737 | . . . 4 β’ ((π β βMod β§ (π΅ Β· πΆ) β π) β ((invgβπ)β(π΅ Β· πΆ)) = (-1 Β· (π΅ Β· πΆ))) |
| 21 | 6, 14, 20 | syl2anc 583 | . . 3 β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β ((invgβπ)β(π΅ Β· πΆ)) = (-1 Β· (π΅ Β· πΆ))) |
| 22 | 21 | oveq2d 7428 | . 2 β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β ((π΄ Β· πΆ) + ((invgβπ)β(π΅ Β· πΆ))) = ((π΄ Β· πΆ) + (-1 Β· (π΅ Β· πΆ)))) |
| 23 | 10, 18, 22 | 3eqtrd 2775 | 1 β’ ((π β βMod β§ (π΄ β πΎ β§ π΅ β πΎ β§ πΆ β π)) β ((π΄ β π΅) Β· πΆ) = ((π΄ Β· πΆ) + (-1 Β· (π΅ Β· πΆ)))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βcfv 6544 (class class class)co 7412 1c1 11114 β cmin 11449 -cneg 11450 Basecbs 17149 +gcplusg 17202 Scalarcsca 17205 Β·π cvsca 17206 invgcminusg 18857 -gcsg 18858 βModcclm 24810 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-addf 11192 ax-mulf 11193 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-seq 13972 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-subrg 20460 df-lmod 20617 df-cnfld 21146 df-clm 24811 |
| This theorem is referenced by: (None) |
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