Theorem ubmelm1fzo 10644

Description: The result of subtracting 1 and an integer of a half-open range of nonnegative integers from the upper bound of this range is contained in this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 30-Oct-2018.)

Assertion

Ref	Expression
ubmelm1fzo	|- ( K e. ( 0..^ N ) -> ( ( N - K ) - 1 ) e. ( 0..^ N ) )

Proof of Theorem ubmelm1fzo

Step	Hyp	Ref	Expression
1		elfzo0 10593	. 2 |- ( K e. ( 0..^ N ) <-> ( K e. NN0 /\ N e. NN /\ K < N ) )
2		nnz 9663	. . . . . . . . 9 |- ( N e. NN -> N e. ZZ )
3	2	adantr 276	. . . . . . . 8 |- ( ( N e. NN /\ K e. NN0 ) -> N e. ZZ )
4		nn0z 9664	. . . . . . . . 9 |- ( K e. NN0 -> K e. ZZ )
5	4	adantl 277	. . . . . . . 8 |- ( ( N e. NN /\ K e. NN0 ) -> K e. ZZ )
6	3, 5	zsubcld 9773	. . . . . . 7 |- ( ( N e. NN /\ K e. NN0 ) -> ( N - K ) e. ZZ )
7	6	ancoms 268	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> ( N - K ) e. ZZ )
8		peano2zm 9682	. . . . . 6 |- ( ( N - K ) e. ZZ -> ( ( N - K ) - 1 ) e. ZZ )
9	7, 8	syl 14	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) e. ZZ )
10	9	3adant3 1048	. . . 4 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. ZZ )
11		simp3 1030	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> K < N )
12	4, 2	anim12i 338	. . . . . . . 8 |- ( ( K e. NN0 /\ N e. NN ) -> ( K e. ZZ /\ N e. ZZ ) )
13	12	3adant3 1048	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( K e. ZZ /\ N e. ZZ ) )
14		znnsub 9696	. . . . . . 7 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K < N <-> ( N - K ) e. NN ) )
15	13, 14	syl 14	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( K < N <-> ( N - K ) e. NN ) )
16	11, 15	mpbid 147	. . . . 5 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( N - K ) e. NN )
17		nnm1ge0 9732	. . . . 5 |- ( ( N - K ) e. NN -> 0 <_ ( ( N - K ) - 1 ) )
18	16, 17	syl 14	. . . 4 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> 0 <_ ( ( N - K ) - 1 ) )
19		elnn0z 9657	. . . 4 |- ( ( ( N - K ) - 1 ) e. NN0 <-> ( ( ( N - K ) - 1 ) e. ZZ /\ 0 <_ ( ( N - K ) - 1 ) ) )
20	10, 18, 19	sylanbrc 421	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. NN0 )
21		simp2 1029	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> N e. NN )
22		nncn 9312	. . . . . . 7 |- ( N e. NN -> N e. CC )
23	22	adantl 277	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> N e. CC )
24		nn0cn 9573	. . . . . . 7 |- ( K e. NN0 -> K e. CC )
25	24	adantr 276	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> K e. CC )
26		1cnd 8342	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> 1 e. CC )
27	23, 25, 26	subsub4d 8668	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) = ( N - ( K + 1 ) ) )
28		nn0p1gt0 9592	. . . . . . 7 |- ( K e. NN0 -> 0 < ( K + 1 ) )
29	28	adantr 276	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> 0 < ( K + 1 ) )
30		nn0re 9572	. . . . . . . 8 |- ( K e. NN0 -> K e. RR )
31		peano2re 8462	. . . . . . . 8 |- ( K e. RR -> ( K + 1 ) e. RR )
32	30, 31	syl 14	. . . . . . 7 |- ( K e. NN0 -> ( K + 1 ) e. RR )
33		nnre 9311	. . . . . . 7 |- ( N e. NN -> N e. RR )
34		ltsubpos 8782	. . . . . . 7 |- ( ( ( K + 1 ) e. RR /\ N e. RR ) -> ( 0 < ( K + 1 ) <-> ( N - ( K + 1 ) ) < N ) )
35	32, 33, 34	syl2an 289	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> ( 0 < ( K + 1 ) <-> ( N - ( K + 1 ) ) < N ) )
36	29, 35	mpbid 147	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( N - ( K + 1 ) ) < N )
37	27, 36	eqbrtrd 4152	. . . 4 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) < N )
38	37	3adant3 1048	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) < N )
39		elfzo0 10593	. . 3 |- ( ( ( N - K ) - 1 ) e. ( 0..^ N ) <-> ( ( ( N - K ) - 1 ) e. NN0 /\ N e. NN /\ ( ( N - K ) - 1 ) < N ) )
40	20, 21, 38, 39	syl3anbrc 1212	. 2 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. ( 0..^ N ) )
41	1, 40	sylbi 121	1 |- ( K e. ( 0..^ N ) -> ( ( N - K ) - 1 ) e. ( 0..^ N ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8497   NNcn 9304   NN0cn0 9563   ZZcz 9644  ..^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)