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Theorem qqhre 24339
Description: The QQHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
Assertion
Ref Expression
qqhre  |-  (QQHom `  (flds  RR ) )  =  (  _I  |`  QQ )

Proof of Theorem qqhre
StepHypRef Expression
1 resubdrg 16705 . . . . . . 7  |-  ( RR  e.  (SubRing ` fld )  /\  (flds  RR )  e.  DivRing )
21simpri 449 . . . . . 6  |-  (flds  RR )  e.  DivRing
3 drngrng 15797 . . . . . . 7  |-  ( (flds  RR )  e.  DivRing  ->  (flds  RR )  e.  Ring )
4 f1oi 5672 . . . . . . . . . . 11  |-  (  _I  |`  ZZ ) : ZZ -1-1-onto-> ZZ
5 f1of1 5632 . . . . . . . . . . 11  |-  ( (  _I  |`  ZZ ) : ZZ -1-1-onto-> ZZ  ->  (  _I  |`  ZZ ) : ZZ -1-1-> ZZ )
64, 5ax-mp 8 . . . . . . . . . 10  |-  (  _I  |`  ZZ ) : ZZ -1-1-> ZZ
7 zssre 10245 . . . . . . . . . 10  |-  ZZ  C_  RR
8 f1ss 5603 . . . . . . . . . 10  |-  ( ( (  _I  |`  ZZ ) : ZZ -1-1-> ZZ  /\  ZZ  C_  RR )  -> 
(  _I  |`  ZZ ) : ZZ -1-1-> RR )
96, 7, 8mp2an 654 . . . . . . . . 9  |-  (  _I  |`  ZZ ) : ZZ -1-1-> RR
10 zrhre 24338 . . . . . . . . . 10  |-  ( ZRHom `  (flds  RR ) )  =  (  _I  |`  ZZ )
11 f1eq1 5593 . . . . . . . . . 10  |-  ( ( ZRHom `  (flds  RR ) )  =  (  _I  |`  ZZ )  ->  ( ( ZRHom `  (flds  RR ) ) : ZZ -1-1-> RR  <->  (  _I  |`  ZZ ) : ZZ -1-1-> RR ) )
1210, 11ax-mp 8 . . . . . . . . 9  |-  ( ( ZRHom `  (flds  RR ) ) : ZZ -1-1-> RR  <->  (  _I  |`  ZZ ) : ZZ -1-1-> RR )
139, 12mpbir 201 . . . . . . . 8  |-  ( ZRHom `  (flds  RR ) ) : ZZ -1-1-> RR
14 eqid 2404 . . . . . . . . . 10  |-  (flds  RR )  =  (flds  RR )
1514rebase 24222 . . . . . . . . 9  |-  RR  =  ( Base `  (flds  RR ) )
16 eqid 2404 . . . . . . . . 9  |-  ( ZRHom `  (flds  RR ) )  =  ( ZRHom `  (flds  RR ) )
1714re0g 24226 . . . . . . . . 9  |-  0  =  ( 0g `  (flds  RR ) )
1815, 16, 17zrhchr 24313 . . . . . . . 8  |-  ( (flds  RR )  e.  Ring  ->  ( (chr
`  (flds  RR ) )  =  0  <-> 
( ZRHom `  (flds  RR )
) : ZZ -1-1-> RR ) )
1913, 18mpbiri 225 . . . . . . 7  |-  ( (flds  RR )  e.  Ring  ->  (chr `  (flds  RR ) )  =  0 )
202, 3, 19mp2b 10 . . . . . 6  |-  (chr `  (flds  RR ) )  =  0
21 eqid 2404 . . . . . . 7  |-  (/r `  (flds  RR )
)  =  (/r `  (flds  RR )
)
2215, 21, 16qqhf 24323 . . . . . 6  |-  ( ( (flds  RR )  e.  DivRing  /\  (chr `  (flds  RR ) )  =  0 )  ->  (QQHom `  (flds  RR )
) : QQ --> RR )
232, 20, 22mp2an 654 . . . . 5  |-  (QQHom `  (flds  RR ) ) : QQ --> RR
2423a1i 11 . . . 4  |-  (  T. 
->  (QQHom `  (flds  RR ) ) : QQ --> RR )
2524feqmptd 5738 . . 3  |-  (  T. 
->  (QQHom `  (flds  RR ) )  =  ( q  e.  QQ  |->  ( (QQHom `  (flds  RR ) ) `  q ) ) )
2625trud 1329 . 2  |-  (QQHom `  (flds  RR ) )  =  ( q  e.  QQ  |->  ( (QQHom `  (flds  RR ) ) `  q ) )
2715, 21, 16qqhvval 24320 . . . . 5  |-  ( ( ( (flds  RR )  e.  DivRing  /\  (chr `  (flds  RR ) )  =  0 )  /\  q  e.  QQ )  ->  (
(QQHom `  (flds  RR ) ) `  q )  =  ( ( ( ZRHom `  (flds  RR ) ) `  (numer `  q ) ) (/r `  (flds  RR ) ) ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) ) ) )
282, 20, 27mpanl12 664 . . . 4  |-  ( q  e.  QQ  ->  (
(QQHom `  (flds  RR ) ) `  q )  =  ( ( ( ZRHom `  (flds  RR ) ) `  (numer `  q ) ) (/r `  (flds  RR ) ) ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) ) ) )
29 f1f 5598 . . . . . . . 8  |-  ( ( ZRHom `  (flds  RR ) ) : ZZ -1-1-> RR  ->  ( ZRHom `  (flds  RR ) ) : ZZ --> RR )
3013, 29ax-mp 8 . . . . . . 7  |-  ( ZRHom `  (flds  RR ) ) : ZZ --> RR
3130a1i 11 . . . . . 6  |-  ( q  e.  QQ  ->  ( ZRHom `  (flds  RR ) ) : ZZ --> RR )
32 qnumcl 13087 . . . . . 6  |-  ( q  e.  QQ  ->  (numer `  q )  e.  ZZ )
3331, 32ffvelrnd 5830 . . . . 5  |-  ( q  e.  QQ  ->  (
( ZRHom `  (flds  RR )
) `  (numer `  q
) )  e.  RR )
34 qdencl 13088 . . . . . . 7  |-  ( q  e.  QQ  ->  (denom `  q )  e.  NN )
3534nnzd 10330 . . . . . 6  |-  ( q  e.  QQ  ->  (denom `  q )  e.  ZZ )
3631, 35ffvelrnd 5830 . . . . 5  |-  ( q  e.  QQ  ->  (
( ZRHom `  (flds  RR )
) `  (denom `  q
) )  e.  RR )
3735anim1i 552 . . . . . . . 8  |-  ( ( q  e.  QQ  /\  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  =  0 )  ->  (
(denom `  q )  e.  ZZ  /\  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  =  0 ) )
3815, 16, 17zrhf1ker 24312 . . . . . . . . . . . 12  |-  ( (flds  RR )  e.  Ring  ->  ( ( ZRHom `  (flds  RR ) ) : ZZ -1-1-> RR  <->  ( `' ( ZRHom `  (flds  RR ) ) " { 0 } )  =  { 0 } ) )
392, 3, 38mp2b 10 . . . . . . . . . . 11  |-  ( ( ZRHom `  (flds  RR ) ) : ZZ -1-1-> RR  <->  ( `' ( ZRHom `  (flds  RR ) ) " { 0 } )  =  { 0 } )
4013, 39mpbi 200 . . . . . . . . . 10  |-  ( `' ( ZRHom `  (flds  RR )
) " { 0 } )  =  {
0 }
4140eleq2i 2468 . . . . . . . . 9  |-  ( (denom `  q )  e.  ( `' ( ZRHom `  (flds  RR ) ) " {
0 } )  <->  (denom `  q
)  e.  { 0 } )
42 ffn 5550 . . . . . . . . . 10  |-  ( ( ZRHom `  (flds  RR ) ) : ZZ --> RR  ->  ( ZRHom `  (flds  RR ) )  Fn  ZZ )
43 fniniseg 5810 . . . . . . . . . 10  |-  ( ( ZRHom `  (flds  RR ) )  Fn  ZZ  ->  ( (denom `  q )  e.  ( `' ( ZRHom `  (flds  RR ) ) " {
0 } )  <->  ( (denom `  q )  e.  ZZ  /\  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  =  0 ) ) )
4430, 42, 43mp2b 10 . . . . . . . . 9  |-  ( (denom `  q )  e.  ( `' ( ZRHom `  (flds  RR ) ) " {
0 } )  <->  ( (denom `  q )  e.  ZZ  /\  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  =  0 ) )
45 fvex 5701 . . . . . . . . . 10  |-  (denom `  q )  e.  _V
4645elsnc 3797 . . . . . . . . 9  |-  ( (denom `  q )  e.  {
0 }  <->  (denom `  q
)  =  0 )
4741, 44, 463bitr3ri 268 . . . . . . . 8  |-  ( (denom `  q )  =  0  <-> 
( (denom `  q
)  e.  ZZ  /\  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  =  0 ) )
4837, 47sylibr 204 . . . . . . 7  |-  ( ( q  e.  QQ  /\  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  =  0 )  ->  (denom `  q )  =  0 )
4934nnne0d 10000 . . . . . . . . 9  |-  ( q  e.  QQ  ->  (denom `  q )  =/=  0
)
5049adantr 452 . . . . . . . 8  |-  ( ( q  e.  QQ  /\  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  =  0 )  ->  (denom `  q )  =/=  0
)
5150neneqd 2583 . . . . . . 7  |-  ( ( q  e.  QQ  /\  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  =  0 )  ->  -.  (denom `  q )  =  0 )
5248, 51pm2.65da 560 . . . . . 6  |-  ( q  e.  QQ  ->  -.  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  =  0 )
5352neneqad 2637 . . . . 5  |-  ( q  e.  QQ  ->  (
( ZRHom `  (flds  RR )
) `  (denom `  q
) )  =/=  0
)
5414redvr 24230 . . . . 5  |-  ( ( ( ( ZRHom `  (flds  RR ) ) `  (numer `  q ) )  e.  RR  /\  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  e.  RR  /\  (
( ZRHom `  (flds  RR )
) `  (denom `  q
) )  =/=  0
)  ->  ( (
( ZRHom `  (flds  RR )
) `  (numer `  q
) ) (/r `  (flds  RR )
) ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) ) )  =  ( ( ( ZRHom `  (flds  RR ) ) `  (numer `  q ) )  /  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) ) ) )
5533, 36, 53, 54syl3anc 1184 . . . 4  |-  ( q  e.  QQ  ->  (
( ( ZRHom `  (flds  RR ) ) `  (numer `  q ) ) (/r `  (flds  RR ) ) ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) ) )  =  ( ( ( ZRHom `  (flds  RR )
) `  (numer `  q
) )  /  (
( ZRHom `  (flds  RR )
) `  (denom `  q
) ) ) )
5610fveq1i 5688 . . . . . . . 8  |-  ( ( ZRHom `  (flds  RR ) ) `  (numer `  q ) )  =  ( (  _I  |`  ZZ ) `  (numer `  q ) )
57 fvresi 5883 . . . . . . . 8  |-  ( (numer `  q )  e.  ZZ  ->  ( (  _I  |`  ZZ ) `
 (numer `  q
) )  =  (numer `  q ) )
5856, 57syl5eq 2448 . . . . . . 7  |-  ( (numer `  q )  e.  ZZ  ->  ( ( ZRHom `  (flds  RR ) ) `  (numer `  q ) )  =  (numer `  q )
)
5932, 58syl 16 . . . . . 6  |-  ( q  e.  QQ  ->  (
( ZRHom `  (flds  RR )
) `  (numer `  q
) )  =  (numer `  q ) )
6010fveq1i 5688 . . . . . . . 8  |-  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  =  ( (  _I  |`  ZZ ) `  (denom `  q ) )
61 fvresi 5883 . . . . . . . 8  |-  ( (denom `  q )  e.  ZZ  ->  ( (  _I  |`  ZZ ) `
 (denom `  q
) )  =  (denom `  q ) )
6260, 61syl5eq 2448 . . . . . . 7  |-  ( (denom `  q )  e.  ZZ  ->  ( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) )  =  (denom `  q )
)
6335, 62syl 16 . . . . . 6  |-  ( q  e.  QQ  ->  (
( ZRHom `  (flds  RR )
) `  (denom `  q
) )  =  (denom `  q ) )
6459, 63oveq12d 6058 . . . . 5  |-  ( q  e.  QQ  ->  (
( ( ZRHom `  (flds  RR ) ) `  (numer `  q ) )  / 
( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) ) )  =  ( (numer `  q )  /  (denom `  q ) ) )
65 qeqnumdivden 13093 . . . . 5  |-  ( q  e.  QQ  ->  q  =  ( (numer `  q )  /  (denom `  q ) ) )
6664, 65eqtr4d 2439 . . . 4  |-  ( q  e.  QQ  ->  (
( ( ZRHom `  (flds  RR ) ) `  (numer `  q ) )  / 
( ( ZRHom `  (flds  RR ) ) `  (denom `  q ) ) )  =  q )
6728, 55, 663eqtrd 2440 . . 3  |-  ( q  e.  QQ  ->  (
(QQHom `  (flds  RR ) ) `  q )  =  q )
6867mpteq2ia 4251 . 2  |-  ( q  e.  QQ  |->  ( (QQHom `  (flds  RR ) ) `  q
) )  =  ( q  e.  QQ  |->  q )
69 mptresid 5154 . 2  |-  ( q  e.  QQ  |->  q )  =  (  _I  |`  QQ )
7026, 68, 693eqtri 2428 1  |-  (QQHom `  (flds  RR ) )  =  (  _I  |`  QQ )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    T. wtru 1322    = wceq 1649    e. wcel 1721    =/= wne 2567    C_ wss 3280   {csn 3774    e. cmpt 4226    _I cid 4453   `'ccnv 4836    |` cres 4839   "cima 4840    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946    / cdiv 9633   ZZcz 10238   QQcq 10530  numercnumer 13080  denomcdenom 13081   ↾s cress 13425   Ringcrg 15615  /rcdvr 15742   DivRingcdr 15790  SubRingcsubrg 15819  ℂfldccnfld 16658   ZRHomczrh 16733  chrcchr 16735  QQHomcqqh 24309
This theorem is referenced by:  rrhre  24340
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-numer 13082  df-denom 13083  df-gz 13253  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-mnd 14645  df-mhm 14693  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-od 15122  df-cmn 15369  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774  df-drng 15792  df-subrg 15821  df-cnfld 16659  df-zrh 16737  df-chr 16739  df-qqh 24310
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