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Theorem 1idpr 8653
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 2549 . . . . 5  |-  ( E. g  e.  1P  x  =  ( f  .Q  g )  <->  E. g
( g  e.  1P  /\  x  =  ( f  .Q  g ) ) )
2 19.42v 1846 . . . . . 6  |-  ( E. g ( x  <Q  f  /\  x  =  ( f  .Q  g ) )  <->  ( x  <Q  f  /\  E. g  x  =  ( f  .Q  g ) ) )
3 elprnq 8615 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  f  e.  A )  ->  f  e.  Q. )
4 breq1 4026 . . . . . . . . . . 11  |-  ( x  =  ( f  .Q  g )  ->  (
x  <Q  f  <->  ( f  .Q  g )  <Q  f
) )
5 df-1p 8606 . . . . . . . . . . . . 13  |-  1P  =  { g  |  g 
<Q  1Q }
65abeq2i 2390 . . . . . . . . . . . 12  |-  ( g  e.  1P  <->  g  <Q  1Q )
7 ltmnq 8596 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
8 mulidnq 8587 . . . . . . . . . . . . . 14  |-  ( f  e.  Q.  ->  (
f  .Q  1Q )  =  f )
98breq2d 4035 . . . . . . . . . . . . 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 1612 . . . . . 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 2560 . . 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 8639 . . . 4  |-  1P  e.  P.
21 df-mp 8608 . . . . 5  |-  .P.  =  ( y  e.  P. ,  z  e.  P.  |->  { w  |  E. u  e.  y  E. v  e.  z  w  =  ( u  .Q  v ) } )
22 mulclnq 8571 . . . . 5  |-  ( ( u  e.  Q.  /\  v  e.  Q. )  ->  ( u  .Q  v
)  e.  Q. )
2321, 22genpelv 8624 . . . 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 8619 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  E. f  e.  A  x  <Q  f )
26 ltrelnq 8550 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
2726brel 4737 . . . . . . . . . 10  |-  ( x 
<Q  f  ->  ( x  e.  Q.  /\  f  e.  Q. ) )
28 vex 2791 . . . . . . . . . . . . . 14  |-  f  e. 
_V
29 vex 2791 . . . . . . . . . . . . . 14  |-  x  e. 
_V
30 fvex 5539 . . . . . . . . . . . . . 14  |-  ( *Q
`  f )  e. 
_V
31 mulcomnq 8577 . . . . . . . . . . . . . 14  |-  ( y  .Q  z )  =  ( z  .Q  y
)
32 mulassnq 8583 . . . . . . . . . . . . . 14  |-  ( ( y  .Q  z )  .Q  w )  =  ( y  .Q  (
z  .Q  w ) )
3328, 29, 30, 31, 32caov12 6048 . . . . . . . . . . . . 13  |-  ( f  .Q  ( x  .Q  ( *Q `  f ) ) )  =  ( x  .Q  ( f  .Q  ( *Q `  f ) ) )
34 recidnq 8589 . . . . . . . . . . . . . 14  |-  ( f  e.  Q.  ->  (
f  .Q  ( *Q
`  f ) )  =  1Q )
3534oveq2d 5874 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
x  .Q  ( f  .Q  ( *Q `  f ) ) )  =  ( x  .Q  1Q ) )
3633, 35syl5eq 2327 . . . . . . . . . . . 12  |-  ( f  e.  Q.  ->  (
f  .Q  ( x  .Q  ( *Q `  f ) ) )  =  ( x  .Q  1Q ) )
37 mulidnq 8587 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  (
x  .Q  1Q )  =  x )
3836, 37sylan9eqr 2337 . . . . . . . . . . 11  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) )  =  x )
3938eqcomd 2288 . . . . . . . . . 10  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  x  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
40 ovex 5883 . . . . . . . . . . 11  |-  ( x  .Q  ( *Q `  f ) )  e. 
_V
41 oveq2 5866 . . . . . . . . . . . 12  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( f  .Q  g )  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
4241eqeq2d 2294 . . . . . . . . . . 11  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( x  =  ( f  .Q  g )  <->  x  =  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) ) ) )
4340, 42spcev 2875 . . . . . . . . . 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 2648 . . . . . 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 8617 . . . . . 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 2667 . . . 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 2281 1  |-  ( A  e.  P.  ->  ( A  .P.  1P )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Q.cnq 8474   1Qc1q 8475    .Q cmq 8478   *Qcrq 8479    <Q cltq 8480   P.cnp 8481   1Pc1p 8482    .P. cmp 8484
This theorem is referenced by:  m1m1sr  8715  1idsr  8720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-rq 8541  df-ltnq 8542  df-np 8605  df-1p 8606  df-mp 8608
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