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Theorem fzonmapblen 10599
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 10593 . . . 4 |- ( A e. ( 0..^ N ) <-> ( A e. NN0 /\ N e. NN /\ A < N ) )
2 nn0re 9572 . . . . . 6 |- ( A e. NN0 -> A e. RR )
3 nnre 9311 . . . . . 6 |- ( N e. NN -> N e. RR )
42, 3anim12i 338 . . . . 5 |- ( ( A e. NN0 /\ N e. NN ) -> ( A e. RR /\ N e. RR ) )
543adant3 1048 . . . 4 |- ( ( A e. NN0 /\ N e. NN /\ A < N ) -> ( A e. RR /\ N e. RR ) )
61, 5sylbi 121 . . 3 |- ( A e. ( 0..^ N ) -> ( A e. RR /\ N e. RR ) )
7 elfzoelz 10554 . . . 4 |- ( B e. ( 0..^ N ) -> B e. ZZ )
87zred 9768 . . 3 |- ( B e. ( 0..^ N ) -> B e. RR )
9 simpr 110 . . . . . . 7 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> B e. RR )
10 simpll 531 . . . . . . 7 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> A e. RR )
11 resubcl 8590 . . . . . . . . 9 |- ( ( N e. RR /\ A e. RR ) -> ( N - A ) e. RR )
1211ancoms 268 . . . . . . . 8 |- ( ( A e. RR /\ N e. RR ) -> ( N - A ) e. RR )
1312adantr 276 . . . . . . 7 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( N - A ) e. RR )
149, 10, 13ltadd1d 8866 . . . . . 6 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( B < A <-> ( B + ( N - A ) ) < ( A + ( N - A ) ) ) )
1514biimpa 296 . . . . 5 |- ( ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) /\ B < A ) -> ( B + ( N - A ) ) < ( A + ( N - A ) ) )
16 recn 8312 . . . . . . . . 9 |- ( A e. RR -> A e. CC )
17 recn 8312 . . . . . . . . 9 |- ( N e. RR -> N e. CC )
1816, 17anim12i 338 . . . . . . . 8 |- ( ( A e. RR /\ N e. RR ) -> ( A e. CC /\ N e. CC ) )
1918adantr 276 . . . . . . 7 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( A e. CC /\ N e. CC ) )
2019adantr 276 . . . . . 6 |- ( ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) /\ B < A ) -> ( A e. CC /\ N e. CC ) )
21 pncan3 8534 . . . . . 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 4156 . . . 4 |- ( ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) /\ B < A ) -> ( B + ( N - A ) ) < N )
2423ex 115 . . 3 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( B < A -> ( B + ( N - A ) ) < N ) )
256, 8, 24syl2an 289 . 2 |- ( ( A e. ( 0..^ N ) /\ B e. ( 0..^ N ) ) -> ( B < A -> ( B + ( N - A ) ) < N ) )
26253impia 1231 1 |- ( ( A e. ( 0..^ N ) /\ B e. ( 0..^ N ) /\ B < A ) -> ( B + ( N - A ) ) < N )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   + caddc 8182   < clt 8360   - cmin 8497   NNcn 9304   NN0cn0 9563  ..^cfzo 10549
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295
This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550
This theorem is used by: (None)
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