Theorem ltsubadd 8535

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 1000	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2		simp2 1001	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
3	1, 2	resubcld 8483	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A - B ) e. RR )
4		simp3 1002	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
5		ltadd1 8532	. . 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 1250	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> ( ( A - B ) + B ) < ( C + B ) ) )
7	1	recnd 8131	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
8	2	recnd 8131	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
9	7, 8	npcand 8417	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) + B ) = A )
10	9	breq1d 4064	. 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 981   e. wcel 2177   class class class wbr 4054  (class class class)co 5962   RRcr 7954   + caddc 7958   < clt 8137   - cmin 8273

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-distr 8059  ax-i2m1 8060  ax-0id 8063  ax-rnegex 8064  ax-cnre 8066  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-iota 5246  df-fun 5287  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-pnf 8139  df-mnf 8140  df-ltxr 8142  df-sub 8275  df-neg 8276

This theorem is referenced by:  ltsubadd2  8536  ltsub23  8545  ltsubpos  8557  ltsub1  8561  ltsub2  8562  ltsubaddi  8608  ltsubaddd  8644  nnsub  9105  iooshf  10104  sincosq2sgn  15384  sincosq3sgn  15385  sincosq4sgn  15386