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Theorem ltresr 8778
Description: Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltresr  |-  ( <. A ,  0R >.  <RR  <. B ,  0R >. 
<->  A  <R  B )

Proof of Theorem ltresr
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelre 8772 . . . 4  |-  <RR  C_  ( RR  X.  RR )
21brel 4753 . . 3  |-  ( <. A ,  0R >.  <RR  <. B ,  0R >.  ->  ( <. A ,  0R >.  e.  RR  /\ 
<. B ,  0R >.  e.  RR ) )
3 opelreal 8768 . . . 4  |-  ( <. A ,  0R >.  e.  RR  <->  A  e.  R. )
4 opelreal 8768 . . . 4  |-  ( <. B ,  0R >.  e.  RR  <->  B  e.  R. )
53, 4anbi12i 678 . . 3  |-  ( (
<. A ,  0R >.  e.  RR  /\  <. B ,  0R >.  e.  RR )  <-> 
( A  e.  R.  /\  B  e.  R. )
)
62, 5sylib 188 . 2  |-  ( <. A ,  0R >.  <RR  <. B ,  0R >.  ->  ( A  e.  R.  /\  B  e. 
R. ) )
7 ltrelsr 8709 . . 3  |-  <R  C_  ( R.  X.  R. )
87brel 4753 . 2  |-  ( A 
<R  B  ->  ( A  e.  R.  /\  B  e.  R. ) )
9 opex 4253 . . . . . . 7  |-  <. A ,  0R >.  e.  _V
10 opex 4253 . . . . . . 7  |-  <. B ,  0R >.  e.  _V
11 eleq1 2356 . . . . . . . . 9  |-  ( x  =  <. A ,  0R >.  ->  ( x  e.  RR  <->  <. A ,  0R >.  e.  RR ) )
1211anbi1d 685 . . . . . . . 8  |-  ( x  =  <. A ,  0R >.  ->  ( ( x  e.  RR  /\  y  e.  RR )  <->  ( <. A ,  0R >.  e.  RR  /\  y  e.  RR ) ) )
13 eqeq1 2302 . . . . . . . . . . 11  |-  ( x  =  <. A ,  0R >.  ->  ( x  = 
<. z ,  0R >.  <->  <. A ,  0R >.  =  <. z ,  0R >. )
)
1413anbi1d 685 . . . . . . . . . 10  |-  ( x  =  <. A ,  0R >.  ->  ( ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  <->  (
<. A ,  0R >.  = 
<. z ,  0R >.  /\  y  =  <. w ,  0R >. ) ) )
1514anbi1d 685 . . . . . . . . 9  |-  ( x  =  <. A ,  0R >.  ->  ( ( ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w )  <->  ( ( <. A ,  0R >.  = 
<. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) )
16152exbidv 1618 . . . . . . . 8  |-  ( x  =  <. A ,  0R >.  ->  ( E. z E. w ( ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w )  <->  E. z E. w ( ( <. A ,  0R >.  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) )
1712, 16anbi12d 691 . . . . . . 7  |-  ( x  =  <. A ,  0R >.  ->  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w
( ( x  = 
<. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) )  <->  ( ( <. A ,  0R >.  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( (
<. A ,  0R >.  = 
<. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) ) )
18 eleq1 2356 . . . . . . . . 9  |-  ( y  =  <. B ,  0R >.  ->  ( y  e.  RR  <->  <. B ,  0R >.  e.  RR ) )
1918anbi2d 684 . . . . . . . 8  |-  ( y  =  <. B ,  0R >.  ->  ( ( <. A ,  0R >.  e.  RR  /\  y  e.  RR )  <-> 
( <. A ,  0R >.  e.  RR  /\  <. B ,  0R >.  e.  RR ) ) )
20 eqeq1 2302 . . . . . . . . . . 11  |-  ( y  =  <. B ,  0R >.  ->  ( y  = 
<. w ,  0R >.  <->  <. B ,  0R >.  =  <. w ,  0R >. )
)
2120anbi2d 684 . . . . . . . . . 10  |-  ( y  =  <. B ,  0R >.  ->  ( ( <. A ,  0R >.  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  <->  ( <. A ,  0R >.  =  <. z ,  0R >.  /\  <. B ,  0R >.  =  <. w ,  0R >. ) ) )
2221anbi1d 685 . . . . . . . . 9  |-  ( y  =  <. B ,  0R >.  ->  ( ( (
<. A ,  0R >.  = 
<. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w )  <->  ( ( <. A ,  0R >.  = 
<. z ,  0R >.  /\ 
<. B ,  0R >.  = 
<. w ,  0R >. )  /\  z  <R  w
) ) )
23222exbidv 1618 . . . . . . . 8  |-  ( y  =  <. B ,  0R >.  ->  ( E. z E. w ( ( <. A ,  0R >.  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w )  <->  E. z E. w ( ( <. A ,  0R >.  =  <. z ,  0R >.  /\  <. B ,  0R >.  =  <. w ,  0R >. )  /\  z  <R  w ) ) )
2419, 23anbi12d 691 . . . . . . 7  |-  ( y  =  <. B ,  0R >.  ->  ( ( (
<. A ,  0R >.  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( (
<. A ,  0R >.  = 
<. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) )  <->  ( ( <. A ,  0R >.  e.  RR  /\  <. B ,  0R >.  e.  RR )  /\  E. z E. w ( ( <. A ,  0R >.  =  <. z ,  0R >.  /\  <. B ,  0R >.  =  <. w ,  0R >. )  /\  z  <R  w ) ) ) )
25 df-lt 8766 . . . . . . 7  |-  <RR  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) }
269, 10, 17, 24, 25brab 4303 . . . . . 6  |-  ( <. A ,  0R >.  <RR  <. B ,  0R >. 
<->  ( ( <. A ,  0R >.  e.  RR  /\  <. B ,  0R >.  e.  RR )  /\  E. z E. w ( ( <. A ,  0R >.  =  <. z ,  0R >.  /\  <. B ,  0R >.  =  <. w ,  0R >. )  /\  z  <R  w ) ) )
2726baib 871 . . . . 5  |-  ( (
<. A ,  0R >.  e.  RR  /\  <. B ,  0R >.  e.  RR )  ->  ( <. A ,  0R >.  <RR  <. B ,  0R >.  <->  E. z E. w ( ( <. A ,  0R >.  =  <. z ,  0R >.  /\  <. B ,  0R >.  =  <. w ,  0R >. )  /\  z  <R  w ) ) )
28 vex 2804 . . . . . . . . . . 11  |-  z  e. 
_V
2928eqresr 8775 . . . . . . . . . 10  |-  ( <.
z ,  0R >.  = 
<. A ,  0R >.  <->  z  =  A )
30 eqcom 2298 . . . . . . . . . 10  |-  ( <. A ,  0R >.  =  <. z ,  0R >.  <->  <. z ,  0R >.  =  <. A ,  0R >. )
31 eqcom 2298 . . . . . . . . . 10  |-  ( A  =  z  <->  z  =  A )
3229, 30, 313bitr4i 268 . . . . . . . . 9  |-  ( <. A ,  0R >.  =  <. z ,  0R >.  <->  A  =  z )
33 vex 2804 . . . . . . . . . . 11  |-  w  e. 
_V
3433eqresr 8775 . . . . . . . . . 10  |-  ( <.
w ,  0R >.  = 
<. B ,  0R >.  <->  w  =  B )
35 eqcom 2298 . . . . . . . . . 10  |-  ( <. B ,  0R >.  =  <. w ,  0R >.  <->  <. w ,  0R >.  =  <. B ,  0R >. )
36 eqcom 2298 . . . . . . . . . 10  |-  ( B  =  w  <->  w  =  B )
3734, 35, 363bitr4i 268 . . . . . . . . 9  |-  ( <. B ,  0R >.  =  <. w ,  0R >.  <->  B  =  w )
3832, 37anbi12i 678 . . . . . . . 8  |-  ( (
<. A ,  0R >.  = 
<. z ,  0R >.  /\ 
<. B ,  0R >.  = 
<. w ,  0R >. )  <-> 
( A  =  z  /\  B  =  w ) )
3928, 33opth2 4264 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. z ,  w >.  <->  ( A  =  z  /\  B  =  w )
)
4038, 39bitr4i 243 . . . . . . 7  |-  ( (
<. A ,  0R >.  = 
<. z ,  0R >.  /\ 
<. B ,  0R >.  = 
<. w ,  0R >. )  <->  <. A ,  B >.  = 
<. z ,  w >. )
4140anbi1i 676 . . . . . 6  |-  ( ( ( <. A ,  0R >.  =  <. z ,  0R >.  /\  <. B ,  0R >.  =  <. w ,  0R >. )  /\  z  <R  w )  <->  ( <. A ,  B >.  =  <. z ,  w >.  /\  z  <R  w ) )
42412exbii 1573 . . . . 5  |-  ( E. z E. w ( ( <. A ,  0R >.  =  <. z ,  0R >.  /\  <. B ,  0R >.  =  <. w ,  0R >. )  /\  z  <R  w )  <->  E. z E. w ( <. A ,  B >.  =  <. z ,  w >.  /\  z  <R  w ) )
4327, 42syl6bb 252 . . . 4  |-  ( (
<. A ,  0R >.  e.  RR  /\  <. B ,  0R >.  e.  RR )  ->  ( <. A ,  0R >.  <RR  <. B ,  0R >.  <->  E. z E. w (
<. A ,  B >.  = 
<. z ,  w >.  /\  z  <R  w )
) )
443, 4, 43syl2anbr 466 . . 3  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( <. A ,  0R >. 
<RR  <. B ,  0R >.  <->  E. z E. w (
<. A ,  B >.  = 
<. z ,  w >.  /\  z  <R  w )
) )
45 breq12 4044 . . . 4  |-  ( ( z  =  A  /\  w  =  B )  ->  ( z  <R  w  <->  A 
<R  B ) )
4645copsex2g 4270 . . 3  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( E. z E. w ( <. A ,  B >.  =  <. z ,  w >.  /\  z  <R  w )  <->  A  <R  B ) )
4744, 46bitrd 244 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( <. A ,  0R >. 
<RR  <. B ,  0R >.  <-> 
A  <R  B ) )
486, 8, 47pm5.21nii 342 1  |-  ( <. A ,  0R >.  <RR  <. B ,  0R >. 
<->  A  <R  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039   R.cnr 8505   0Rc0r 8506    <R cltr 8511   RRcr 8752    <RR cltrr 8757
This theorem is referenced by:  ltresr2  8779  axpre-lttri  8803  axpre-lttrn  8804  axpre-ltadd  8805  axpre-mulgt0  8806  axpre-sup  8807
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-ec 6678  df-qs 6682  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-enr 8697  df-nr 8698  df-ltr 8701  df-0r 8702  df-r 8763  df-lt 8766
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