MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qsssubdrg Structured version   Unicode version

Theorem qsssubdrg 18238
Description: The rational numbers are a subset of any subfield of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
qsssubdrg  |-  ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  ->  QQ  C_  R )

Proof of Theorem qsssubdrg
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elq 11173 . . 3  |-  ( z  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  z  =  ( x  /  y ) )
2 drngrng 17179 . . . . . . . 8  |-  ( (flds  R )  e.  DivRing  ->  (flds  R )  e.  Ring )
32ad2antlr 726 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  (flds  R )  e.  Ring )
4 zsssubrg 18237 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  ZZ  C_  R )
54ad2antrr 725 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ZZ  C_  R
)
6 eqid 2460 . . . . . . . . . . 11  |-  (flds  R )  =  (flds  R )
76subrgbas 17214 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  R  =  ( Base `  (flds  R )
) )
87ad2antrr 725 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  R  =  ( Base `  (flds  R ) ) )
95, 8sseqtrd 3533 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ZZ  C_  ( Base `  (flds  R ) ) )
10 simprl 755 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  ZZ )
119, 10sseldd 3498 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  ( Base `  (flds  R ) ) )
12 nnz 10875 . . . . . . . . . 10  |-  ( y  e.  NN  ->  y  e.  ZZ )
1312ad2antll 728 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  ZZ )
149, 13sseldd 3498 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  ( Base `  (flds  R ) ) )
15 nnne0 10557 . . . . . . . . . 10  |-  ( y  e.  NN  ->  y  =/=  0 )
1615ad2antll 728 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  =/=  0 )
17 cnfld0 18206 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
186, 17subrg0 17212 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  0  =  ( 0g `  (flds  R )
) )
1918ad2antrr 725 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  0  =  ( 0g `  (flds  R ) ) )
2016, 19neeqtrd 2755 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  =/=  ( 0g `  (flds  R ) ) )
21 eqid 2460 . . . . . . . . . 10  |-  ( Base `  (flds  R ) )  =  (
Base `  (flds  R ) )
22 eqid 2460 . . . . . . . . . 10  |-  (Unit `  (flds  R
) )  =  (Unit `  (flds  R ) )
23 eqid 2460 . . . . . . . . . 10  |-  ( 0g
`  (flds  R ) )  =  ( 0g `  (flds  R ) )
2421, 22, 23drngunit 17177 . . . . . . . . 9  |-  ( (flds  R )  e.  DivRing  ->  ( y  e.  (Unit `  (flds  R ) )  <->  ( y  e.  ( Base `  (flds  R )
)  /\  y  =/=  ( 0g `  (flds  R ) ) ) ) )
2524ad2antlr 726 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( y  e.  (Unit `  (flds  R ) )  <->  ( y  e.  ( Base `  (flds  R )
)  /\  y  =/=  ( 0g `  (flds  R ) ) ) ) )
2614, 20, 25mpbir2and 915 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  (Unit `  (flds  R ) ) )
27 eqid 2460 . . . . . . . 8  |-  (/r `  (flds  R )
)  =  (/r `  (flds  R )
)
2821, 22, 27dvrcl 17112 . . . . . . 7  |-  ( ( (flds  R )  e.  Ring  /\  x  e.  ( Base `  (flds  R )
)  /\  y  e.  (Unit `  (flds  R ) ) )  -> 
( x (/r `  (flds  R )
) y )  e.  ( Base `  (flds  R )
) )
293, 11, 26, 28syl3anc 1223 . . . . . 6  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( x
(/r `  (flds  R ) ) y )  e.  ( Base `  (flds  R )
) )
30 simpll 753 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  R  e.  (SubRing ` fld ) )
315, 10sseldd 3498 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  R )
32 cnflddiv 18212 . . . . . . . 8  |-  /  =  (/r
` fld
)
336, 32, 22, 27subrgdv 17222 . . . . . . 7  |-  ( ( R  e.  (SubRing ` fld )  /\  x  e.  R  /\  y  e.  (Unit `  (flds  R ) ) )  ->  ( x  / 
y )  =  ( x (/r `  (flds  R ) ) y ) )
3430, 31, 26, 33syl3anc 1223 . . . . . 6  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( x  /  y )  =  ( x (/r `  (flds  R )
) y ) )
3529, 34, 83eltr4d 2563 . . . . 5  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( x  /  y )  e.  R )
36 eleq1 2532 . . . . 5  |-  ( z  =  ( x  / 
y )  ->  (
z  e.  R  <->  ( x  /  y )  e.  R ) )
3735, 36syl5ibrcom 222 . . . 4  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( z  =  ( x  / 
y )  ->  z  e.  R ) )
3837rexlimdvva 2955 . . 3  |-  ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  ->  ( E. x  e.  ZZ  E. y  e.  NN  z  =  ( x  / 
y )  ->  z  e.  R ) )
391, 38syl5bi 217 . 2  |-  ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  ->  (
z  e.  QQ  ->  z  e.  R ) )
4039ssrdv 3503 1  |-  ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  ->  QQ  C_  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808    C_ wss 3469   ` cfv 5579  (class class class)co 6275   0cc0 9481    / cdiv 10195   NNcn 10525   ZZcz 10853   QQcq 11171   Basecbs 14479   ↾s cress 14480   0gc0g 14684   Ringcrg 16979  Unitcui 17065  /rcdvr 17108   DivRingcdr 17172  SubRingcsubrg 17201  ℂfldccnfld 18184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-fz 11662  df-seq 12064  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-mulg 15854  df-subg 15986  df-cmn 16589  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109  df-drng 17174  df-subrg 17203  df-cnfld 18185
This theorem is referenced by:  cphqss  21363  resscdrg  21526
  Copyright terms: Public domain W3C validator