MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1idpr Structured version   Unicode version

Theorem 1idpr 8906
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 2711 . . . . 5  |-  ( E. g  e.  1P  x  =  ( f  .Q  g )  <->  E. g
( g  e.  1P  /\  x  =  ( f  .Q  g ) ) )
2 19.42v 1928 . . . . . 6  |-  ( E. g ( x  <Q  f  /\  x  =  ( f  .Q  g ) )  <->  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) )
3 elprnq 8868 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  f  e.  Q. )
4 breq1 4215 . . . . . . . . . . 11  |-  ( x  =  ( f  .Q  g )  ->  (
x  <Q  f  <->  ( f  .Q  g )  <Q  f
) )
5 df-1p 8859 . . . . . . . . . . . . 13  |-  1P  =  { g  |  g 
<Q  1Q }
65abeq2i 2543 . . . . . . . . . . . 12  |-  ( g  e.  1P  <->  g  <Q  1Q )
7 ltmnq 8849 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
8 mulidnq 8840 . . . . . . . . . . . . . 14  |-  ( f  e.  Q.  ->  (
f  .Q  1Q )  =  f )
98breq2d 4224 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
( f  .Q  g
)  <Q  ( f  .Q  1Q )  <->  ( f  .Q  g )  <Q  f
) )
107, 9bitrd 245 . . . . . . . . . . . 12  |-  ( f  e.  Q.  ->  (
g  <Q  1Q  <->  ( f  .Q  g )  <Q  f
) )
116, 10syl5rbb 250 . . . . . . . . . . 11  |-  ( f  e.  Q.  ->  (
( f  .Q  g
)  <Q  f  <->  g  e.  1P ) )
124, 11sylan9bbr 682 . . . . . . . . . 10  |-  ( ( f  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  1P ) )
133, 12sylan 458 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  f  e.  A )  /\  x  =  ( f  .Q  g ) )  ->  ( x  <Q  f  <->  g  e.  1P ) )
1413ex 424 . . . . . . . 8  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( x  =  ( f  .Q  g )  ->  ( x  <Q  f  <-> 
g  e.  1P ) ) )
1514pm5.32rd 622 . . . . . . 7  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( ( x  <Q  f  /\  x  =  ( f  .Q  g ) )  <->  ( g  e.  1P  /\  x  =  ( f  .Q  g
) ) ) )
1615exbidv 1636 . . . . . 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 252 . . . . 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 249 . . . 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 2722 . . 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 8892 . . . 4  |-  1P  e.  P.
21 df-mp 8861 . . . . 5  |-  .P.  =  ( y  e.  P. ,  z  e.  P.  |->  { w  |  E. u  e.  y  E. v  e.  z  w  =  ( u  .Q  v ) } )
22 mulclnq 8824 . . . . 5  |-  ( ( u  e.  Q.  /\  v  e.  Q. )  ->  ( u  .Q  v
)  e.  Q. )
2321, 22genpelv 8877 . . . 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 653 . . 3  |-  ( A  e.  P.  ->  (
x  e.  ( A  .P.  1P )  <->  E. f  e.  A  E. g  e.  1P  x  =  ( f  .Q  g ) ) )
25 prnmax 8872 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  E. f  e.  A  x  <Q  f )
26 ltrelnq 8803 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
2726brel 4926 . . . . . . . . . 10  |-  ( x 
<Q  f  ->  ( x  e.  Q.  /\  f  e.  Q. ) )
28 vex 2959 . . . . . . . . . . . . . 14  |-  f  e. 
_V
29 vex 2959 . . . . . . . . . . . . . 14  |-  x  e. 
_V
30 fvex 5742 . . . . . . . . . . . . . 14  |-  ( *Q
`  f )  e. 
_V
31 mulcomnq 8830 . . . . . . . . . . . . . 14  |-  ( y  .Q  z )  =  ( z  .Q  y
)
32 mulassnq 8836 . . . . . . . . . . . . . 14  |-  ( ( y  .Q  z )  .Q  w )  =  ( y  .Q  (
z  .Q  w ) )
3328, 29, 30, 31, 32caov12 6275 . . . . . . . . . . . . 13  |-  ( f  .Q  ( x  .Q  ( *Q `  f ) ) )  =  ( x  .Q  ( f  .Q  ( *Q `  f ) ) )
34 recidnq 8842 . . . . . . . . . . . . . 14  |-  ( f  e.  Q.  ->  (
f  .Q  ( *Q
`  f ) )  =  1Q )
3534oveq2d 6097 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
x  .Q  ( f  .Q  ( *Q `  f ) ) )  =  ( x  .Q  1Q ) )
3633, 35syl5eq 2480 . . . . . . . . . . . 12  |-  ( f  e.  Q.  ->  (
f  .Q  ( x  .Q  ( *Q `  f ) ) )  =  ( x  .Q  1Q ) )
37 mulidnq 8840 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  (
x  .Q  1Q )  =  x )
3836, 37sylan9eqr 2490 . . . . . . . . . . 11  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) )  =  x )
3938eqcomd 2441 . . . . . . . . . 10  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  x  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
40 ovex 6106 . . . . . . . . . . 11  |-  ( x  .Q  ( *Q `  f ) )  e. 
_V
41 oveq2 6089 . . . . . . . . . . . 12  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( f  .Q  g )  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
4241eqeq2d 2447 . . . . . . . . . . 11  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( x  =  ( f  .Q  g )  <->  x  =  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) ) ) )
4340, 42spcev 3043 . . . . . . . . . 10  |-  ( x  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) )  ->  E. g  x  =  ( f  .Q  g ) )
4427, 39, 433syl 19 . . . . . . . . 9  |-  ( x 
<Q  f  ->  E. g  x  =  ( f  .Q  g ) )
4544a1i 11 . . . . . . . 8  |-  ( f  e.  A  ->  (
x  <Q  f  ->  E. g  x  =  ( f  .Q  g ) ) )
4645ancld 537 . . . . . . 7  |-  ( f  e.  A  ->  (
x  <Q  f  ->  (
x  <Q  f  /\  E. g  x  =  (
f  .Q  g ) ) ) )
4746reximia 2811 . . . . . 6  |-  ( E. f  e.  A  x 
<Q  f  ->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) )
4825, 47syl 16 . . . . 5  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) )
4948ex 424 . . . 4  |-  ( A  e.  P.  ->  (
x  e.  A  ->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) ) )
50 prcdnq 8870 . . . . . 6  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( x  <Q  f  ->  x  e.  A ) )
5150adantrd 455 . . . . 5  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) )  ->  x  e.  A )
)
5251rexlimdva 2830 . . . 4  |-  ( A  e.  P.  ->  ( E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) )  ->  x  e.  A ) )
5349, 52impbid 184 . . 3  |-  ( A  e.  P.  ->  (
x  e.  A  <->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) ) )
5419, 24, 533bitr4d 277 . 2  |-  ( A  e.  P.  ->  (
x  e.  ( A  .P.  1P )  <->  x  e.  A ) )
5554eqrdv 2434 1  |-  ( A  e.  P.  ->  ( A  .P.  1P )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Q.cnq 8727   1Qc1q 8728    .Q cmq 8731   *Qcrq 8732    <Q cltq 8733   P.cnp 8734   1Pc1p 8735    .P. cmp 8737
This theorem is referenced by:  m1m1sr  8968  1idsr  8973
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ni 8749  df-pli 8750  df-mi 8751  df-lti 8752  df-plpq 8785  df-mpq 8786  df-ltpq 8787  df-enq 8788  df-nq 8789  df-erq 8790  df-plq 8791  df-mq 8792  df-1nq 8793  df-rq 8794  df-ltnq 8795  df-np 8858  df-1p 8859  df-mp 8861
  Copyright terms: Public domain W3C validator