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Theorem fzmmmeqm 9204
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 9164 . . 3  |-  ( M  e.  ( L ... N )  <->  ( ( L  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  /\  ( L  <_  M  /\  M  <_  N ) ) )
2 zcn 8489 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
3 zcn 8489 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  CC )
4 zcn 8489 . . . . . 6  |-  ( L  e.  ZZ  ->  L  e.  CC )
52, 3, 43anim123i 1124 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  L  e.  ZZ )  ->  ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC ) )
653comr 1147 . . . 4  |-  ( ( L  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC ) )
76adantr 270 . . 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 119 . 2  |-  ( M  e.  ( L ... N )  ->  ( N  e.  CC  /\  M  e.  CC  /\  L  e.  CC ) )
9 nnncan2 7464 . 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 102    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3805  (class class class)co 5563   CCcc 7093    <_ cle 7268    - cmin 7398   ZZcz 8484   ...cfz 9157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-sub 7400  df-neg 7401  df-z 8485  df-fz 9158
This theorem is referenced by: (None)
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