Theorem fzrev3 9555

Description: The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Assertion

Ref	Expression
fzrev3	|- ( K e. ZZ -> ( K e. ( M ... N ) <-> ( ( M + N ) - K ) e. ( M ... N ) ) )

Proof of Theorem fzrev3

Step	Hyp	Ref	Expression
1		simpl 108	. . 3 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> K e. ZZ )
2		elfzel1 9493	. . . 4 |- ( K e. ( M ... N ) -> M e. ZZ )
3	2	adantl 272	. . 3 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> M e. ZZ )
4		elfzel2 9492	. . . 4 |- ( K e. ( M ... N ) -> N e. ZZ )
5	4	adantl 272	. . 3 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> N e. ZZ )
6	1, 3, 5	3jca 1124	. 2 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) )
7		simpl 108	. . 3 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> K e. ZZ )
8		elfzel1 9493	. . . 4 |- ( ( ( M + N ) - K ) e. ( M ... N ) -> M e. ZZ )
9	8	adantl 272	. . 3 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> M e. ZZ )
10		elfzel2 9492	. . . 4 |- ( ( ( M + N ) - K ) e. ( M ... N ) -> N e. ZZ )
11	10	adantl 272	. . 3 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> N e. ZZ )
12	7, 9, 11	3jca 1124	. 2 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) )
13		zcn 8809	. . . . . 6 |- ( M e. ZZ -> M e. CC )
14		zcn 8809	. . . . . 6 |- ( N e. ZZ -> N e. CC )
15		pncan 7742	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - N ) = M )
16		pncan2 7743	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - M ) = N )
17	15, 16	oveq12d 5684	. . . . . 6 |- ( ( M e. CC /\ N e. CC ) -> ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) = ( M ... N ) )
18	13, 14, 17	syl2an 284	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) = ( M ... N ) )
19	18	eleq2d 2158	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> K e. ( M ... N ) ) )
20	19	3adant1 962	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> K e. ( M ... N ) ) )
21		3simpc 943	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
22		zaddcl 8844	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
23	22	3adant1 962	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
24		simp1 944	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
25		fzrev 9552	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( ( M + N ) e. ZZ /\ K e. ZZ ) ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> ( ( M + N ) - K ) e. ( M ... N ) ) )
26	21, 23, 24, 25	syl12anc 1173	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> ( ( M + N ) - K ) e. ( M ... N ) ) )
27	20, 26	bitr3d 189	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... N ) <-> ( ( M + N ) - K ) e. ( M ... N ) ) )
28	6, 12, 27	pm5.21nd 864	1 |- ( K e. ZZ -> ( K e. ( M ... N ) <-> ( ( M + N ) - K ) e. ( M ... N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 925   = wceq 1290   e. wcel 1439  (class class class)co 5666   CCcc 7402   + caddc 7407   - cmin 7707   ZZcz 8804   ...cfz 9478

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-ltadd 7515

This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-inn 8477  df-n0 8728  df-z 8805  df-uz 9074  df-fz 9479

This theorem is referenced by:  fzrev3i  9556