Theorem ltresr2 9962

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

Step	Hyp	Ref	Expression
1		elreal2 9953	. . . 4 |- ( A e. RR <-> ( ( 1st ` A ) e. R. /\ A = <. ( 1st ` A ) , 0R >. ) )
2	1	simprbi 480	. . 3 |- ( A e. RR -> A = <. ( 1st ` A ) , 0R >. )
3		elreal2 9953	. . . 4 |- ( B e. RR <-> ( ( 1st ` B ) e. R. /\ B = <. ( 1st ` B ) , 0R >. ) )
4	3	simprbi 480	. . 3 |- ( B e. RR -> B = <. ( 1st ` B ) , 0R >. )
5	2, 4	breqan12d 4669	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A <RR B <-> <. ( 1st ` A ) , 0R >. <RR <. ( 1st ` B ) , 0R >. ) )
6		ltresr 9961	. 2 |- ( <. ( 1st ` A ) , 0R >. <RR <. ( 1st ` B ) , 0R >. <-> ( 1st ` A ) <R ( 1st ` B ) )
7	5, 6	syl6bb 276	1 |- ( ( A e. RR /\ B e. RR ) -> ( A <RR B <-> ( 1st ` A ) <R ( 1st ` B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888   1stc1st 7166   R.cnr 9687   0Rc0r 9688   <R cltr 9693   RRcr 9935   <RR cltrr 9940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-1p 9804  df-enr 9877  df-nr 9878  df-ltr 9881  df-0r 9882  df-r 9946  df-lt 9949

This theorem is referenced by:  axpre-sup  9990