Theorem reapneg 8859

Description: Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)

Assertion

Ref	Expression
reapneg	|- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> -u A # -u B ) )

Proof of Theorem reapneg

Step	Hyp	Ref	Expression
1		reaplt 8850	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
2		ltneg 8724	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) )
3		ltneg 8724	. . . . . 6 |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> -u A < -u B ) )
4	3	ancoms 268	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( B < A <-> -u A < -u B ) )
5	2, 4	orbi12d 801	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ B < A ) <-> ( -u B < -u A \/ -u A < -u B ) ) )
6	1, 5	bitrd 188	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( -u B < -u A \/ -u A < -u B ) ) )
7		orcom 736	. . 3 |- ( ( -u B < -u A \/ -u A < -u B ) <-> ( -u A < -u B \/ -u B < -u A ) )
8	6, 7	bitrdi 196	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( -u A < -u B \/ -u B < -u A ) ) )
9		simpl 109	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
10	9	renegcld 8641	. . 3 |- ( ( A e. RR /\ B e. RR ) -> -u A e. RR )
11		simpr 110	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
12	11	renegcld 8641	. . 3 |- ( ( A e. RR /\ B e. RR ) -> -u B e. RR )
13		reaplt 8850	. . 3 |- ( ( -u A e. RR /\ -u B e. RR ) -> ( -u A # -u B <-> ( -u A < -u B \/ -u B < -u A ) ) )
14	10, 12, 13	syl2anc 411	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( -u A # -u B <-> ( -u A < -u B \/ -u B < -u A ) ) )
15	8, 14	bitr4d 191	1 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> -u A # -u B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   e. wcel 2203   class class class wbr 4102   RRcr 8114   < clt 8296   -ucneg 8433   # cap 8843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650  ax-cnex 8206  ax-resscn 8207  ax-1cn 8208  ax-1re 8209  ax-icn 8210  ax-addcl 8211  ax-addrcl 8212  ax-mulcl 8213  ax-mulrcl 8214  ax-addcom 8215  ax-mulcom 8216  ax-addass 8217  ax-mulass 8218  ax-distr 8219  ax-i2m1 8220  ax-0lt1 8221  ax-1rid 8222  ax-0id 8223  ax-rnegex 8224  ax-precex 8225  ax-cnre 8226  ax-pre-ltirr 8227  ax-pre-lttrn 8229  ax-pre-apti 8230  ax-pre-ltadd 8231  ax-pre-mulgt0 8232

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-br 4103  df-opab 4165  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-iota 5303  df-fun 5345  df-fv 5351  df-riota 5994  df-ov 6044  df-oprab 6045  df-mpo 6046  df-pnf 8298  df-mnf 8299  df-ltxr 8301  df-sub 8434  df-neg 8435  df-reap 8837  df-ap 8844

This theorem is referenced by:  apneg  8873