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Theorem fzmmmeqm 10057
Description: Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)
Assertion
Ref Expression
fzmmmeqm  |-  ( M  e.  ( L ... N )  ->  (
( N  -  L
)  -  ( M  -  L ) )  =  ( N  -  M ) )

Proof of Theorem fzmmmeqm
StepHypRef Expression
1 elfz2 10014 . . 3  |-  ( M  e.  ( L ... N )  <->  ( ( L  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  /\  ( L  <_  M  /\  M  <_  N ) ) )
2 zcn 9257 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
3 zcn 9257 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  CC )
4 zcn 9257 . . . . . 6  |-  ( L  e.  ZZ  ->  L  e.  CC )
52, 3, 43anim123i 1184 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  L  e.  ZZ )  ->  ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC ) )
653comr 1211 . . . 4  |-  ( ( L  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC ) )
76adantr 276 . . 3  |-  ( ( ( L  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  /\  ( L  <_  M  /\  M  <_  N ) )  ->  ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC ) )
81, 7sylbi 121 . 2  |-  ( M  e.  ( L ... N )  ->  ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC ) )
9 nnncan2 8193 . 2  |-  ( ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC )  ->  (
( N  -  L
)  -  ( M  -  L ) )  =  ( N  -  M ) )
108, 9syl 14 1  |-  ( M  e.  ( L ... N )  ->  (
( N  -  L
)  -  ( M  -  L ) )  =  ( N  -  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4003  (class class class)co 5874   CCcc 7808    <_ cle 7992    - cmin 8127   ZZcz 9252   ...cfz 10007
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-sub 8129  df-neg 8130  df-z 9253  df-fz 10008
This theorem is referenced by: (None)
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