Theorem fzonmapblen 9964

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

Step	Hyp	Ref	Expression
1		elfzo0 9959	. . . 4 |- ( A e. ( 0..^ N ) <-> ( A e. NN0 /\ N e. NN /\ A < N ) )
2		nn0re 8986	. . . . . 6 |- ( A e. NN0 -> A e. RR )
3		nnre 8727	. . . . . 6 |- ( N e. NN -> N e. RR )
4	2, 3	anim12i 336	. . . . 5 |- ( ( A e. NN0 /\ N e. NN ) -> ( A e. RR /\ N e. RR ) )
5	4	3adant3 1001	. . . 4 |- ( ( A e. NN0 /\ N e. NN /\ A < N ) -> ( A e. RR /\ N e. RR ) )
6	1, 5	sylbi 120	. . 3 |- ( A e. ( 0..^ N ) -> ( A e. RR /\ N e. RR ) )
7		elfzoelz 9924	. . . 4 |- ( B e. ( 0..^ N ) -> B e. ZZ )
8	7	zred 9173	. . 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 518	. . . . . . 7 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> A e. RR )
11		resubcl 8026	. . . . . . . . 9 |- ( ( N e. RR /\ A e. RR ) -> ( N - A ) e. RR )
12	11	ancoms 266	. . . . . . . 8 |- ( ( A e. RR /\ N e. RR ) -> ( N - A ) e. RR )
13	12	adantr 274	. . . . . . 7 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( N - A ) e. RR )
14	9, 10, 13	ltadd1d 8300	. . . . . 6 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( B < A <-> ( B + ( N - A ) ) < ( A + ( N - A ) ) ) )
15	14	biimpa 294	. . . . 5 |- ( ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) /\ B < A ) -> ( B + ( N - A ) ) < ( A + ( N - A ) ) )
16		recn 7753	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
17		recn 7753	. . . . . . . . 9 |- ( N e. RR -> N e. CC )
18	16, 17	anim12i 336	. . . . . . . 8 |- ( ( A e. RR /\ N e. RR ) -> ( A e. CC /\ N e. CC ) )
19	18	adantr 274	. . . . . . 7 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( A e. CC /\ N e. CC ) )
20	19	adantr 274	. . . . . 6 |- ( ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) /\ B < A ) -> ( A e. CC /\ N e. CC ) )
21		pncan3 7970	. . . . . 6 |- ( ( A e. CC /\ N e. CC ) -> ( A + ( N - A ) ) = N )
22	20, 21	syl 14	. . . . 5 |- ( ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) /\ B < A ) -> ( A + ( N - A ) ) = N )
23	15, 22	breqtrd 3954	. . . 4 |- ( ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) /\ B < A ) -> ( B + ( N - A ) ) < N )
24	23	ex 114	. . 3 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( B < A -> ( B + ( N - A ) ) < N ) )
25	6, 8, 24	syl2an 287	. 2 |- ( ( A e. ( 0..^ N ) /\ B e. ( 0..^ N ) ) -> ( B < A -> ( B + ( N - A ) ) < N ) )
26	25	3impia 1178	1 |- ( ( A e. ( 0..^ N ) /\ B e. ( 0..^ N ) /\ B < A ) -> ( B + ( N - A ) ) < N )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   + caddc 7623   < clt 7800   - cmin 7933   NNcn 8720   NN0cn0 8977  ..^cfzo 9919

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-fz 9791  df-fzo 9920

This theorem is referenced by: (None)