Theorem fzrevral2 10062

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 9253	. . . . 5 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K - N ) e. ZZ )
2	1	3adant2 1011	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K - N ) e. ZZ )
3		zsubcl 9253	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K - M ) e. ZZ )
4	3	3adant3 1012	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K - M ) e. ZZ )
5		simp1 992	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
6		fzrevral 10061	. . . 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 1233	. . 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 9217	. . . . 5 |- ( K e. ZZ -> K e. CC )
9		zcn 9217	. . . . 5 |- ( M e. ZZ -> M e. CC )
10		zcn 9217	. . . . 5 |- ( N e. ZZ -> N e. CC )
11		nncan 8148	. . . . . . 7 |- ( ( K e. CC /\ M e. CC ) -> ( K - ( K - M ) ) = M )
12	11	3adant3 1012	. . . . . 6 |- ( ( K e. CC /\ M e. CC /\ N e. CC ) -> ( K - ( K - M ) ) = M )
13		nncan 8148	. . . . . . 7 |- ( ( K e. CC /\ N e. CC ) -> ( K - ( K - N ) ) = N )
14	13	3adant2 1011	. . . . . 6 |- ( ( K e. CC /\ M e. CC /\ N e. CC ) -> ( K - ( K - N ) ) = N )
15	12, 14	oveq12d 5871	. . . . 5 |- ( ( K e. CC /\ M e. CC /\ N e. CC ) -> ( ( K - ( K - M ) ) ... ( K - ( K - N ) ) ) = ( M ... N ) )
16	8, 9, 10, 15	syl3an 1275	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K - ( K - M ) ) ... ( K - ( K - N ) ) ) = ( M ... N ) )
17	16	raleqdv 2671	. . 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 1205	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 973   = wceq 1348   e. wcel 2141   A.wral 2448   [.wsbc 2955  (class class class)co 5853   CCcc 7772   - cmin 8090   ZZcz 9212   ...cfz 9965

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966

This theorem is referenced by: (None)