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Theorem ltresr2 9561
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
ltresr2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  ( 1st `  A )  <R  ( 1st `  B ) ) )

Proof of Theorem ltresr2
StepHypRef Expression
1 elreal2 9552 . . . 4  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
21simprbi 465 . . 3  |-  ( A  e.  RR  ->  A  =  <. ( 1st `  A
) ,  0R >. )
3 elreal2 9552 . . . 4  |-  ( B  e.  RR  <->  ( ( 1st `  B )  e. 
R.  /\  B  =  <. ( 1st `  B
) ,  0R >. ) )
43simprbi 465 . . 3  |-  ( B  e.  RR  ->  B  =  <. ( 1st `  B
) ,  0R >. )
52, 4breqan12d 4433 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  <. ( 1st `  A ) ,  0R >. 
<RR  <. ( 1st `  B
) ,  0R >. ) )
6 ltresr 9560 . 2  |-  ( <.
( 1st `  A
) ,  0R >.  <RR  <. ( 1st `  B
) ,  0R >.  <->  ( 1st `  A )  <R 
( 1st `  B
) )
75, 6syl6bb 264 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  ( 1st `  A )  <R  ( 1st `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   <.cop 3999   class class class wbr 4417   ` cfv 5593   1stc1st 6797   R.cnr 9286   0Rc0r 9287    <R cltr 9292   RRcr 9534    <RR cltrr 9539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-inf2 8144
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-tr 4513  df-eprel 4757  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-pred 5391  df-ord 5437  df-on 5438  df-lim 5439  df-suc 5440  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-om 6699  df-1st 6799  df-2nd 6800  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-omul 7187  df-er 7363  df-ec 7365  df-qs 7369  df-ni 9293  df-pli 9294  df-mi 9295  df-lti 9296  df-plpq 9329  df-mpq 9330  df-ltpq 9331  df-enq 9332  df-nq 9333  df-erq 9334  df-plq 9335  df-mq 9336  df-1nq 9337  df-rq 9338  df-ltnq 9339  df-np 9402  df-1p 9403  df-enr 9476  df-nr 9477  df-ltr 9480  df-0r 9481  df-r 9545  df-lt 9548
This theorem is referenced by:  axpre-sup  9589
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