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Theorem qqhf 28864
Description: QQHom as a function. (Contributed by Thierry Arnoux, 28-Oct-2017.)
Hypotheses
Ref Expression
qqhval2.0  |-  B  =  ( Base `  R
)
qqhval2.1  |-  ./  =  (/r
`  R )
qqhval2.2  |-  L  =  ( ZRHom `  R
)
Assertion
Ref Expression
qqhf  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )

Proof of Theorem qqhf
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 qqhval2.0 . . 3  |-  B  =  ( Base `  R
)
2 qqhval2.1 . . 3  |-  ./  =  (/r
`  R )
3 qqhval2.2 . . 3  |-  L  =  ( ZRHom `  R
)
41, 2, 3qqhval2 28860 . 2  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  =  ( q  e.  QQ  |->  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) )
5 drngring 18060 . . . . 5  |-  ( R  e.  DivRing  ->  R  e.  Ring )
65adantr 472 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  R  e.  Ring )
76adantr 472 . . 3  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  R  e.  Ring )
83zrhrhm 19160 . . . . 5  |-  ( R  e.  Ring  ->  L  e.  (ring RingHom  R ) )
9 zringbas 19122 . . . . . 6  |-  ZZ  =  ( Base ` ring )
109, 1rhmf 18032 . . . . 5  |-  ( L  e.  (ring RingHom  R )  ->  L : ZZ --> B )
117, 8, 103syl 18 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  L : ZZ --> B )
12 qnumcl 14768 . . . . 5  |-  ( q  e.  QQ  ->  (numer `  q )  e.  ZZ )
1312adantl 473 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  (numer `  q )  e.  ZZ )
1411, 13ffvelrnd 6038 . . 3  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  ( L `  (numer `  q
) )  e.  B
)
15 simpll 768 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  R  e.  DivRing )
16 qdencl 14769 . . . . . . 7  |-  ( q  e.  QQ  ->  (denom `  q )  e.  NN )
1716adantl 473 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  (denom `  q )  e.  NN )
1817nnzd 11062 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  (denom `  q )  e.  ZZ )
1911, 18ffvelrnd 6038 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  ( L `  (denom `  q
) )  e.  B
)
2017nnne0d 10676 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  (denom `  q )  =/=  0
)
2120neneqd 2648 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  -.  (denom `  q )  =  0 )
22 fvex 5889 . . . . . . . . . 10  |-  (denom `  q )  e.  _V
2322elsnc 3984 . . . . . . . . 9  |-  ( (denom `  q )  e.  {
0 }  <->  (denom `  q
)  =  0 )
2421, 23sylnibr 312 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  -.  (denom `  q )  e. 
{ 0 } )
25 eqid 2471 . . . . . . . . . . . 12  |-  ( 0g
`  R )  =  ( 0g `  R
)
261, 3, 25zrhker 28855 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( (chr
`  R )  =  0  <->  ( `' L " { ( 0g `  R ) } )  =  { 0 } ) )
2726biimpa 492 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  (chr `  R )  =  0 )  ->  ( `' L " { ( 0g
`  R ) } )  =  { 0 } )
285, 27sylan 479 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( `' L " { ( 0g
`  R ) } )  =  { 0 } )
2928adantr 472 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  ( `' L " { ( 0g `  R ) } )  =  {
0 } )
3024, 29neleqtrrd 2571 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  -.  (denom `  q )  e.  ( `' L " { ( 0g `  R ) } ) )
31 ffn 5739 . . . . . . . . . . . 12  |-  ( L : ZZ --> B  ->  L  Fn  ZZ )
328, 10, 313syl 18 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  L  Fn  ZZ )
33 elpreima 6017 . . . . . . . . . . 11  |-  ( L  Fn  ZZ  ->  (
(denom `  q )  e.  ( `' L " { ( 0g `  R ) } )  <-> 
( (denom `  q
)  e.  ZZ  /\  ( L `  (denom `  q ) )  e. 
{ ( 0g `  R ) } ) ) )
345, 32, 333syl 18 . . . . . . . . . 10  |-  ( R  e.  DivRing  ->  ( (denom `  q )  e.  ( `' L " { ( 0g `  R ) } )  <->  ( (denom `  q )  e.  ZZ  /\  ( L `  (denom `  q ) )  e. 
{ ( 0g `  R ) } ) ) )
3534biimpar 493 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  (
(denom `  q )  e.  ZZ  /\  ( L `
 (denom `  q
) )  e.  {
( 0g `  R
) } ) )  ->  (denom `  q
)  e.  ( `' L " { ( 0g `  R ) } ) )
3635expr 626 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  (denom `  q )  e.  ZZ )  ->  ( ( L `
 (denom `  q
) )  e.  {
( 0g `  R
) }  ->  (denom `  q )  e.  ( `' L " { ( 0g `  R ) } ) ) )
3736con3dimp 448 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (denom `  q )  e.  ZZ )  /\  -.  (denom `  q )  e.  ( `' L " { ( 0g `  R ) } ) )  ->  -.  ( L `  (denom `  q
) )  e.  {
( 0g `  R
) } )
3815, 18, 30, 37syl21anc 1291 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  -.  ( L `  (denom `  q ) )  e. 
{ ( 0g `  R ) } )
39 fvex 5889 . . . . . . 7  |-  ( L `
 (denom `  q
) )  e.  _V
4039elsnc 3984 . . . . . 6  |-  ( ( L `  (denom `  q ) )  e. 
{ ( 0g `  R ) }  <->  ( L `  (denom `  q )
)  =  ( 0g
`  R ) )
4138, 40sylnib 311 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  -.  ( L `  (denom `  q ) )  =  ( 0g `  R
) )
4241neqned 2650 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  ( L `  (denom `  q
) )  =/=  ( 0g `  R ) )
43 eqid 2471 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
441, 43, 25drngunit 18058 . . . . 5  |-  ( R  e.  DivRing  ->  ( ( L `
 (denom `  q
) )  e.  (Unit `  R )  <->  ( ( L `  (denom `  q
) )  e.  B  /\  ( L `  (denom `  q ) )  =/=  ( 0g `  R
) ) ) )
4544biimpar 493 . . . 4  |-  ( ( R  e.  DivRing  /\  (
( L `  (denom `  q ) )  e.  B  /\  ( L `
 (denom `  q
) )  =/=  ( 0g `  R ) ) )  ->  ( L `  (denom `  q )
)  e.  (Unit `  R ) )
4615, 19, 42, 45syl12anc 1290 . . 3  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  ( L `  (denom `  q
) )  e.  (Unit `  R ) )
471, 43, 2dvrcl 17992 . . 3  |-  ( ( R  e.  Ring  /\  ( L `  (numer `  q
) )  e.  B  /\  ( L `  (denom `  q ) )  e.  (Unit `  R )
)  ->  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) )  e.  B )
487, 14, 46, 47syl3anc 1292 . 2  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  (
( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) )  e.  B )
494, 48fmpt3d 6062 1  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {csn 3959   `'ccnv 4838   "cima 4842    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   NNcn 10631   ZZcz 10961   QQcq 11287  numercnumer 14761  denomcdenom 14762   Basecbs 15199   0gc0g 15416   Ringcrg 17858  Unitcui 17945  /rcdvr 17988   RingHom crh 18018   DivRingcdr 18053  ℤringzring 19116   ZRHomczrh 19148  chrcchr 19150  QQHomcqqh 28850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-numer 14763  df-denom 14764  df-gz 14953  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-ghm 16959  df-od 17250  df-cmn 17510  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-rnghom 18021  df-drng 18055  df-subrg 18084  df-cnfld 19048  df-zring 19117  df-zrh 19152  df-chr 19154  df-qqh 28851
This theorem is referenced by:  qqhghm  28866  qqhrhm  28867  qqhcn  28869  qqhucn  28870  qqhre  28898
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