Theorem fprodrev 11582

Description: Reversal of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)

Hypotheses

Ref	Expression
fprodshft.1	|- ( ph -> K e. ZZ )
fprodshft.2	|- ( ph -> M e. ZZ )
fprodshft.3	|- ( ph -> N e. ZZ )
fprodshft.4	|- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
fprodrev.5	|- ( j = ( K - k ) -> A = B )

Assertion

Ref	Expression
fprodrev	|- ( ph -> prod_ j e. ( M ... N ) A = prod_ k e. ( ( K - N ) ... ( K - M ) ) B )

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

Allowed substitution hints:   A( j)   B( k)

Proof of Theorem fprodrev

Step	Hyp	Ref	Expression
1		fprodrev.5	. 2 |- ( j = ( K - k ) -> A = B )
2		fprodshft.1	. . . 4 |- ( ph -> K e. ZZ )
3		fprodshft.3	. . . 4 |- ( ph -> N e. ZZ )
4	2, 3	zsubcld 9339	. . 3 |- ( ph -> ( K - N ) e. ZZ )
5		fprodshft.2	. . . 4 |- ( ph -> M e. ZZ )
6	2, 5	zsubcld 9339	. . 3 |- ( ph -> ( K - M ) e. ZZ )
7	4, 6	fzfigd 10387	. 2 |- ( ph -> ( ( K - N ) ... ( K - M ) ) e. Fin )
8		eqid 2170	. . 3 |- ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) = ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) )
9	2	adantr 274	. . . 4 |- ( ( ph /\ j e. ( ( K - N ) ... ( K - M ) ) ) -> K e. ZZ )
10		elfzelz 9981	. . . . 5 |- ( j e. ( ( K - N ) ... ( K - M ) ) -> j e. ZZ )
11	10	adantl 275	. . . 4 |- ( ( ph /\ j e. ( ( K - N ) ... ( K - M ) ) ) -> j e. ZZ )
12	9, 11	zsubcld 9339	. . 3 |- ( ( ph /\ j e. ( ( K - N ) ... ( K - M ) ) ) -> ( K - j ) e. ZZ )
13	2	adantr 274	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> K e. ZZ )
14		elfzelz 9981	. . . . 5 |- ( k e. ( M ... N ) -> k e. ZZ )
15	14	adantl 275	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> k e. ZZ )
16	13, 15	zsubcld 9339	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> ( K - k ) e. ZZ )
17		simprr 527	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> k = ( K - j ) )
18		simprl 526	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. ( ( K - N ) ... ( K - M ) ) )
19	5	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> M e. ZZ )
20	3	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> N e. ZZ )
21	2	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> K e. ZZ )
22	10	ad2antrl 487	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. ZZ )
23		fzrev 10040	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ j e. ZZ ) ) -> ( j e. ( ( K - N ) ... ( K - M ) ) <-> ( K - j ) e. ( M ... N ) ) )
24	19, 20, 21, 22, 23	syl22anc 1234	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( j e. ( ( K - N ) ... ( K - M ) ) <-> ( K - j ) e. ( M ... N ) ) )
25	18, 24	mpbid 146	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - j ) e. ( M ... N ) )
26	17, 25	eqeltrd 2247	. . . . 5 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> k e. ( M ... N ) )
27		oveq2 5861	. . . . . . 7 |- ( k = ( K - j ) -> ( K - k ) = ( K - ( K - j ) ) )
28	27	ad2antll 488	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - k ) = ( K - ( K - j ) ) )
29	2	zcnd 9335	. . . . . . . 8 |- ( ph -> K e. CC )
30	29	adantr 274	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> K e. CC )
31	10	zcnd 9335	. . . . . . . 8 |- ( j e. ( ( K - N ) ... ( K - M ) ) -> j e. CC )
32	31	ad2antrl 487	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. CC )
33	30, 32	nncand 8235	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - ( K - j ) ) = j )
34	28, 33	eqtr2d 2204	. . . . 5 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j = ( K - k ) )
35	26, 34	jca 304	. . . 4 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( k e. ( M ... N ) /\ j = ( K - k ) ) )
36		simprr 527	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> j = ( K - k ) )
37		simprl 526	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. ( M ... N ) )
38	5	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> M e. ZZ )
39	3	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> N e. ZZ )
40	2	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> K e. ZZ )
41	14	ad2antrl 487	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. ZZ )
42		fzrev2 10041	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ k e. ZZ ) ) -> ( k e. ( M ... N ) <-> ( K - k ) e. ( ( K - N ) ... ( K - M ) ) ) )
43	38, 39, 40, 41, 42	syl22anc 1234	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( k e. ( M ... N ) <-> ( K - k ) e. ( ( K - N ) ... ( K - M ) ) ) )
44	37, 43	mpbid 146	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - k ) e. ( ( K - N ) ... ( K - M ) ) )
45	36, 44	eqeltrd 2247	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> j e. ( ( K - N ) ... ( K - M ) ) )
46		oveq2 5861	. . . . . . 7 |- ( j = ( K - k ) -> ( K - j ) = ( K - ( K - k ) ) )
47	46	ad2antll 488	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - j ) = ( K - ( K - k ) ) )
48	29	adantr 274	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> K e. CC )
49	14	zcnd 9335	. . . . . . . 8 |- ( k e. ( M ... N ) -> k e. CC )
50	49	ad2antrl 487	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. CC )
51	48, 50	nncand 8235	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - ( K - k ) ) = k )
52	47, 51	eqtr2d 2204	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k = ( K - j ) )
53	45, 52	jca 304	. . . 4 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) )
54	35, 53	impbida 591	. . 3 |- ( ph -> ( ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) <-> ( k e. ( M ... N ) /\ j = ( K - k ) ) ) )
55	8, 12, 16, 54	f1od 6052	. 2 |- ( ph -> ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) : ( ( K - N ) ... ( K - M ) ) -1-1-onto-> ( M ... N ) )
56		oveq2 5861	. . 3 |- ( j = k -> ( K - j ) = ( K - k ) )
57		simpr 109	. . 3 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> k e. ( ( K - N ) ... ( K - M ) ) )
58	2	adantr 274	. . . 4 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> K e. ZZ )
59		elfzelz 9981	. . . . 5 |- ( k e. ( ( K - N ) ... ( K - M ) ) -> k e. ZZ )
60	59	adantl 275	. . . 4 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> k e. ZZ )
61	58, 60	zsubcld 9339	. . 3 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> ( K - k ) e. ZZ )
62	8, 56, 57, 61	fvmptd3 5589	. 2 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> ( ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) ` k ) = ( K - k ) )
63		fprodshft.4	. 2 |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
64	1, 7, 55, 62, 63	fprodf1o 11551	1 |- ( ph -> prod_ j e. ( M ... N ) A = prod_ k e. ( ( K - N ) ... ( K - M ) ) B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   |-> cmpt 4050  (class class class)co 5853   CCcc 7772   - cmin 8090   ZZcz 9212   ...cfz 9965   prod_cprod 11513

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-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  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-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  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-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  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-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-proddc 11514

This theorem is referenced by: (None)