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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lincval1 | Structured version Visualization version GIF version |
Description: The linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.) |
Ref | Expression |
---|---|
lincval1.b | β’ π΅ = (Baseβπ) |
lincval1.s | β’ π = (Scalarβπ) |
lincval1.r | β’ π = (Baseβπ) |
lincval1.f | β’ πΉ = {β¨π, (0gβπ)β©} |
Ref | Expression |
---|---|
lincval1 | β’ ((π β LMod β§ π β π΅) β (πΉ( linC βπ){π}) = (0gβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lincval1.s | . . . . 5 β’ π = (Scalarβπ) | |
2 | lincval1.r | . . . . 5 β’ π = (Baseβπ) | |
3 | eqid 2731 | . . . . 5 β’ (0gβπ) = (0gβπ) | |
4 | 1, 2, 3 | lmod0cl 20643 | . . . 4 β’ (π β LMod β (0gβπ) β π ) |
5 | 4 | adantr 480 | . . 3 β’ ((π β LMod β§ π β π΅) β (0gβπ) β π ) |
6 | lincval1.b | . . . 4 β’ π΅ = (Baseβπ) | |
7 | eqid 2731 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
8 | lincval1.f | . . . 4 β’ πΉ = {β¨π, (0gβπ)β©} | |
9 | 6, 1, 2, 7, 8 | lincvalsn 47186 | . . 3 β’ ((π β LMod β§ π β π΅ β§ (0gβπ) β π ) β (πΉ( linC βπ){π}) = ((0gβπ)( Β·π βπ)π)) |
10 | 5, 9 | mpd3an3 1461 | . 2 β’ ((π β LMod β§ π β π΅) β (πΉ( linC βπ){π}) = ((0gβπ)( Β·π βπ)π)) |
11 | eqid 2731 | . . 3 β’ (0gβπ) = (0gβπ) | |
12 | 6, 1, 7, 3, 11 | lmod0vs 20650 | . 2 β’ ((π β LMod β§ π β π΅) β ((0gβπ)( Β·π βπ)π) = (0gβπ)) |
13 | 10, 12 | eqtrd 2771 | 1 β’ ((π β LMod β§ π β π΅) β (πΉ( linC βπ){π}) = (0gβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 {csn 4628 β¨cop 4634 βcfv 6543 (class class class)co 7412 Basecbs 17149 Scalarcsca 17205 Β·π cvsca 17206 0gc0g 17390 LModclmod 20615 linC clinc 47173 |
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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-oi 9509 df-card 9938 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-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-fzo 13633 df-seq 13972 df-hash 14296 df-0g 17392 df-gsum 17393 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-mulg 18988 df-cntz 19223 df-ring 20130 df-lmod 20617 df-linc 47175 |
This theorem is referenced by: lcosn0 47189 |
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