Theorem fzrev3 10283

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 109	. . 3 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> K e. ZZ )
2		elfzel1 10220	. . . 4 |- ( K e. ( M ... N ) -> M e. ZZ )
3	2	adantl 277	. . 3 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> M e. ZZ )
4		elfzel2 10219	. . . 4 |- ( K e. ( M ... N ) -> N e. ZZ )
5	4	adantl 277	. . 3 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> N e. ZZ )
6	1, 3, 5	3jca 1201	. 2 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) )
7		simpl 109	. . 3 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> K e. ZZ )
8		elfzel1 10220	. . . 4 |- ( ( ( M + N ) - K ) e. ( M ... N ) -> M e. ZZ )
9	8	adantl 277	. . 3 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> M e. ZZ )
10		elfzel2 10219	. . . 4 |- ( ( ( M + N ) - K ) e. ( M ... N ) -> N e. ZZ )
11	10	adantl 277	. . 3 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> N e. ZZ )
12	7, 9, 11	3jca 1201	. 2 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) )
13		zcn 9451	. . . . . 6 |- ( M e. ZZ -> M e. CC )
14		zcn 9451	. . . . . 6 |- ( N e. ZZ -> N e. CC )
15		pncan 8352	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - N ) = M )
16		pncan2 8353	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - M ) = N )
17	15, 16	oveq12d 6019	. . . . . 6 |- ( ( M e. CC /\ N e. CC ) -> ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) = ( M ... N ) )
18	13, 14, 17	syl2an 289	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) = ( M ... N ) )
19	18	eleq2d 2299	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> K e. ( M ... N ) ) )
20	19	3adant1 1039	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> K e. ( M ... N ) ) )
21		3simpc 1020	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
22		zaddcl 9486	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
23	22	3adant1 1039	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
24		simp1 1021	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
25		fzrev 10280	. . . 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 1269	. . 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 190	. 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 921	1 |- ( K e. ZZ -> ( K e. ( M ... N ) <-> ( ( M + N ) - K ) e. ( M ... N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200  (class class class)co 6001   CCcc 7997   + caddc 8002   - cmin 8317   ZZcz 9446   ...cfz 10204

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-fz 10205

This theorem is referenced by:  fzrev3i  10284