Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > lmodvs1 | GIF version | ||
| Description: Scalar product with the ring unity. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
| Ref | Expression |
|---|---|
| lmodvs1.v | β’ π = (Baseβπ) |
| lmodvs1.f | β’ πΉ = (Scalarβπ) |
| lmodvs1.s | β’ Β· = ( Β·π βπ) |
| lmodvs1.u | β’ 1 = (1rβπΉ) |
| Ref | Expression |
|---|---|
| lmodvs1 | β’ ((π β LMod β§ π β π) β ( 1 Β· π) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl 109 | . 2 β’ ((π β LMod β§ π β π) β π β LMod) | |
| 2 | lmodvs1.f | . . . 4 β’ πΉ = (Scalarβπ) | |
| 3 | eqid 2177 | . . . 4 β’ (BaseβπΉ) = (BaseβπΉ) | |
| 4 | lmodvs1.u | . . . 4 β’ 1 = (1rβπΉ) | |
| 5 | 2, 3, 4 | lmod1cl 13410 | . . 3 β’ (π β LMod β 1 β (BaseβπΉ)) |
| 6 | 5 | adantr 276 | . 2 β’ ((π β LMod β§ π β π) β 1 β (BaseβπΉ)) |
| 7 | simpr 110 | . 2 β’ ((π β LMod β§ π β π) β π β π) | |
| 8 | lmodvs1.v | . . . 4 β’ π = (Baseβπ) | |
| 9 | eqid 2177 | . . . 4 β’ (+gβπ) = (+gβπ) | |
| 10 | lmodvs1.s | . . . 4 β’ Β· = ( Β·π βπ) | |
| 11 | eqid 2177 | . . . 4 β’ (+gβπΉ) = (+gβπΉ) | |
| 12 | eqid 2177 | . . . 4 β’ (.rβπΉ) = (.rβπΉ) | |
| 13 | 8, 9, 10, 2, 3, 11, 12, 4 | lmodlema 13387 | . . 3 β’ ((π β LMod β§ ( 1 β (BaseβπΉ) β§ 1 β (BaseβπΉ)) β§ (π β π β§ π β π)) β ((( 1 Β· π) β π β§ ( 1 Β· (π(+gβπ)π)) = (( 1 Β· π)(+gβπ)( 1 Β· π)) β§ (( 1 (+gβπΉ) 1 ) Β· π) = (( 1 Β· π)(+gβπ)( 1 Β· π))) β§ ((( 1 (.rβπΉ) 1 ) Β· π) = ( 1 Β· ( 1 Β· π)) β§ ( 1 Β· π) = π))) |
| 14 | 13 | simprrd 532 | . 2 β’ ((π β LMod β§ ( 1 β (BaseβπΉ) β§ 1 β (BaseβπΉ)) β§ (π β π β§ π β π)) β ( 1 Β· π) = π) |
| 15 | 1, 6, 6, 7, 7, 14 | syl122anc 1247 | 1 β’ ((π β LMod β§ π β π) β ( 1 Β· π) = π) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 β§ w3a 978 = wceq 1353 β wcel 2148 βcfv 5218 (class class class)co 5877 Basecbs 12464 +gcplusg 12538 .rcmulr 12539 Scalarcsca 12541 Β·π cvsca 12542 1rcur 13147 LModclmod 13382 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-pre-ltirr 7925 ax-pre-ltadd 7929 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-ltxr 7999 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-5 8983 df-6 8984 df-ndx 12467 df-slot 12468 df-base 12470 df-sets 12471 df-plusg 12551 df-mulr 12552 df-sca 12554 df-vsca 12555 df-0g 12712 df-mgm 12780 df-sgrp 12813 df-mnd 12823 df-mgp 13136 df-ur 13148 df-ring 13186 df-lmod 13384 |
| This theorem is referenced by: lmodfopne 13421 lmodvneg1 13425 lmodcom 13428 lssvacl 13466 islss3 13471 |
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