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Theorem fzonmapblen 9932
Description: The result of subtracting a nonnegative integer from a positive integer and adding another nonnegative integer which is less than the first one is less then the positive integer. (Contributed by Alexander van der Vekens, 19-May-2018.)
Assertion
Ref Expression
fzonmapblen  |-  ( ( A  e.  ( 0..^ N )  /\  B  e.  ( 0..^ N )  /\  B  <  A
)  ->  ( B  +  ( N  -  A ) )  < 
N )

Proof of Theorem fzonmapblen
StepHypRef Expression
1 elfzo0 9927 . . . 4  |-  ( A  e.  ( 0..^ N )  <->  ( A  e. 
NN0  /\  N  e.  NN  /\  A  <  N
) )
2 nn0re 8954 . . . . . 6  |-  ( A  e.  NN0  ->  A  e.  RR )
3 nnre 8695 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR )
42, 3anim12i 336 . . . . 5  |-  ( ( A  e.  NN0  /\  N  e.  NN )  ->  ( A  e.  RR  /\  N  e.  RR ) )
543adant3 986 . . . 4  |-  ( ( A  e.  NN0  /\  N  e.  NN  /\  A  <  N )  ->  ( A  e.  RR  /\  N  e.  RR ) )
61, 5sylbi 120 . . 3  |-  ( A  e.  ( 0..^ N )  ->  ( A  e.  RR  /\  N  e.  RR ) )
7 elfzoelz 9892 . . . 4  |-  ( B  e.  ( 0..^ N )  ->  B  e.  ZZ )
87zred 9141 . . 3  |-  ( B  e.  ( 0..^ N )  ->  B  e.  RR )
9 simpr 109 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  B  e.  RR )
10 simpll 503 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  A  e.  RR )
11 resubcl 7994 . . . . . . . . 9  |-  ( ( N  e.  RR  /\  A  e.  RR )  ->  ( N  -  A
)  e.  RR )
1211ancoms 266 . . . . . . . 8  |-  ( ( A  e.  RR  /\  N  e.  RR )  ->  ( N  -  A
)  e.  RR )
1312adantr 274 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( N  -  A )  e.  RR )
149, 10, 13ltadd1d 8268 . . . . . 6  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( B  < 
A  <->  ( B  +  ( N  -  A
) )  <  ( A  +  ( N  -  A ) ) ) )
1514biimpa 294 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( B  +  ( N  -  A ) )  < 
( A  +  ( N  -  A ) ) )
16 recn 7721 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
17 recn 7721 . . . . . . . . 9  |-  ( N  e.  RR  ->  N  e.  CC )
1816, 17anim12i 336 . . . . . . . 8  |-  ( ( A  e.  RR  /\  N  e.  RR )  ->  ( A  e.  CC  /\  N  e.  CC ) )
1918adantr 274 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( A  e.  CC  /\  N  e.  CC ) )
2019adantr 274 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( A  e.  CC  /\  N  e.  CC ) )
21 pncan3 7938 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  +  ( N  -  A ) )  =  N )
2220, 21syl 14 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( A  +  ( N  -  A ) )  =  N )
2315, 22breqtrd 3924 . . . 4  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( B  +  ( N  -  A ) )  < 
N )
2423ex 114 . . 3  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( B  < 
A  ->  ( B  +  ( N  -  A ) )  < 
N ) )
256, 8, 24syl2an 287 . 2  |-  ( ( A  e.  ( 0..^ N )  /\  B  e.  ( 0..^ N ) )  ->  ( B  <  A  ->  ( B  +  ( N  -  A ) )  < 
N ) )
26253impia 1163 1  |-  ( ( A  e.  ( 0..^ N )  /\  B  e.  ( 0..^ N )  /\  B  <  A
)  ->  ( B  +  ( N  -  A ) )  < 
N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316    e. wcel 1465   class class class wbr 3899  (class class class)co 5742   CCcc 7586   RRcr 7587   0cc0 7588    + caddc 7591    < clt 7768    - cmin 7901   NNcn 8688   NN0cn0 8945  ..^cfzo 9887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-fz 9759  df-fzo 9888
This theorem is referenced by: (None)
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