Theorem ltsubadd 8590

Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
ltsubadd	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> A < ( C + B ) ) )

Proof of Theorem ltsubadd

Step	Hyp	Ref	Expression
1		simp1 1021	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2		simp2 1022	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
3	1, 2	resubcld 8538	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A - B ) e. RR )
4		simp3 1023	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
5		ltadd1 8587	. . 3 |- ( ( ( A - B ) e. RR /\ C e. RR /\ B e. RR ) -> ( ( A - B ) < C <-> ( ( A - B ) + B ) < ( C + B ) ) )
6	3, 4, 2, 5	syl3anc 1271	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> ( ( A - B ) + B ) < ( C + B ) ) )
7	1	recnd 8186	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
8	2	recnd 8186	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
9	7, 8	npcand 8472	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) + B ) = A )
10	9	breq1d 4093	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( A - B ) + B ) < ( C + B ) <-> A < ( C + B ) ) )
11	6, 10	bitrd 188	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> A < ( C + B ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   e. wcel 2200   class class class wbr 4083  (class class class)co 6007   RRcr 8009   + caddc 8013   < clt 8192   - cmin 8328

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-sub 8330  df-neg 8331

This theorem is referenced by:  ltsubadd2  8591  ltsub23  8600  ltsubpos  8612  ltsub1  8616  ltsub2  8617  ltsubaddi  8663  ltsubaddd  8699  nnsub  9160  iooshf  10160  sincosq2sgn  15516  sincosq3sgn  15517  sincosq4sgn  15518