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Theorem qsssubdrg 16447
Description: The rational numbers are a subset of any subfield of the complexes. (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 10334 . . 3  |-  ( z  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  z  =  ( x  /  y ) )
2 drngrng 15535 . . . . . . . 8  |-  ( (flds  R )  e.  DivRing  ->  (flds  R )  e.  Ring )
32ad2antlr 707 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  (flds  R )  e.  Ring )
4 zsssubrg 16446 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  ZZ  C_  R )
54ad2antrr 706 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ZZ  C_  R
)
6 eqid 2296 . . . . . . . . . . 11  |-  (flds  R )  =  (flds  R )
76subrgbas 15570 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  R  =  ( Base `  (flds  R )
) )
87ad2antrr 706 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  R  =  ( Base `  (flds  R ) ) )
95, 8sseqtrd 3227 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ZZ  C_  ( Base `  (flds  R ) ) )
10 simprl 732 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  ZZ )
119, 10sseldd 3194 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  ( Base `  (flds  R ) ) )
12 nnz 10061 . . . . . . . . . 10  |-  ( y  e.  NN  ->  y  e.  ZZ )
1312ad2antll 709 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  ZZ )
149, 13sseldd 3194 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  ( Base `  (flds  R ) ) )
15 nnne0 9794 . . . . . . . . . 10  |-  ( y  e.  NN  ->  y  =/=  0 )
1615ad2antll 709 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  =/=  0 )
17 cnfld0 16414 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
186, 17subrg0 15568 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  0  =  ( 0g `  (flds  R )
) )
1918ad2antrr 706 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  0  =  ( 0g `  (flds  R ) ) )
2016, 19neeqtrd 2481 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  =/=  ( 0g `  (flds  R ) ) )
21 eqid 2296 . . . . . . . . . 10  |-  ( Base `  (flds  R ) )  =  (
Base `  (flds  R ) )
22 eqid 2296 . . . . . . . . . 10  |-  (Unit `  (flds  R
) )  =  (Unit `  (flds  R ) )
23 eqid 2296 . . . . . . . . . 10  |-  ( 0g
`  (flds  R ) )  =  ( 0g `  (flds  R ) )
2421, 22, 23drngunit 15533 . . . . . . . . 9  |-  ( (flds  R )  e.  DivRing  ->  ( y  e.  (Unit `  (flds  R ) )  <->  ( y  e.  ( Base `  (flds  R )
)  /\  y  =/=  ( 0g `  (flds  R ) ) ) ) )
2524ad2antlr 707 . . . . . . . 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 888 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  (Unit `  (flds  R ) ) )
27 eqid 2296 . . . . . . . 8  |-  (/r `  (flds  R )
)  =  (/r `  (flds  R )
)
2821, 22, 27dvrcl 15484 . . . . . . 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 1182 . . . . . 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 730 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  R  e.  (SubRing ` fld ) )
315, 10sseldd 3194 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  R )
32 cnflddiv 16420 . . . . . . . . 9  |-  /  =  (/r
` fld
)
336, 32, 22, 27subrgdv 15578 . . . . . . . 8  |-  ( ( R  e.  (SubRing ` fld )  /\  x  e.  R  /\  y  e.  (Unit `  (flds  R ) ) )  ->  ( x  / 
y )  =  ( x (/r `  (flds  R ) ) y ) )
3430, 31, 26, 33syl3anc 1182 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( x  /  y )  =  ( x (/r `  (flds  R )
) y ) )
3534, 8eleq12d 2364 . . . . . 6  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( (
x  /  y )  e.  R  <->  ( x
(/r `  (flds  R ) ) y )  e.  ( Base `  (flds  R )
) ) )
3629, 35mpbird 223 . . . . 5  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( x  /  y )  e.  R )
37 eleq1 2356 . . . . 5  |-  ( z  =  ( x  / 
y )  ->  (
z  e.  R  <->  ( x  /  y )  e.  R ) )
3836, 37syl5ibrcom 213 . . . 4  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( z  =  ( x  / 
y )  ->  z  e.  R ) )
3938rexlimdvva 2687 . . 3  |-  ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  ->  ( E. x  e.  ZZ  E. y  e.  NN  z  =  ( x  / 
y )  ->  z  e.  R ) )
401, 39syl5bi 208 . 2  |-  ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  ->  (
z  e.  QQ  ->  z  e.  R ) )
4140ssrdv 3198 1  |-  ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  ->  QQ  C_  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    C_ wss 3165   ` cfv 5271  (class class class)co 5874   0cc0 8753    / cdiv 9439   NNcn 9762   ZZcz 10040   QQcq 10332   Basecbs 13164   ↾s cress 13165   0gc0g 13416   Ringcrg 15353  Unitcui 15437  /rcdvr 15480   DivRingcdr 15528  SubRingcsubrg 15557  ℂfldccnfld 16393
This theorem is referenced by:  cphqss  18640  resscdrg  18791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-fz 10799  df-seq 11063  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mulg 14508  df-subg 14634  df-cmn 15107  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-subrg 15559  df-cnfld 16394
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