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Mirrors > Home > MPE Home > Th. List > clmvsubval2 | Structured version Visualization version GIF version |
Description: Value of vector subtraction on a subcomplex module. (Contributed by Mario Carneiro, 19-Nov-2013.) (Revised by AV, 7-Oct-2021.) |
Ref | Expression |
---|---|
clmvsubval.v | β’ π = (Baseβπ) |
clmvsubval.p | β’ + = (+gβπ) |
clmvsubval.m | β’ β = (-gβπ) |
clmvsubval.f | β’ πΉ = (Scalarβπ) |
clmvsubval.s | β’ Β· = ( Β·π βπ) |
Ref | Expression |
---|---|
clmvsubval2 | β’ ((π β βMod β§ π΄ β π β§ π΅ β π) β (π΄ β π΅) = ((-1 Β· π΅) + π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmvsubval.v | . . 3 β’ π = (Baseβπ) | |
2 | clmvsubval.p | . . 3 β’ + = (+gβπ) | |
3 | clmvsubval.m | . . 3 β’ β = (-gβπ) | |
4 | clmvsubval.f | . . 3 β’ πΉ = (Scalarβπ) | |
5 | clmvsubval.s | . . 3 β’ Β· = ( Β·π βπ) | |
6 | 1, 2, 3, 4, 5 | clmvsubval 24378 | . 2 β’ ((π β βMod β§ π΄ β π β§ π΅ β π) β (π΄ β π΅) = (π΄ + (-1 Β· π΅))) |
7 | clmabl 24338 | . . . 4 β’ (π β βMod β π β Abel) | |
8 | 7 | 3ad2ant1 1132 | . . 3 β’ ((π β βMod β§ π΄ β π β§ π΅ β π) β π β Abel) |
9 | simp2 1136 | . . 3 β’ ((π β βMod β§ π΄ β π β§ π΅ β π) β π΄ β π) | |
10 | simpl 483 | . . . . 5 β’ ((π β βMod β§ π΅ β π) β π β βMod) | |
11 | eqid 2736 | . . . . . . 7 β’ (BaseβπΉ) = (BaseβπΉ) | |
12 | 4, 11 | clmneg1 24351 | . . . . . 6 β’ (π β βMod β -1 β (BaseβπΉ)) |
13 | 12 | adantr 481 | . . . . 5 β’ ((π β βMod β§ π΅ β π) β -1 β (BaseβπΉ)) |
14 | simpr 485 | . . . . 5 β’ ((π β βMod β§ π΅ β π) β π΅ β π) | |
15 | 1, 4, 5, 11 | clmvscl 24357 | . . . . 5 β’ ((π β βMod β§ -1 β (BaseβπΉ) β§ π΅ β π) β (-1 Β· π΅) β π) |
16 | 10, 13, 14, 15 | syl3anc 1370 | . . . 4 β’ ((π β βMod β§ π΅ β π) β (-1 Β· π΅) β π) |
17 | 16 | 3adant2 1130 | . . 3 β’ ((π β βMod β§ π΄ β π β§ π΅ β π) β (-1 Β· π΅) β π) |
18 | 1, 2 | ablcom 19499 | . . 3 β’ ((π β Abel β§ π΄ β π β§ (-1 Β· π΅) β π) β (π΄ + (-1 Β· π΅)) = ((-1 Β· π΅) + π΄)) |
19 | 8, 9, 17, 18 | syl3anc 1370 | . 2 β’ ((π β βMod β§ π΄ β π β§ π΅ β π) β (π΄ + (-1 Β· π΅)) = ((-1 Β· π΅) + π΄)) |
20 | 6, 19 | eqtrd 2776 | 1 β’ ((π β βMod β§ π΄ β π β§ π΅ β π) β (π΄ β π΅) = ((-1 Β· π΅) + π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1086 = wceq 1540 β wcel 2105 βcfv 6479 (class class class)co 7337 1c1 10973 -cneg 11307 Basecbs 17009 +gcplusg 17059 Scalarcsca 17062 Β·π cvsca 17063 -gcsg 18675 Abelcabl 19482 βModcclm 24331 |
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 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-addf 11051 ax-mulf 11052 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-n0 12335 df-z 12421 df-dec 12539 df-uz 12684 df-fz 13341 df-seq 13823 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-starv 17074 df-tset 17078 df-ple 17079 df-ds 17081 df-unif 17082 df-0g 17249 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-grp 18676 df-minusg 18677 df-sbg 18678 df-mulg 18797 df-subg 18848 df-cmn 19483 df-abl 19484 df-mgp 19816 df-ur 19833 df-ring 19880 df-cring 19881 df-subrg 20127 df-lmod 20231 df-cnfld 20704 df-clm 24332 |
This theorem is referenced by: clmvz 24380 |
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