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Mirrors > Home > MPE Home > Th. List > clmsscn | Structured version Visualization version GIF version |
Description: The scalar ring of a subcomplex module is a subset of the complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
clm0.f | ⊢ 𝐹 = (Scalar‘𝑊) |
clmsub.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
clmsscn | ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clm0.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | clmsub.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
3 | 1, 2 | clmsubrg 25079 | . 2 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld)) |
4 | cnfldbas 21341 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
5 | 4 | subrgss 20550 | . 2 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 ⊆ ℂ) |
6 | 3, 5 | syl 17 | 1 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6544 ℂcc 11145 Basecbs 17206 Scalarcsca 17262 SubRingcsubrg 20545 ℂfldccnfld 21337 ℂModcclm 25075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-z 12603 df-dec 12722 df-uz 12867 df-fz 13531 df-struct 17142 df-slot 17177 df-ndx 17189 df-base 17207 df-plusg 17272 df-mulr 17273 df-starv 17274 df-tset 17278 df-ple 17279 df-ds 17281 df-unif 17282 df-subrg 20547 df-cnfld 21338 df-clm 25076 |
This theorem is referenced by: clmneg 25094 clmvscom 25103 cvsi 25143 cvsmuleqdivd 25147 cvsdiveqd 25148 cphassr 25226 cph2ass 25227 tcphcphlem3 25247 ipcau2 25248 tcphcphlem1 25249 tcphcphlem2 25250 nmparlem 25253 ipcn 25260 |
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