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Theorem fzonmapblen 10333
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 10328 . . . 4 |- ( A e. ( 0..^ N ) <-> ( A e. NN0 /\ N e. NN /\ A < N ) )
2 nn0re 9324 . . . . . 6 |- ( A e. NN0 -> A e. RR )
3 nnre 9063 . . . . . 6 |- ( N e. NN -> N e. RR )
42, 3anim12i 338 . . . . 5 |- ( ( A e. NN0 /\ N e. NN ) -> ( A e. RR /\ N e. RR ) )
543adant3 1020 . . . 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 10289 . . . 4 |- ( B e. ( 0..^ N ) -> B e. ZZ )
87zred 9515 . . 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 527 . . . . . . 7 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> A e. RR )
11 resubcl 8356 . . . . . . . . 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 8631 . . . . . 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 8078 . . . . . . . . 9 |- ( A e. RR -> A e. CC )
17 recn 8078 . . . . . . . . 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 8300 . . . . . 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 4077 . . . 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 1203 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 104   /\ w3a 981   = wceq 1373   e. wcel 2177   class class class wbr 4051  (class class class)co 5957   CCcc 7943   RRcr 7944   0cc0 7945   + caddc 7948   < clt 8127   - cmin 8263   NNcn 9056   NN0cn0 9315  ..^cfzo 10284
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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-ltadd 8061
This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-inn 9057  df-n0 9316  df-z 9393  df-uz 9669  df-fz 10151  df-fzo 10285
This theorem is referenced by: (None)
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