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

Theorem qsssubdrg 18795
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 11228 . . 3  |-  ( z  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  z  =  ( x  /  y ) )
2 drngring 17721 . . . . . . . 8  |-  ( (flds  R )  e.  DivRing  ->  (flds  R )  e.  Ring )
32ad2antlr 725 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  (flds  R )  e.  Ring )
4 zsssubrg 18794 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  ZZ  C_  R )
54ad2antrr 724 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ZZ  C_  R
)
6 eqid 2402 . . . . . . . . . . 11  |-  (flds  R )  =  (flds  R )
76subrgbas 17756 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  R  =  ( Base `  (flds  R )
) )
87ad2antrr 724 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  R  =  ( Base `  (flds  R ) ) )
95, 8sseqtrd 3477 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ZZ  C_  ( Base `  (flds  R ) ) )
10 simprl 756 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  ZZ )
119, 10sseldd 3442 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  ( Base `  (flds  R ) ) )
12 nnz 10926 . . . . . . . . . 10  |-  ( y  e.  NN  ->  y  e.  ZZ )
1312ad2antll 727 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  ZZ )
149, 13sseldd 3442 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  ( Base `  (flds  R ) ) )
15 nnne0 10608 . . . . . . . . . 10  |-  ( y  e.  NN  ->  y  =/=  0 )
1615ad2antll 727 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  =/=  0 )
17 cnfld0 18760 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
186, 17subrg0 17754 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  0  =  ( 0g `  (flds  R )
) )
1918ad2antrr 724 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  0  =  ( 0g `  (flds  R ) ) )
2016, 19neeqtrd 2698 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  =/=  ( 0g `  (flds  R ) ) )
21 eqid 2402 . . . . . . . . . 10  |-  ( Base `  (flds  R ) )  =  (
Base `  (flds  R ) )
22 eqid 2402 . . . . . . . . . 10  |-  (Unit `  (flds  R
) )  =  (Unit `  (flds  R ) )
23 eqid 2402 . . . . . . . . . 10  |-  ( 0g
`  (flds  R ) )  =  ( 0g `  (flds  R ) )
2421, 22, 23drngunit 17719 . . . . . . . . 9  |-  ( (flds  R )  e.  DivRing  ->  ( y  e.  (Unit `  (flds  R ) )  <->  ( y  e.  ( Base `  (flds  R )
)  /\  y  =/=  ( 0g `  (flds  R ) ) ) ) )
2524ad2antlr 725 . . . . . . . 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 923 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  (Unit `  (flds  R ) ) )
27 eqid 2402 . . . . . . . 8  |-  (/r `  (flds  R )
)  =  (/r `  (flds  R )
)
2821, 22, 27dvrcl 17653 . . . . . . 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 1230 . . . . . 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 752 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  R  e.  (SubRing ` fld ) )
315, 10sseldd 3442 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  R )
32 cnflddiv 18766 . . . . . . . 8  |-  /  =  (/r
` fld
)
336, 32, 22, 27subrgdv 17764 . . . . . . 7  |-  ( ( R  e.  (SubRing ` fld )  /\  x  e.  R  /\  y  e.  (Unit `  (flds  R ) ) )  ->  ( x  / 
y )  =  ( x (/r `  (flds  R ) ) y ) )
3430, 31, 26, 33syl3anc 1230 . . . . . 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 2505 . . . . 5  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( x  /  y )  e.  R )
36 eleq1 2474 . . . . 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 2902 . . 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 3447 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754    C_ wss 3413   ` cfv 5568  (class class class)co 6277   0cc0 9521    / cdiv 10246   NNcn 10575   ZZcz 10904   QQcq 11226   Basecbs 14839   ↾s cress 14840   0gc0g 15052   Ringcrg 17516  Unitcui 17606  /rcdvr 17649   DivRingcdr 17714  SubRingcsubrg 17743  ℂfldccnfld 18738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-addf 9600  ax-mulf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-tpos 6957  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-q 11227  df-fz 11725  df-seq 12150  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-minusg 16380  df-mulg 16382  df-subg 16520  df-cmn 17122  df-mgp 17460  df-ur 17472  df-ring 17518  df-cring 17519  df-oppr 17590  df-dvdsr 17608  df-unit 17609  df-invr 17639  df-dvr 17650  df-drng 17716  df-subrg 17745  df-cnfld 18739
This theorem is referenced by:  cphqss  21925  resscdrg  22088
  Copyright terms: Public domain W3C validator