Theorem fzrev3 10384

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 10321	. . . 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 10320	. . . 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 1204	. 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 10321	. . . 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 10320	. . . 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 1204	. 2 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) )
13		zcn 9545	. . . . . 6 |- ( M e. ZZ -> M e. CC )
14		zcn 9545	. . . . . 6 |- ( N e. ZZ -> N e. CC )
15		pncan 8444	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - N ) = M )
16		pncan2 8445	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - M ) = N )
17	15, 16	oveq12d 6046	. . . . . 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 2301	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> K e. ( M ... N ) ) )
20	19	3adant1 1042	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> K e. ( M ... N ) ) )
21		3simpc 1023	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
22		zaddcl 9580	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
23	22	3adant1 1042	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
24		simp1 1024	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
25		fzrev 10381	. . . 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 1272	. . 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 924	1 |- ( K e. ZZ -> ( K e. ( M ... N ) <-> ( ( M + N ) - K ) e. ( M ... N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202  (class class class)co 6028   CCcc 8090   + caddc 8095   - cmin 8409   ZZcz 9540   ...cfz 10305

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306

This theorem is referenced by:  fzrev3i  10385