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Theorem ltsubsubaddltsub 32684
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 455 . . 3  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  J  e.  RR )
2 resubcl 9818 . . . . . 6  |-  ( ( L  e.  RR  /\  M  e.  RR )  ->  ( L  -  M
)  e.  RR )
323adant3 1014 . . . . 5  |-  ( ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )  ->  ( L  -  M )  e.  RR )
4 simp3 996 . . . . 5  |-  ( ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )  ->  N  e.  RR )
53, 4resubcld 9927 . . . 4  |-  ( ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )  ->  (
( L  -  M
)  -  N )  e.  RR )
65adantl 464 . . 3  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  ( ( L  -  M )  -  N )  e.  RR )
7 simpr2 1001 . . 3  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  M  e.  RR )
81, 6, 7ltadd1d 10084 . 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 9515 . . . . 5  |-  ( L  e.  RR  ->  L  e.  CC )
10 recn 9515 . . . . 5  |-  ( M  e.  RR  ->  M  e.  CC )
11 recn 9515 . . . . 5  |-  ( N  e.  RR  ->  N  e.  CC )
12 nnpcan 9777 . . . . 5  |-  ( ( L  e.  CC  /\  M  e.  CC  /\  N  e.  CC )  ->  (
( ( L  -  M )  -  N
)  +  M )  =  ( L  -  N ) )
139, 10, 11, 12syl3an 1268 . . . 4  |-  ( ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )  ->  (
( ( L  -  M )  -  N
)  +  M )  =  ( L  -  N ) )
1413adantl 464 . . 3  |-  ( ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
)  ->  ( (
( L  -  M
)  -  N )  +  M )  =  ( L  -  N
) )
1514breq2d 4396 . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1836   class class class wbr 4384  (class class class)co 6218   CCcc 9423   RRcr 9424    + caddc 9428    < clt 9561    - cmin 9740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-po 4731  df-so 4732  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-pnf 9563  df-mnf 9564  df-ltxr 9566  df-sub 9742  df-neg 9743
This theorem is referenced by: (None)
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