Theorem fzrev3 10244

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 10181	. . . 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 10180	. . . 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 1180	. 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 10181	. . . 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 10180	. . . 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 1180	. 2 |- ( ( K e. ZZ /\ ( ( M + N ) - K ) e. ( M ... N ) ) -> ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) )
13		zcn 9412	. . . . . 6 |- ( M e. ZZ -> M e. CC )
14		zcn 9412	. . . . . 6 |- ( N e. ZZ -> N e. CC )
15		pncan 8313	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - N ) = M )
16		pncan2 8314	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - M ) = N )
17	15, 16	oveq12d 5985	. . . . . 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 2277	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> K e. ( M ... N ) ) )
20	19	3adant1 1018	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( ( ( M + N ) - N ) ... ( ( M + N ) - M ) ) <-> K e. ( M ... N ) ) )
21		3simpc 999	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
22		zaddcl 9447	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
23	22	3adant1 1018	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
24		simp1 1000	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
25		fzrev 10241	. . . 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 1248	. . 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 918	1 |- ( K e. ZZ -> ( K e. ( M ... N ) <-> ( ( M + N ) - K ) e. ( M ... N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178  (class class class)co 5967   CCcc 7958   + caddc 7963   - cmin 8278   ZZcz 9407   ...cfz 10165

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166

This theorem is referenced by:  fzrev3i  10245