Theorem fzrevral2 9123

Description: Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Assertion

Ref	Expression
fzrevral2	|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( A. j e. ( ( K - N ) ... ( K - M ) ) ph <-> A. k e. ( M ... N ) [. ( K - k ) / j ]. ph ) )

Distinct variable groups:   j, k, K   j, M, k   j, N, k   ph, k

Allowed substitution hint:   ph( j)

Proof of Theorem fzrevral2

Step	Hyp	Ref	Expression
1		zsubcl 8392	. . . . 5 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K - N ) e. ZZ )
2	1	3adant2 957	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K - N ) e. ZZ )
3		zsubcl 8392	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K - M ) e. ZZ )
4	3	3adant3 958	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K - M ) e. ZZ )
5		simp1 938	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
6		fzrevral 9122	. . . 4 |- ( ( ( K - N ) e. ZZ /\ ( K - M ) e. ZZ /\ K e. ZZ ) -> ( A. j e. ( ( K - N ) ... ( K - M ) ) ph <-> A. k e. ( ( K - ( K - M ) ) ... ( K - ( K - N ) ) ) [. ( K - k ) / j ]. ph ) )
7	2, 4, 5, 6	syl3anc 1169	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( A. j e. ( ( K - N ) ... ( K - M ) ) ph <-> A. k e. ( ( K - ( K - M ) ) ... ( K - ( K - N ) ) ) [. ( K - k ) / j ]. ph ) )
8		zcn 8356	. . . . 5 |- ( K e. ZZ -> K e. CC )
9		zcn 8356	. . . . 5 |- ( M e. ZZ -> M e. CC )
10		zcn 8356	. . . . 5 |- ( N e. ZZ -> N e. CC )
11		nncan 7337	. . . . . . 7 |- ( ( K e. CC /\ M e. CC ) -> ( K - ( K - M ) ) = M )
12	11	3adant3 958	. . . . . 6 |- ( ( K e. CC /\ M e. CC /\ N e. CC ) -> ( K - ( K - M ) ) = M )
13		nncan 7337	. . . . . . 7 |- ( ( K e. CC /\ N e. CC ) -> ( K - ( K - N ) ) = N )
14	13	3adant2 957	. . . . . 6 |- ( ( K e. CC /\ M e. CC /\ N e. CC ) -> ( K - ( K - N ) ) = N )
15	12, 14	oveq12d 5550	. . . . 5 |- ( ( K e. CC /\ M e. CC /\ N e. CC ) -> ( ( K - ( K - M ) ) ... ( K - ( K - N ) ) ) = ( M ... N ) )
16	8, 9, 10, 15	syl3an 1211	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K - ( K - M ) ) ... ( K - ( K - N ) ) ) = ( M ... N ) )
17	16	raleqdv 2555	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( A. k e. ( ( K - ( K - M ) ) ... ( K - ( K - N ) ) ) [. ( K - k ) / j ]. ph <-> A. k e. ( M ... N ) [. ( K - k ) / j ]. ph ) )
18	7, 17	bitrd 186	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( A. j e. ( ( K - N ) ... ( K - M ) ) ph <-> A. k e. ( M ... N ) [. ( K - k ) / j ]. ph ) )
19	18	3coml 1145	1 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( A. j e. ( ( K - N ) ... ( K - M ) ) ph <-> A. k e. ( M ... N ) [. ( K - k ) / j ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   A.wral 2348   [.wsbc 2815  (class class class)co 5532   CCcc 6979   - 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: (None)