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Theorem 1idpr 8669
Description: 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
1idpr  |-  ( A  e.  P.  ->  ( A  .P.  1P )  =  A )

Proof of Theorem 1idpr
Dummy variables  x  y  z  w  v  u  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 2562 . . . . 5  |-  ( E. g  e.  1P  x  =  ( f  .Q  g )  <->  E. g
( g  e.  1P  /\  x  =  ( f  .Q  g ) ) )
2 19.42v 1858 . . . . . 6  |-  ( E. g ( x  <Q  f  /\  x  =  ( f  .Q  g ) )  <->  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) )
3 elprnq 8631 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  f  e.  Q. )
4 breq1 4042 . . . . . . . . . . 11  |-  ( x  =  ( f  .Q  g )  ->  (
x  <Q  f  <->  ( f  .Q  g )  <Q  f
) )
5 df-1p 8622 . . . . . . . . . . . . 13  |-  1P  =  { g  |  g 
<Q  1Q }
65abeq2i 2403 . . . . . . . . . . . 12  |-  ( g  e.  1P  <->  g  <Q  1Q )
7 ltmnq 8612 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
8 mulidnq 8603 . . . . . . . . . . . . . 14  |-  ( f  e.  Q.  ->  (
f  .Q  1Q )  =  f )
98breq2d 4051 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
( f  .Q  g
)  <Q  ( f  .Q  1Q )  <->  ( f  .Q  g )  <Q  f
) )
107, 9bitrd 244 . . . . . . . . . . . 12  |-  ( f  e.  Q.  ->  (
g  <Q  1Q  <->  ( f  .Q  g )  <Q  f
) )
116, 10syl5rbb 249 . . . . . . . . . . 11  |-  ( f  e.  Q.  ->  (
( f  .Q  g
)  <Q  f  <->  g  e.  1P ) )
124, 11sylan9bbr 681 . . . . . . . . . 10  |-  ( ( f  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  1P ) )
133, 12sylan 457 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  f  e.  A )  /\  x  =  ( f  .Q  g ) )  ->  ( x  <Q  f  <->  g  e.  1P ) )
1413ex 423 . . . . . . . 8  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( x  =  ( f  .Q  g )  ->  ( x  <Q  f  <-> 
g  e.  1P ) ) )
1514pm5.32rd 621 . . . . . . 7  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( ( x  <Q  f  /\  x  =  ( f  .Q  g ) )  <->  ( g  e.  1P  /\  x  =  ( f  .Q  g
) ) ) )
1615exbidv 1616 . . . . . 6  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( E. g ( x  <Q  f  /\  x  =  ( f  .Q  g ) )  <->  E. g
( g  e.  1P  /\  x  =  ( f  .Q  g ) ) ) )
172, 16syl5rbbr 251 . . . . 5  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( E. g ( g  e.  1P  /\  x  =  ( f  .Q  g ) )  <->  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) ) )
181, 17syl5bb 248 . . . 4  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( E. g  e.  1P  x  =  ( f  .Q  g )  <-> 
( x  <Q  f  /\  E. g  x  =  ( f  .Q  g
) ) ) )
1918rexbidva 2573 . . 3  |-  ( A  e.  P.  ->  ( E. f  e.  A  E. g  e.  1P  x  =  ( f  .Q  g )  <->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) ) )
20 1pr 8655 . . . 4  |-  1P  e.  P.
21 df-mp 8624 . . . . 5  |-  .P.  =  ( y  e.  P. ,  z  e.  P.  |->  { w  |  E. u  e.  y  E. v  e.  z  w  =  ( u  .Q  v ) } )
22 mulclnq 8587 . . . . 5  |-  ( ( u  e.  Q.  /\  v  e.  Q. )  ->  ( u  .Q  v
)  e.  Q. )
2321, 22genpelv 8640 . . . 4  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( x  e.  ( A  .P.  1P )  <->  E. f  e.  A  E. g  e.  1P  x  =  ( f  .Q  g ) ) )
2420, 23mpan2 652 . . 3  |-  ( A  e.  P.  ->  (
x  e.  ( A  .P.  1P )  <->  E. f  e.  A  E. g  e.  1P  x  =  ( f  .Q  g ) ) )
25 prnmax 8635 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  E. f  e.  A  x  <Q  f )
26 ltrelnq 8566 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
2726brel 4753 . . . . . . . . . 10  |-  ( x 
<Q  f  ->  ( x  e.  Q.  /\  f  e.  Q. ) )
28 vex 2804 . . . . . . . . . . . . . 14  |-  f  e. 
_V
29 vex 2804 . . . . . . . . . . . . . 14  |-  x  e. 
_V
30 fvex 5555 . . . . . . . . . . . . . 14  |-  ( *Q
`  f )  e. 
_V
31 mulcomnq 8593 . . . . . . . . . . . . . 14  |-  ( y  .Q  z )  =  ( z  .Q  y
)
32 mulassnq 8599 . . . . . . . . . . . . . 14  |-  ( ( y  .Q  z )  .Q  w )  =  ( y  .Q  (
z  .Q  w ) )
3328, 29, 30, 31, 32caov12 6064 . . . . . . . . . . . . 13  |-  ( f  .Q  ( x  .Q  ( *Q `  f ) ) )  =  ( x  .Q  ( f  .Q  ( *Q `  f ) ) )
34 recidnq 8605 . . . . . . . . . . . . . 14  |-  ( f  e.  Q.  ->  (
f  .Q  ( *Q
`  f ) )  =  1Q )
3534oveq2d 5890 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
x  .Q  ( f  .Q  ( *Q `  f ) ) )  =  ( x  .Q  1Q ) )
3633, 35syl5eq 2340 . . . . . . . . . . . 12  |-  ( f  e.  Q.  ->  (
f  .Q  ( x  .Q  ( *Q `  f ) ) )  =  ( x  .Q  1Q ) )
37 mulidnq 8603 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  (
x  .Q  1Q )  =  x )
3836, 37sylan9eqr 2350 . . . . . . . . . . 11  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) )  =  x )
3938eqcomd 2301 . . . . . . . . . 10  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  x  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
40 ovex 5899 . . . . . . . . . . 11  |-  ( x  .Q  ( *Q `  f ) )  e. 
_V
41 oveq2 5882 . . . . . . . . . . . 12  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( f  .Q  g )  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
4241eqeq2d 2307 . . . . . . . . . . 11  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( x  =  ( f  .Q  g )  <->  x  =  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) ) ) )
4340, 42spcev 2888 . . . . . . . . . 10  |-  ( x  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) )  ->  E. g  x  =  ( f  .Q  g ) )
4427, 39, 433syl 18 . . . . . . . . 9  |-  ( x 
<Q  f  ->  E. g  x  =  ( f  .Q  g ) )
4544a1i 10 . . . . . . . 8  |-  ( f  e.  A  ->  (
x  <Q  f  ->  E. g  x  =  ( f  .Q  g ) ) )
4645ancld 536 . . . . . . 7  |-  ( f  e.  A  ->  (
x  <Q  f  ->  (
x  <Q  f  /\  E. g  x  =  (
f  .Q  g ) ) ) )
4746reximia 2661 . . . . . 6  |-  ( E. f  e.  A  x 
<Q  f  ->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) )
4825, 47syl 15 . . . . 5  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) )
4948ex 423 . . . 4  |-  ( A  e.  P.  ->  (
x  e.  A  ->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) ) )
50 prcdnq 8633 . . . . . 6  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( x  <Q  f  ->  x  e.  A ) )
5150adantrd 454 . . . . 5  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) )  ->  x  e.  A )
)
5251rexlimdva 2680 . . . 4  |-  ( A  e.  P.  ->  ( E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) )  ->  x  e.  A ) )
5349, 52impbid 183 . . 3  |-  ( A  e.  P.  ->  (
x  e.  A  <->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) ) )
5419, 24, 533bitr4d 276 . 2  |-  ( A  e.  P.  ->  (
x  e.  ( A  .P.  1P )  <->  x  e.  A ) )
5554eqrdv 2294 1  |-  ( A  e.  P.  ->  ( A  .P.  1P )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Q.cnq 8490   1Qc1q 8491    .Q cmq 8494   *Qcrq 8495    <Q cltq 8496   P.cnp 8497   1Pc1p 8498    .P. cmp 8500
This theorem is referenced by:  m1m1sr  8731  1idsr  8736
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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-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-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-1p 8622  df-mp 8624
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