Theorem fzrevral2 10041

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 9232	. . . . 5 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K - N ) e. ZZ )
2	1	3adant2 1006	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K - N ) e. ZZ )
3		zsubcl 9232	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K - M ) e. ZZ )
4	3	3adant3 1007	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K - M ) e. ZZ )
5		simp1 987	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
6		fzrevral 10040	. . . 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 1228	. . 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 9196	. . . . 5 |- ( K e. ZZ -> K e. CC )
9		zcn 9196	. . . . 5 |- ( M e. ZZ -> M e. CC )
10		zcn 9196	. . . . 5 |- ( N e. ZZ -> N e. CC )
11		nncan 8127	. . . . . . 7 |- ( ( K e. CC /\ M e. CC ) -> ( K - ( K - M ) ) = M )
12	11	3adant3 1007	. . . . . 6 |- ( ( K e. CC /\ M e. CC /\ N e. CC ) -> ( K - ( K - M ) ) = M )
13		nncan 8127	. . . . . . 7 |- ( ( K e. CC /\ N e. CC ) -> ( K - ( K - N ) ) = N )
14	13	3adant2 1006	. . . . . 6 |- ( ( K e. CC /\ M e. CC /\ N e. CC ) -> ( K - ( K - N ) ) = N )
15	12, 14	oveq12d 5860	. . . . 5 |- ( ( K e. CC /\ M e. CC /\ N e. CC ) -> ( ( K - ( K - M ) ) ... ( K - ( K - N ) ) ) = ( M ... N ) )
16	8, 9, 10, 15	syl3an 1270	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K - ( K - M ) ) ... ( K - ( K - N ) ) ) = ( M ... N ) )
17	16	raleqdv 2667	. . 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 187	. 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 1200	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 104   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   [.wsbc 2951  (class class class)co 5842   CCcc 7751   - cmin 8069   ZZcz 9191   ...cfz 9944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945

This theorem is referenced by: (None)