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Theorem fzmmmeqm 10061
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 10018 . . 3  |-  ( M  e.  ( L ... N )  <->  ( ( L  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  /\  ( L  <_  M  /\  M  <_  N ) ) )
2 zcn 9261 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
3 zcn 9261 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  CC )
4 zcn 9261 . . . . . 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 8197 . 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 4005  (class class class)co 5878   CCcc 7812    <_ cle 7996    - cmin 8131   ZZcz 9256   ...cfz 10011
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 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-distr 7918  ax-i2m1 7919  ax-0id 7922  ax-rnegex 7923  ax-cnre 7925
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 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-sub 8133  df-neg 8134  df-z 9257  df-fz 10012
This theorem is referenced by: (None)
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