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Mirrors > Home > MPE Home > Th. List > asclf | Structured version Visualization version GIF version |
Description: The algebra scalars function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
asclf.a | β’ π΄ = (algScβπ) |
asclf.f | β’ πΉ = (Scalarβπ) |
asclf.r | β’ (π β π β Ring) |
asclf.l | β’ (π β π β LMod) |
asclf.k | β’ πΎ = (BaseβπΉ) |
asclf.b | β’ π΅ = (Baseβπ) |
Ref | Expression |
---|---|
asclf | β’ (π β π΄:πΎβΆπ΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | asclf.l | . . . 4 β’ (π β π β LMod) | |
2 | 1 | adantr 480 | . . 3 β’ ((π β§ π₯ β πΎ) β π β LMod) |
3 | simpr 484 | . . 3 β’ ((π β§ π₯ β πΎ) β π₯ β πΎ) | |
4 | asclf.r | . . . . 5 β’ (π β π β Ring) | |
5 | asclf.b | . . . . . 6 β’ π΅ = (Baseβπ) | |
6 | eqid 2728 | . . . . . 6 β’ (1rβπ) = (1rβπ) | |
7 | 5, 6 | ringidcl 20202 | . . . . 5 β’ (π β Ring β (1rβπ) β π΅) |
8 | 4, 7 | syl 17 | . . . 4 β’ (π β (1rβπ) β π΅) |
9 | 8 | adantr 480 | . . 3 β’ ((π β§ π₯ β πΎ) β (1rβπ) β π΅) |
10 | asclf.f | . . . 4 β’ πΉ = (Scalarβπ) | |
11 | eqid 2728 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
12 | asclf.k | . . . 4 β’ πΎ = (BaseβπΉ) | |
13 | 5, 10, 11, 12 | lmodvscl 20761 | . . 3 β’ ((π β LMod β§ π₯ β πΎ β§ (1rβπ) β π΅) β (π₯( Β·π βπ)(1rβπ)) β π΅) |
14 | 2, 3, 9, 13 | syl3anc 1369 | . 2 β’ ((π β§ π₯ β πΎ) β (π₯( Β·π βπ)(1rβπ)) β π΅) |
15 | asclf.a | . . 3 β’ π΄ = (algScβπ) | |
16 | 15, 10, 12, 11, 6 | asclfval 21812 | . 2 β’ π΄ = (π₯ β πΎ β¦ (π₯( Β·π βπ)(1rβπ))) |
17 | 14, 16 | fmptd 7124 | 1 β’ (π β π΄:πΎβΆπ΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βΆwf 6544 βcfv 6548 (class class class)co 7420 Basecbs 17180 Scalarcsca 17236 Β·π cvsca 17237 1rcur 20121 Ringcrg 20173 LModclmod 20743 algSccascl 21786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mgp 20075 df-ur 20122 df-ring 20175 df-lmod 20745 df-ascl 21789 |
This theorem is referenced by: asclghm 21816 ascldimul 21821 aspval2 21831 psrasclcl 21922 mplasclf 22009 subrgasclcl 22011 mpfconst 22047 ply1sclf 22204 cply1coe0bi 22221 lply1binomsc 22230 evls1sca 22242 evl1scvarpw 22282 evls1fpws 22288 mat2pmatbas 22641 chpscmat 22757 chpscmatgsumbin 22759 irngnzply1lem 33368 |
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