Theorem ltsubadd 8654

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 1024	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2		simp2 1025	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
3	1, 2	resubcld 8602	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A - B ) e. RR )
4		simp3 1026	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
5		ltadd1 8651	. . 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 1274	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> ( ( A - B ) + B ) < ( C + B ) ) )
7	1	recnd 8250	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
8	2	recnd 8250	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
9	7, 8	npcand 8536	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) + B ) = A )
10	9	breq1d 4103	. 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 1005   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   RRcr 8074   + caddc 8078   < clt 8256   - cmin 8392

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-sub 8394  df-neg 8395

This theorem is referenced by:  ltsubadd2  8655  ltsub23  8664  ltsubpos  8676  ltsub1  8680  ltsub2  8681  ltsubaddi  8727  ltsubaddd  8763  nnsub  9224  iooshf  10231  sincosq2sgn  15621  sincosq3sgn  15622  sincosq4sgn  15623