Theorem fzonmapblen 10425

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 10420	. . . 4 |- ( A e. ( 0..^ N ) <-> ( A e. NN0 /\ N e. NN /\ A < N ) )
2		nn0re 9410	. . . . . 6 |- ( A e. NN0 -> A e. RR )
3		nnre 9149	. . . . . 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 1043	. . . 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 10381	. . . 4 |- ( B e. ( 0..^ N ) -> B e. ZZ )
8	7	zred 9601	. . 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 8442	. . . . . . . . 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 8717	. . . . . 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 8164	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
17		recn 8164	. . . . . . . . 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 8386	. . . . . 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 4114	. . . 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 1226	1 |- ( ( A e. ( 0..^ N ) /\ B e. ( 0..^ N ) /\ B < A ) -> ( B + ( N - A ) ) < N )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   + caddc 8034   < clt 8213   - cmin 8349   NNcn 9142   NN0cn0 9401  ..^cfzo 10376

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377

This theorem is referenced by: (None)