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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 27991 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 472 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ LMod) | |
2 | lmodvs1.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2651 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
4 | lmodvs1.u | . . . 4 ⊢ 1 = (1r‘𝐹) | |
5 | 2, 3, 4 | lmod1cl 18938 | . . 3 ⊢ (𝑊 ∈ LMod → 1 ∈ (Base‘𝐹)) |
6 | 5 | adantr 480 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 1 ∈ (Base‘𝐹)) |
7 | simpr 476 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
8 | lmodvs1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
9 | eqid 2651 | . . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
10 | lmodvs1.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
11 | eqid 2651 | . . . 4 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
12 | eqid 2651 | . . . 4 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
13 | 8, 9, 10, 2, 3, 11, 12, 4 | lmodlema 18916 | . . 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 812 | . 2 ⊢ ((𝑊 ∈ LMod ∧ ( 1 ∈ (Base‘𝐹) ∧ 1 ∈ (Base‘𝐹)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ( 1 · 𝑋) = 𝑋) |
15 | 1, 6, 6, 7, 7, 14 | syl122anc 1375 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 1 · 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 .rcmulr 15989 Scalarcsca 15991 ·𝑠 cvsca 15992 1rcur 18547 LModclmod 18911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mgp 18536 df-ur 18548 df-ring 18595 df-lmod 18913 |
This theorem is referenced by: lmodfopne 18949 lmodvneg1 18954 lmodcom 18957 lssvacl 19002 islss3 19007 prdslmodd 19017 lspsn 19050 islmhm2 19086 lbsind2 19129 lvecvs0or 19156 lssvs0or 19158 lvecinv 19161 lspsnvs 19162 lspsneq 19170 lspfixed 19176 lspexch 19177 lspsolv 19191 asclrhm 19390 assamulgscmlem1 19396 coe1pwmul 19697 ply1scl1 19710 ply1idvr1 19711 frlmup2 20186 lindfind2 20205 scmatid 20368 scmatmhm 20388 matinv 20531 decpmatid 20623 idpm2idmp 20654 chfacfscmulgsum 20713 cpmadugsumlemF 20729 clmvs1 22939 cvsi 22976 deg1pwle 23924 deg1pw 23925 ply1remlem 23967 lfl0 34670 lfladd 34671 dochfl1 37082 lcfl7lem 37105 mapdpglem21 37298 mapdpglem30 37308 mapdpglem31 37309 hgmapval1 37502 mendlmod 38080 lmod0rng 42193 ascl1 42491 ply1vr1smo 42494 linc1 42539 ldepspr 42587 lincresunit3lem3 42588 islindeps2 42597 |
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