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Mirrors > Home > MPE Home > Th. List > lmodsca | Structured version Visualization version GIF version |
Description: The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
lvecfn.w | ⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) |
Ref | Expression |
---|---|
lmodsca | ⊢ (𝐹 ∈ 𝑋 → 𝐹 = (Scalar‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecfn.w | . . 3 ⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) | |
2 | 1 | lmodstr 16217 | . 2 ⊢ 𝑊 Struct 〈1, 6〉 |
3 | scaid 16214 | . 2 ⊢ Scalar = Slot (Scalar‘ndx) | |
4 | snsstp3 4492 | . . 3 ⊢ {〈(Scalar‘ndx), 𝐹〉} ⊆ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} | |
5 | ssun1 3917 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ⊆ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) | |
6 | 5, 1 | sseqtr4i 3777 | . . 3 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ⊆ 𝑊 |
7 | 4, 6 | sstri 3751 | . 2 ⊢ {〈(Scalar‘ndx), 𝐹〉} ⊆ 𝑊 |
8 | 2, 3, 7 | strfv 16107 | 1 ⊢ (𝐹 ∈ 𝑋 → 𝐹 = (Scalar‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1630 ∈ wcel 2137 ∪ cun 3711 {csn 4319 {ctp 4323 〈cop 4325 ‘cfv 6047 1c1 10127 6c6 11264 ndxcnx 16054 Basecbs 16057 +gcplusg 16141 Scalarcsca 16144 ·𝑠 cvsca 16145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 ax-cnex 10182 ax-resscn 10183 ax-1cn 10184 ax-icn 10185 ax-addcl 10186 ax-addrcl 10187 ax-mulcl 10188 ax-mulrcl 10189 ax-mulcom 10190 ax-addass 10191 ax-mulass 10192 ax-distr 10193 ax-i2m1 10194 ax-1ne0 10195 ax-1rid 10196 ax-rnegex 10197 ax-rrecex 10198 ax-cnre 10199 ax-pre-lttri 10200 ax-pre-lttrn 10201 ax-pre-ltadd 10202 ax-pre-mulgt0 10203 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-nel 3034 df-ral 3053 df-rex 3054 df-reu 3055 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-pss 3729 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-tp 4324 df-op 4326 df-uni 4587 df-int 4626 df-iun 4672 df-br 4803 df-opab 4863 df-mpt 4880 df-tr 4903 df-id 5172 df-eprel 5177 df-po 5185 df-so 5186 df-fr 5223 df-we 5225 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-pred 5839 df-ord 5885 df-on 5886 df-lim 5887 df-suc 5888 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-riota 6772 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-om 7229 df-1st 7331 df-2nd 7332 df-wrecs 7574 df-recs 7635 df-rdg 7673 df-1o 7727 df-oadd 7731 df-er 7909 df-en 8120 df-dom 8121 df-sdom 8122 df-fin 8123 df-pnf 10266 df-mnf 10267 df-xr 10268 df-ltxr 10269 df-le 10270 df-sub 10458 df-neg 10459 df-nn 11211 df-2 11269 df-3 11270 df-4 11271 df-5 11272 df-6 11273 df-n0 11483 df-z 11568 df-uz 11878 df-fz 12518 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-plusg 16154 df-sca 16157 df-vsca 16158 |
This theorem is referenced by: cnlmodlem3 23136 cnstrcvs 23139 ldualsca 34920 dvasca 36794 dvhsca 36871 lmod1lem3 42786 lmod1lem4 42787 lmod1lem5 42788 lmod1 42789 lmod1zrnlvec 42791 |
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