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Mirrors > Home > MPE Home > Th. List > lmodvs1 | Structured version Visualization version GIF version |
Description: Scalar product with ring unit. (ax-hvmulid 28710 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodvs1.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodvs1.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodvs1.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodvs1.u | ⊢ 1 = (1r‘𝐹) |
Ref | Expression |
---|---|
lmodvs1 | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 1 · 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ LMod) | |
2 | lmodvs1.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2818 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
4 | lmodvs1.u | . . . 4 ⊢ 1 = (1r‘𝐹) | |
5 | 2, 3, 4 | lmod1cl 19590 | . . 3 ⊢ (𝑊 ∈ LMod → 1 ∈ (Base‘𝐹)) |
6 | 5 | adantr 481 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 1 ∈ (Base‘𝐹)) |
7 | simpr 485 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
8 | lmodvs1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
9 | eqid 2818 | . . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
10 | lmodvs1.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
11 | eqid 2818 | . . . 4 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
12 | eqid 2818 | . . . 4 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
13 | 8, 9, 10, 2, 3, 11, 12, 4 | lmodlema 19568 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ ( 1 ∈ (Base‘𝐹) ∧ 1 ∈ (Base‘𝐹)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((( 1 · 𝑋) ∈ 𝑉 ∧ ( 1 · (𝑋(+g‘𝑊)𝑋)) = (( 1 · 𝑋)(+g‘𝑊)( 1 · 𝑋)) ∧ (( 1 (+g‘𝐹) 1 ) · 𝑋) = (( 1 · 𝑋)(+g‘𝑊)( 1 · 𝑋))) ∧ ((( 1 (.r‘𝐹) 1 ) · 𝑋) = ( 1 · ( 1 · 𝑋)) ∧ ( 1 · 𝑋) = 𝑋))) |
14 | 13 | simprrd 770 | . 2 ⊢ ((𝑊 ∈ LMod ∧ ( 1 ∈ (Base‘𝐹) ∧ 1 ∈ (Base‘𝐹)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ( 1 · 𝑋) = 𝑋) |
15 | 1, 6, 6, 7, 7, 14 | syl122anc 1371 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 1 · 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 1rcur 19180 LModclmod 19563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mgp 19169 df-ur 19181 df-ring 19228 df-lmod 19565 |
This theorem is referenced by: lmodfopne 19601 lmodvneg1 19606 lmodcom 19609 lssvacl 19655 islss3 19660 prdslmodd 19670 lspsn 19703 islmhm2 19739 lbsind2 19782 lvecvs0or 19809 lssvs0or 19811 lvecinv 19814 lspsnvs 19815 lspsneq 19823 lspfixed 19829 lspexch 19830 lspsolv 19844 asclrhm 20047 assamulgscmlem1 20056 coe1pwmul 20375 ply1scl1 20388 ply1idvr1 20389 frlmup2 20871 lindfind2 20890 scmatid 21051 scmatmhm 21071 matinv 21214 decpmatid 21306 idpm2idmp 21337 chfacfscmulgsum 21396 cpmadugsumlemF 21412 clmvs1 23624 deg1pwle 24640 deg1pw 24641 ply1remlem 24683 imaslmod 30849 lfl0 36081 lfladd 36082 dochfl1 38492 lcfl7lem 38515 mapdpglem21 38708 mapdpglem30 38718 mapdpglem31 38719 hgmapval1 38909 prjsperref 39134 mendlmod 39671 lmod0rng 44067 ascl1 44360 ply1vr1smo 44363 linc1 44408 ldepspr 44456 lincresunit3lem3 44457 islindeps2 44466 |
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