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Theorem ltsubsubaddltsub 30616
Description: If the result of subtracting two numbers is greater than a number, the result of adding one of these subtracted numbers to the number is less than the result of subtracting the other subtracted number only. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
Assertion
Ref Expression
ltsubsubaddltsub  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  ( J  <  ( ( L  -  M )  -  N
)  <->  ( J  +  M )  <  ( L  -  N )
) )

Proof of Theorem ltsubsubaddltsub
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  J  e.  RR )
2 resubcl 9785 . . . . . 6  |-  ( ( L  e.  RR  /\  M  e.  RR )  ->  ( L  -  M
)  e.  RR )
323adant3 1008 . . . . 5  |-  ( ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )  ->  ( L  -  M )  e.  RR )
4 simp3 990 . . . . 5  |-  ( ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )  ->  N  e.  RR )
53, 4resubcld 9888 . . . 4  |-  ( ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )  ->  (
( L  -  M
)  -  N )  e.  RR )
65adantl 466 . . 3  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  ( ( L  -  M )  -  N )  e.  RR )
7 simpr2 995 . . 3  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  M  e.  RR )
81, 6, 7ltadd1d 10044 . 2  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  ( J  <  ( ( L  -  M )  -  N
)  <->  ( J  +  M )  <  (
( ( L  -  M )  -  N
)  +  M ) ) )
9 recn 9484 . . . . 5  |-  ( L  e.  RR  ->  L  e.  CC )
10 recn 9484 . . . . 5  |-  ( M  e.  RR  ->  M  e.  CC )
11 recn 9484 . . . . 5  |-  ( N  e.  RR  ->  N  e.  CC )
12 nnpcan 9744 . . . . 5  |-  ( ( L  e.  CC  /\  M  e.  CC  /\  N  e.  CC )  ->  (
( ( L  -  M )  -  N
)  +  M )  =  ( L  -  N ) )
139, 10, 11, 12syl3an 1261 . . . 4  |-  ( ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )  ->  (
( ( L  -  M )  -  N
)  +  M )  =  ( L  -  N ) )
1413adantl 466 . . 3  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  ( (
( L  -  M
)  -  N )  +  M )  =  ( L  -  N
) )
1514breq2d 4413 . 2  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  ( ( J  +  M )  <  ( ( ( L  -  M )  -  N )  +  M
)  <->  ( J  +  M )  <  ( L  -  N )
) )
168, 15bitrd 253 1  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  ( J  <  ( ( L  -  M )  -  N
)  <->  ( J  +  M )  <  ( L  -  N )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4401  (class class class)co 6201   CCcc 9392   RRcr 9393    + caddc 9397    < clt 9530    - cmin 9707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-ltxr 9535  df-sub 9709  df-neg 9710
This theorem is referenced by: (None)
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