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Theorem fzonmapblen 11845
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 than 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 11840 . . . 4  |-  ( A  e.  ( 0..^ N )  <->  ( A  e. 
NN0  /\  N  e.  NN  /\  A  <  N
) )
2 nn0re 10800 . . . . . 6  |-  ( A  e.  NN0  ->  A  e.  RR )
3 nnre 10538 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR )
42, 3anim12i 564 . . . . 5  |-  ( ( A  e.  NN0  /\  N  e.  NN )  ->  ( A  e.  RR  /\  N  e.  RR ) )
543adant3 1014 . . . 4  |-  ( ( A  e.  NN0  /\  N  e.  NN  /\  A  <  N )  ->  ( A  e.  RR  /\  N  e.  RR ) )
61, 5sylbi 195 . . 3  |-  ( A  e.  ( 0..^ N )  ->  ( A  e.  RR  /\  N  e.  RR ) )
7 elfzoelz 11804 . . . 4  |-  ( B  e.  ( 0..^ N )  ->  B  e.  ZZ )
87zred 10965 . . 3  |-  ( B  e.  ( 0..^ N )  ->  B  e.  RR )
9 simpr 459 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  B  e.  RR )
10 simpll 751 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  A  e.  RR )
11 resubcl 9874 . . . . . . . . 9  |-  ( ( N  e.  RR  /\  A  e.  RR )  ->  ( N  -  A
)  e.  RR )
1211ancoms 451 . . . . . . . 8  |-  ( ( A  e.  RR  /\  N  e.  RR )  ->  ( N  -  A
)  e.  RR )
1312adantr 463 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( N  -  A )  e.  RR )
149, 10, 13ltadd1d 10141 . . . . . 6  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( B  < 
A  <->  ( B  +  ( N  -  A
) )  <  ( A  +  ( N  -  A ) ) ) )
1514biimpa 482 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( B  +  ( N  -  A ) )  < 
( A  +  ( N  -  A ) ) )
16 recn 9571 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
17 recn 9571 . . . . . . . . 9  |-  ( N  e.  RR  ->  N  e.  CC )
1816, 17anim12i 564 . . . . . . . 8  |-  ( ( A  e.  RR  /\  N  e.  RR )  ->  ( A  e.  CC  /\  N  e.  CC ) )
1918adantr 463 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( A  e.  CC  /\  N  e.  CC ) )
2019adantr 463 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( A  e.  CC  /\  N  e.  CC ) )
21 pncan3 9819 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  +  ( N  -  A ) )  =  N )
2220, 21syl 16 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( A  +  ( N  -  A ) )  =  N )
2315, 22breqtrd 4463 . . . 4  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( B  +  ( N  -  A ) )  < 
N )
2423ex 432 . . 3  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( B  < 
A  ->  ( B  +  ( N  -  A ) )  < 
N ) )
256, 8, 24syl2an 475 . 2  |-  ( ( A  e.  ( 0..^ N )  /\  B  e.  ( 0..^ N ) )  ->  ( B  <  A  ->  ( B  +  ( N  -  A ) )  < 
N ) )
26253impia 1191 1  |-  ( ( A  e.  ( 0..^ N )  /\  B  e.  ( 0..^ N )  /\  B  <  A
)  ->  ( B  +  ( N  -  A ) )  < 
N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484    < clt 9617    - cmin 9796   NNcn 10531   NN0cn0 10791  ..^cfzo 11799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800
This theorem is referenced by:  cshwshashlem2  14665
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