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Theorem 1idpr 8648
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 )
Dummy variables  x  y  z  w  v  u  f  g are mutually distinct and distinct from all other variables.

Proof of Theorem 1idpr
StepHypRef Expression
1 df-rex 2550 . . . . 5  |-  ( E. g  e.  1P  x  =  ( f  .Q  g )  <->  E. g
( g  e.  1P  /\  x  =  ( f  .Q  g ) ) )
2 19.42v 1847 . . . . . 6  |-  ( E. g ( x  <Q  f  /\  x  =  ( f  .Q  g ) )  <->  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) )
3 elprnq 8610 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  f  e.  Q. )
4 breq1 4027 . . . . . . . . . . 11  |-  ( x  =  ( f  .Q  g )  ->  (
x  <Q  f  <->  ( f  .Q  g )  <Q  f
) )
5 df-1p 8601 . . . . . . . . . . . . 13  |-  1P  =  { g  |  g 
<Q  1Q }
65abeq2i 2391 . . . . . . . . . . . 12  |-  ( g  e.  1P  <->  g  <Q  1Q )
7 ltmnq 8591 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
8 mulidnq 8582 . . . . . . . . . . . . . 14  |-  ( f  e.  Q.  ->  (
f  .Q  1Q )  =  f )
98breq2d 4036 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
( f  .Q  g
)  <Q  ( f  .Q  1Q )  <->  ( f  .Q  g )  <Q  f
) )
107, 9bitrd 246 . . . . . . . . . . . 12  |-  ( f  e.  Q.  ->  (
g  <Q  1Q  <->  ( f  .Q  g )  <Q  f
) )
116, 10syl5rbb 251 . . . . . . . . . . 11  |-  ( f  e.  Q.  ->  (
( f  .Q  g
)  <Q  f  <->  g  e.  1P ) )
124, 11sylan9bbr 683 . . . . . . . . . 10  |-  ( ( f  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  1P ) )
133, 12sylan 459 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  f  e.  A )  /\  x  =  ( f  .Q  g ) )  ->  ( x  <Q  f  <->  g  e.  1P ) )
1413ex 425 . . . . . . . 8  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( x  =  ( f  .Q  g )  ->  ( x  <Q  f  <-> 
g  e.  1P ) ) )
1514pm5.32rd 623 . . . . . . 7  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( ( x  <Q  f  /\  x  =  ( f  .Q  g ) )  <->  ( g  e.  1P  /\  x  =  ( f  .Q  g
) ) ) )
1615exbidv 1613 . . . . . 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 253 . . . . 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 250 . . . 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 2561 . . 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 8634 . . . 4  |-  1P  e.  P.
21 df-mp 8603 . . . . 5  |-  .P.  =  ( y  e.  P. ,  z  e.  P.  |->  { w  |  E. u  e.  y  E. v  e.  z  w  =  ( u  .Q  v ) } )
22 mulclnq 8566 . . . . 5  |-  ( ( u  e.  Q.  /\  v  e.  Q. )  ->  ( u  .Q  v
)  e.  Q. )
2321, 22genpelv 8619 . . . 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 654 . . 3  |-  ( A  e.  P.  ->  (
x  e.  ( A  .P.  1P )  <->  E. f  e.  A  E. g  e.  1P  x  =  ( f  .Q  g ) ) )
25 prnmax 8614 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  E. f  e.  A  x  <Q  f )
26 ltrelnq 8545 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
2726brel 4736 . . . . . . . . . 10  |-  ( x 
<Q  f  ->  ( x  e.  Q.  /\  f  e.  Q. ) )
28 vex 2792 . . . . . . . . . . . . . 14  |-  f  e. 
_V
29 vex 2792 . . . . . . . . . . . . . 14  |-  x  e. 
_V
30 fvex 5499 . . . . . . . . . . . . . 14  |-  ( *Q
`  f )  e. 
_V
31 mulcomnq 8572 . . . . . . . . . . . . . 14  |-  ( y  .Q  z )  =  ( z  .Q  y
)
32 mulassnq 8578 . . . . . . . . . . . . . 14  |-  ( ( y  .Q  z )  .Q  w )  =  ( y  .Q  (
z  .Q  w ) )
3328, 29, 30, 31, 32caov12 6009 . . . . . . . . . . . . 13  |-  ( f  .Q  ( x  .Q  ( *Q `  f ) ) )  =  ( x  .Q  ( f  .Q  ( *Q `  f ) ) )
34 recidnq 8584 . . . . . . . . . . . . . 14  |-  ( f  e.  Q.  ->  (
f  .Q  ( *Q
`  f ) )  =  1Q )
3534oveq2d 5835 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
x  .Q  ( f  .Q  ( *Q `  f ) ) )  =  ( x  .Q  1Q ) )
3633, 35syl5eq 2328 . . . . . . . . . . . 12  |-  ( f  e.  Q.  ->  (
f  .Q  ( x  .Q  ( *Q `  f ) ) )  =  ( x  .Q  1Q ) )
37 mulidnq 8582 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  (
x  .Q  1Q )  =  x )
3836, 37sylan9eqr 2338 . . . . . . . . . . 11  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) )  =  x )
3938eqcomd 2289 . . . . . . . . . 10  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  x  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
40 ovex 5844 . . . . . . . . . . 11  |-  ( x  .Q  ( *Q `  f ) )  e. 
_V
41 oveq2 5827 . . . . . . . . . . . 12  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( f  .Q  g )  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
4241eqeq2d 2295 . . . . . . . . . . 11  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( x  =  ( f  .Q  g )  <->  x  =  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) ) ) )
4340, 42spcev 2876 . . . . . . . . . 10  |-  ( x  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) )  ->  E. g  x  =  ( f  .Q  g ) )
4427, 39, 433syl 20 . . . . . . . . 9  |-  ( x 
<Q  f  ->  E. g  x  =  ( f  .Q  g ) )
4544a1i 12 . . . . . . . 8  |-  ( f  e.  A  ->  (
x  <Q  f  ->  E. g  x  =  ( f  .Q  g ) ) )
4645ancld 538 . . . . . . 7  |-  ( f  e.  A  ->  (
x  <Q  f  ->  (
x  <Q  f  /\  E. g  x  =  (
f  .Q  g ) ) ) )
4746reximia 2649 . . . . . 6  |-  ( E. f  e.  A  x 
<Q  f  ->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) )
4825, 47syl 17 . . . . 5  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) )
4948ex 425 . . . 4  |-  ( A  e.  P.  ->  (
x  e.  A  ->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) ) )
50 prcdnq 8612 . . . . . 6  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( x  <Q  f  ->  x  e.  A ) )
5150adantrd 456 . . . . 5  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  ( ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) )  ->  x  e.  A )
)
5251rexlimdva 2668 . . . 4  |-  ( A  e.  P.  ->  ( E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) )  ->  x  e.  A ) )
5349, 52impbid 185 . . 3  |-  ( A  e.  P.  ->  (
x  e.  A  <->  E. f  e.  A  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) ) )
5419, 24, 533bitr4d 278 . 2  |-  ( A  e.  P.  ->  (
x  e.  ( A  .P.  1P )  <->  x  e.  A ) )
5554eqrdv 2282 1  |-  ( A  e.  P.  ->  ( A  .P.  1P )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   E.wex 1529    = wceq 1624    e. wcel 1685   E.wrex 2545   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Q.cnq 8469   1Qc1q 8470    .Q cmq 8473   *Qcrq 8474    <Q cltq 8475   P.cnp 8476   1Pc1p 8477    .P. cmp 8479
This theorem is referenced by:  m1m1sr  8710  1idsr  8715
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6655  df-ni 8491  df-pli 8492  df-mi 8493  df-lti 8494  df-plpq 8527  df-mpq 8528  df-ltpq 8529  df-enq 8530  df-nq 8531  df-erq 8532  df-plq 8533  df-mq 8534  df-1nq 8535  df-rq 8536  df-ltnq 8537  df-np 8600  df-1p 8601  df-mp 8603
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