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Mirrors > Home > MPE Home > Th. List > asclrhm | Structured version Visualization version GIF version |
Description: The scalar injection is a ring homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.) |
Ref | Expression |
---|---|
asclrhm.a | β’ π΄ = (algScβπ) |
asclrhm.f | β’ πΉ = (Scalarβπ) |
Ref | Expression |
---|---|
asclrhm | β’ (π β AssAlg β π΄ β (πΉ RingHom π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . 2 β’ (BaseβπΉ) = (BaseβπΉ) | |
2 | eqid 2727 | . 2 β’ (1rβπΉ) = (1rβπΉ) | |
3 | eqid 2727 | . 2 β’ (1rβπ) = (1rβπ) | |
4 | eqid 2727 | . 2 β’ (.rβπΉ) = (.rβπΉ) | |
5 | eqid 2727 | . 2 β’ (.rβπ) = (.rβπ) | |
6 | asclrhm.f | . . 3 β’ πΉ = (Scalarβπ) | |
7 | 6 | assasca 21801 | . 2 β’ (π β AssAlg β πΉ β Ring) |
8 | assaring 21800 | . 2 β’ (π β AssAlg β π β Ring) | |
9 | asclrhm.a | . . 3 β’ π΄ = (algScβπ) | |
10 | assalmod 21799 | . . 3 β’ (π β AssAlg β π β LMod) | |
11 | 9, 6, 10, 8 | ascl1 21823 | . 2 β’ (π β AssAlg β (π΄β(1rβπΉ)) = (1rβπ)) |
12 | 9, 6, 1, 5, 4 | ascldimul 21826 | . . 3 β’ ((π β AssAlg β§ π₯ β (BaseβπΉ) β§ π¦ β (BaseβπΉ)) β (π΄β(π₯(.rβπΉ)π¦)) = ((π΄βπ₯)(.rβπ)(π΄βπ¦))) |
13 | 12 | 3expb 1117 | . 2 β’ ((π β AssAlg β§ (π₯ β (BaseβπΉ) β§ π¦ β (BaseβπΉ))) β (π΄β(π₯(.rβπΉ)π¦)) = ((π΄βπ₯)(.rβπ)(π΄βπ¦))) |
14 | 9, 6, 8, 10 | asclghm 21821 | . 2 β’ (π β AssAlg β π΄ β (πΉ GrpHom π)) |
15 | 1, 2, 3, 4, 5, 7, 8, 11, 13, 14 | isrhm2d 20431 | 1 β’ (π β AssAlg β π΄ β (πΉ RingHom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6551 (class class class)co 7424 Basecbs 17185 .rcmulr 17239 Scalarcsca 17241 1rcur 20126 RingHom crh 20413 AssAlgcasa 21789 algSccascl 21791 |
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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 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 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-plusg 17251 df-0g 17428 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18745 df-grp 18898 df-ghm 19173 df-mgp 20080 df-ur 20127 df-ring 20180 df-rhm 20416 df-lmod 20750 df-assa 21792 df-ascl 21794 |
This theorem is referenced by: rnasclsubrg 21831 mplind 22019 evlslem1 22033 mpfind 22058 ply1fermltlchr 22236 pf1ind 22279 mat2pmatmul 22651 mat2pmatlin 22655 ply1asclunit 33264 selvcllem2 41814 selvvvval 41821 evlselv 41823 |
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