Theorem reapneg 8495

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 8486	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
2		ltneg 8360	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) )
3		ltneg 8360	. . . . . 6 |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> -u A < -u B ) )
4	3	ancoms 266	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( B < A <-> -u A < -u B ) )
5	2, 4	orbi12d 783	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ B < A ) <-> ( -u B < -u A \/ -u A < -u B ) ) )
6	1, 5	bitrd 187	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( -u B < -u A \/ -u A < -u B ) ) )
7		orcom 718	. . 3 |- ( ( -u B < -u A \/ -u A < -u B ) <-> ( -u A < -u B \/ -u B < -u A ) )
8	6, 7	bitrdi 195	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( -u A < -u B \/ -u B < -u A ) ) )
9		simpl 108	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
10	9	renegcld 8278	. . 3 |- ( ( A e. RR /\ B e. RR ) -> -u A e. RR )
11		simpr 109	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
12	11	renegcld 8278	. . 3 |- ( ( A e. RR /\ B e. RR ) -> -u B e. RR )
13		reaplt 8486	. . 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 409	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( -u A # -u B <-> ( -u A < -u B \/ -u B < -u A ) ) )
15	8, 14	bitr4d 190	1 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> -u A # -u B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   e. wcel 2136   class class class wbr 3982   RRcr 7752   < clt 7933   -ucneg 8070   # cap 8479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-ltxr 7938  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480

This theorem is referenced by:  apneg  8509