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Theorem qsssubdrg 16682
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 10509 . . 3  |-  ( z  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  z  =  ( x  /  y ) )
2 drngrng 15770 . . . . . . . 8  |-  ( (flds  R )  e.  DivRing  ->  (flds  R )  e.  Ring )
32ad2antlr 708 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  (flds  R )  e.  Ring )
4 zsssubrg 16681 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  ZZ  C_  R )
54ad2antrr 707 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ZZ  C_  R
)
6 eqid 2388 . . . . . . . . . . 11  |-  (flds  R )  =  (flds  R )
76subrgbas 15805 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  R  =  ( Base `  (flds  R )
) )
87ad2antrr 707 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  R  =  ( Base `  (flds  R ) ) )
95, 8sseqtrd 3328 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ZZ  C_  ( Base `  (flds  R ) ) )
10 simprl 733 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  ZZ )
119, 10sseldd 3293 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  ( Base `  (flds  R ) ) )
12 nnz 10236 . . . . . . . . . 10  |-  ( y  e.  NN  ->  y  e.  ZZ )
1312ad2antll 710 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  ZZ )
149, 13sseldd 3293 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  ( Base `  (flds  R ) ) )
15 nnne0 9965 . . . . . . . . . 10  |-  ( y  e.  NN  ->  y  =/=  0 )
1615ad2antll 710 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  =/=  0 )
17 cnfld0 16649 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
186, 17subrg0 15803 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  0  =  ( 0g `  (flds  R )
) )
1918ad2antrr 707 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  0  =  ( 0g `  (flds  R ) ) )
2016, 19neeqtrd 2573 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  =/=  ( 0g `  (flds  R ) ) )
21 eqid 2388 . . . . . . . . . 10  |-  ( Base `  (flds  R ) )  =  (
Base `  (flds  R ) )
22 eqid 2388 . . . . . . . . . 10  |-  (Unit `  (flds  R
) )  =  (Unit `  (flds  R ) )
23 eqid 2388 . . . . . . . . . 10  |-  ( 0g
`  (flds  R ) )  =  ( 0g `  (flds  R ) )
2421, 22, 23drngunit 15768 . . . . . . . . 9  |-  ( (flds  R )  e.  DivRing  ->  ( y  e.  (Unit `  (flds  R ) )  <->  ( y  e.  ( Base `  (flds  R )
)  /\  y  =/=  ( 0g `  (flds  R ) ) ) ) )
2524ad2antlr 708 . . . . . . . 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 889 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  (Unit `  (flds  R ) ) )
27 eqid 2388 . . . . . . . 8  |-  (/r `  (flds  R )
)  =  (/r `  (flds  R )
)
2821, 22, 27dvrcl 15719 . . . . . . 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 1184 . . . . . 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 731 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  R  e.  (SubRing ` fld ) )
315, 10sseldd 3293 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  R )
32 cnflddiv 16655 . . . . . . . 8  |-  /  =  (/r
` fld
)
336, 32, 22, 27subrgdv 15813 . . . . . . 7  |-  ( ( R  e.  (SubRing ` fld )  /\  x  e.  R  /\  y  e.  (Unit `  (flds  R ) ) )  ->  ( x  / 
y )  =  ( x (/r `  (flds  R ) ) y ) )
3430, 31, 26, 33syl3anc 1184 . . . . . 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 2469 . . . . 5  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( x  /  y )  e.  R )
36 eleq1 2448 . . . . 5  |-  ( z  =  ( x  / 
y )  ->  (
z  e.  R  <->  ( x  /  y )  e.  R ) )
3735, 36syl5ibrcom 214 . . . 4  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( z  =  ( x  / 
y )  ->  z  e.  R ) )
3837rexlimdvva 2781 . . 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 209 . 2  |-  ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  ->  (
z  e.  QQ  ->  z  e.  R ) )
4039ssrdv 3298 1  |-  ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  ->  QQ  C_  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651    C_ wss 3264   ` cfv 5395  (class class class)co 6021   0cc0 8924    / cdiv 9610   NNcn 9933   ZZcz 10215   QQcq 10507   Basecbs 13397   ↾s cress 13398   0gc0g 13651   Ringcrg 15588  Unitcui 15672  /rcdvr 15715   DivRingcdr 15763  SubRingcsubrg 15792  ℂfldccnfld 16627
This theorem is referenced by:  cphqss  19023  resscdrg  19180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-fz 10977  df-seq 11252  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-mulg 14743  df-subg 14869  df-cmn 15342  df-mgp 15577  df-rng 15591  df-cring 15592  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-dvr 15716  df-drng 15765  df-subrg 15794  df-cnfld 16628
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