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| Mirrors > Home > MPE Home > Th. List > asclmul1 | Structured version Visualization version GIF version | ||
| Description: Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.) |
| Ref | Expression |
|---|---|
| asclmul1.a | β’ π΄ = (algScβπ) |
| asclmul1.f | β’ πΉ = (Scalarβπ) |
| asclmul1.k | β’ πΎ = (BaseβπΉ) |
| asclmul1.v | β’ π = (Baseβπ) |
| asclmul1.t | β’ Γ = (.rβπ) |
| asclmul1.s | β’ Β· = ( Β·π βπ) |
| Ref | Expression |
|---|---|
| asclmul1 | β’ ((π β AssAlg β§ π β πΎ β§ π β π) β ((π΄βπ ) Γ π) = (π Β· π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | asclmul1.a | . . . . 5 β’ π΄ = (algScβπ) | |
| 2 | asclmul1.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
| 3 | asclmul1.k | . . . . 5 β’ πΎ = (BaseβπΉ) | |
| 4 | asclmul1.s | . . . . 5 β’ Β· = ( Β·π βπ) | |
| 5 | eqid 2732 | . . . . 5 β’ (1rβπ) = (1rβπ) | |
| 6 | 1, 2, 3, 4, 5 | asclval 21653 | . . . 4 β’ (π β πΎ β (π΄βπ ) = (π Β· (1rβπ))) |
| 7 | 6 | 3ad2ant2 1134 | . . 3 β’ ((π β AssAlg β§ π β πΎ β§ π β π) β (π΄βπ ) = (π Β· (1rβπ))) |
| 8 | 7 | oveq1d 7426 | . 2 β’ ((π β AssAlg β§ π β πΎ β§ π β π) β ((π΄βπ ) Γ π) = ((π Β· (1rβπ)) Γ π)) |
| 9 | simp1 1136 | . . 3 β’ ((π β AssAlg β§ π β πΎ β§ π β π) β π β AssAlg) | |
| 10 | simp2 1137 | . . 3 β’ ((π β AssAlg β§ π β πΎ β§ π β π) β π β πΎ) | |
| 11 | assaring 21635 | . . . . 5 β’ (π β AssAlg β π β Ring) | |
| 12 | 11 | 3ad2ant1 1133 | . . . 4 β’ ((π β AssAlg β§ π β πΎ β§ π β π) β π β Ring) |
| 13 | asclmul1.v | . . . . 5 β’ π = (Baseβπ) | |
| 14 | 13, 5 | ringidcl 20154 | . . . 4 β’ (π β Ring β (1rβπ) β π) |
| 15 | 12, 14 | syl 17 | . . 3 β’ ((π β AssAlg β§ π β πΎ β§ π β π) β (1rβπ) β π) |
| 16 | simp3 1138 | . . 3 β’ ((π β AssAlg β§ π β πΎ β§ π β π) β π β π) | |
| 17 | asclmul1.t | . . . 4 β’ Γ = (.rβπ) | |
| 18 | 13, 2, 3, 4, 17 | assaass 21632 | . . 3 β’ ((π β AssAlg β§ (π β πΎ β§ (1rβπ) β π β§ π β π)) β ((π Β· (1rβπ)) Γ π) = (π Β· ((1rβπ) Γ π))) |
| 19 | 9, 10, 15, 16, 18 | syl13anc 1372 | . 2 β’ ((π β AssAlg β§ π β πΎ β§ π β π) β ((π Β· (1rβπ)) Γ π) = (π Β· ((1rβπ) Γ π))) |
| 20 | 13, 17, 5 | ringlidm 20157 | . . . 4 β’ ((π β Ring β§ π β π) β ((1rβπ) Γ π) = π) |
| 21 | 12, 16, 20 | syl2anc 584 | . . 3 β’ ((π β AssAlg β§ π β πΎ β§ π β π) β ((1rβπ) Γ π) = π) |
| 22 | 21 | oveq2d 7427 | . 2 β’ ((π β AssAlg β§ π β πΎ β§ π β π) β (π Β· ((1rβπ) Γ π)) = (π Β· π)) |
| 23 | 8, 19, 22 | 3eqtrd 2776 | 1 β’ ((π β AssAlg β§ π β πΎ β§ π β π) β ((π΄βπ ) Γ π) = (π Β· π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7411 Basecbs 17148 .rcmulr 17202 Scalarcsca 17204 Β·π cvsca 17205 1rcur 20075 Ringcrg 20127 AssAlgcasa 21624 algSccascl 21626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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-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-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-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mgp 20029 df-ur 20076 df-ring 20129 df-assa 21627 df-ascl 21629 |
| This theorem is referenced by: ascldimul 21661 issubassa2 21665 mplind 21850 evl1vsd 22083 evl1scvarpw 22102 chpscmatgsumbin 22566 fta1blem 25910 evls1fpws 32908 selvvvval 41459 |
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