Theorem fzonmapblen 10189

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 10184	. . . 4 |- ( A e. ( 0..^ N ) <-> ( A e. NN0 /\ N e. NN /\ A < N ) )
2		nn0re 9187	. . . . . 6 |- ( A e. NN0 -> A e. RR )
3		nnre 8928	. . . . . 6 |- ( N e. NN -> N e. RR )
4	2, 3	anim12i 338	. . . . 5 |- ( ( A e. NN0 /\ N e. NN ) -> ( A e. RR /\ N e. RR ) )
5	4	3adant3 1017	. . . 4 |- ( ( A e. NN0 /\ N e. NN /\ A < N ) -> ( A e. RR /\ N e. RR ) )
6	1, 5	sylbi 121	. . 3 |- ( A e. ( 0..^ N ) -> ( A e. RR /\ N e. RR ) )
7		elfzoelz 10149	. . . 4 |- ( B e. ( 0..^ N ) -> B e. ZZ )
8	7	zred 9377	. . 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 8223	. . . . . . . . 9 |- ( ( N e. RR /\ A e. RR ) -> ( N - A ) e. RR )
12	11	ancoms 268	. . . . . . . 8 |- ( ( A e. RR /\ N e. RR ) -> ( N - A ) e. RR )
13	12	adantr 276	. . . . . . 7 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( N - A ) e. RR )
14	9, 10, 13	ltadd1d 8497	. . . . . 6 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( B < A <-> ( B + ( N - A ) ) < ( A + ( N - A ) ) ) )
15	14	biimpa 296	. . . . 5 |- ( ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) /\ B < A ) -> ( B + ( N - A ) ) < ( A + ( N - A ) ) )
16		recn 7946	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
17		recn 7946	. . . . . . . . 9 |- ( N e. RR -> N e. CC )
18	16, 17	anim12i 338	. . . . . . . 8 |- ( ( A e. RR /\ N e. RR ) -> ( A e. CC /\ N e. CC ) )
19	18	adantr 276	. . . . . . 7 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( A e. CC /\ N e. CC ) )
20	19	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) /\ B < A ) -> ( A e. CC /\ N e. CC ) )
21		pncan3 8167	. . . . . 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 4031	. . . 4 |- ( ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) /\ B < A ) -> ( B + ( N - A ) ) < N )
24	23	ex 115	. . 3 |- ( ( ( A e. RR /\ N e. RR ) /\ B e. RR ) -> ( B < A -> ( B + ( N - A ) ) < N ) )
25	6, 8, 24	syl2an 289	. 2 |- ( ( A e. ( 0..^ N ) /\ B e. ( 0..^ N ) ) -> ( B < A -> ( B + ( N - A ) ) < N ) )
26	25	3impia 1200	1 |- ( ( A e. ( 0..^ N ) /\ B e. ( 0..^ N ) /\ B < A ) -> ( B + ( N - A ) ) < N )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   CCcc 7811   RRcr 7812   0cc0 7813   + caddc 7816   < clt 7994   - cmin 8130   NNcn 8921   NN0cn0 9178  ..^cfzo 10144

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011  df-fzo 10145

This theorem is referenced by: (None)