Theorem fzrev3 9104

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 107	. . 3 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> K e. ZZ )
2		elfzel1 9044	. . . 4 |- ( K e. ( M ... N ) -> M e. ZZ )
3	2	adantl 271	. . 3 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> M e. ZZ )
4		elfzel2 9043	. . . 4 |- ( K e. ( M ... N ) -> N e. ZZ )
5	4	adantl 271	. . 3 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> N e. ZZ )
6	1, 3, 5	3jca 1118	. 2 |- ( ( K e. ZZ /\ K e. ( M ... N ) ) -> ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) )
7		simpl 107	. . 3 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> K e. ZZ )
8		elfzel1 9044	. . . 4 |- ( ( ( M + N ) - K ) e. ( M ... N ) -> M e. ZZ )
9	8	adantl 271	. . 3 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> M e. ZZ )
10		elfzel2 9043	. . . 4 |- ( ( ( M + N ) - K ) e. ( M ... N ) -> N e. ZZ )
11	10	adantl 271	. . 3 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> N e. ZZ )
12	7, 9, 11	3jca 1118	. 2 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) )
13		zcn 8356	. . . . . 6 |- ( M e. ZZ -> M e. CC )
14		zcn 8356	. . . . . 6 |- ( N e. ZZ -> N e. CC )
15		pncan 7314	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - N ) = M )
16		pncan2 7315	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - M ) = N )
17	15, 16	oveq12d 5550	. . . . . 6 |- ( ( M e. CC /\ N e. CC ) -> ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) = ( M ... N ) )
18	13, 14, 17	syl2an 283	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) = ( M ... N ) )
19	18	eleq2d 2148	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> K e. ( M ... N ) ) )
20	19	3adant1 956	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> K e. ( M ... N ) ) )
21		3simpc 937	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
22		zaddcl 8391	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
23	22	3adant1 956	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
24		simp1 938	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
25		fzrev 9101	. . . 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 1167	. . 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 188	. 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 858	1 |- ( K e. ZZ -> ( K e. ( M ... N ) <-> ( ( M + N ) - K ) e. ( M ... N ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433  (class class class)co 5532   CCcc 6979   + caddc 6984   - cmin 7279   ZZcz 8351   ...cfz 9029

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030

This theorem is referenced by:  fzrev3i  9105