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Theorem qsssubdrg 16750
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 10568 . . 3  |-  ( z  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  z  =  ( x  /  y ) )
2 drngrng 15834 . . . . . . . 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 16749 . . . . . . . . . 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 2435 . . . . . . . . . . 11  |-  (flds  R )  =  (flds  R )
76subrgbas 15869 . . . . . . . . . 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 3376 . . . . . . . 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 3341 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  ( Base `  (flds  R ) ) )
12 nnz 10295 . . . . . . . . . 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 3341 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  e.  ( Base `  (flds  R ) ) )
15 nnne0 10024 . . . . . . . . . 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 16717 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
186, 17subrg0 15867 . . . . . . . . . 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 2620 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  y  =/=  ( 0g `  (flds  R ) ) )
21 eqid 2435 . . . . . . . . . 10  |-  ( Base `  (flds  R ) )  =  (
Base `  (flds  R ) )
22 eqid 2435 . . . . . . . . . 10  |-  (Unit `  (flds  R
) )  =  (Unit `  (flds  R ) )
23 eqid 2435 . . . . . . . . . 10  |-  ( 0g
`  (flds  R ) )  =  ( 0g `  (flds  R ) )
2421, 22, 23drngunit 15832 . . . . . . . . 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 2435 . . . . . . . 8  |-  (/r `  (flds  R )
)  =  (/r `  (flds  R )
)
2821, 22, 27dvrcl 15783 . . . . . . 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 3341 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  x  e.  R )
32 cnflddiv 16723 . . . . . . . 8  |-  /  =  (/r
` fld
)
336, 32, 22, 27subrgdv 15877 . . . . . . 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 2516 . . . . 5  |-  ( ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing )  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  ->  ( x  /  y )  e.  R )
36 eleq1 2495 . . . . 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 2829 . . 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 3346 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    C_ wss 3312   ` cfv 5446  (class class class)co 6073   0cc0 8982    / cdiv 9669   NNcn 9992   ZZcz 10274   QQcq 10566   Basecbs 13461   ↾s cress 13462   0gc0g 13715   Ringcrg 15652  Unitcui 15736  /rcdvr 15779   DivRingcdr 15827  SubRingcsubrg 15856  ℂfldccnfld 16695
This theorem is referenced by:  cphqss  19143  resscdrg  19304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-fz 11036  df-seq 11316  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-mulg 14807  df-subg 14933  df-cmn 15406  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-subrg 15858  df-cnfld 16696
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