Theorem fprodrev 11560

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 9318	. . 3 |- ( ph -> ( K - N ) e. ZZ )
5		fprodshft.2	. . . 4 |- ( ph -> M e. ZZ )
6	2, 5	zsubcld 9318	. . 3 |- ( ph -> ( K - M ) e. ZZ )
7	4, 6	fzfigd 10366	. 2 |- ( ph -> ( ( K - N ) ... ( K - M ) ) e. Fin )
8		eqid 2165	. . 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 9960	. . . . 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 9318	. . 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 9960	. . . . 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 9318	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> ( K - k ) e. ZZ )
17		simprr 522	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> k = ( K - j ) )
18		simprl 521	. . . . . . 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 482	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. ZZ )
23		fzrev 10019	. . . . . . . 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 1229	. . . . . . 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 2243	. . . . 5 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> k e. ( M ... N ) )
27		oveq2 5850	. . . . . . 7 |- ( k = ( K - j ) -> ( K - k ) = ( K - ( K - j ) ) )
28	27	ad2antll 483	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - k ) = ( K - ( K - j ) ) )
29	2	zcnd 9314	. . . . . . . 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 9314	. . . . . . . 8 |- ( j e. ( ( K - N ) ... ( K - M ) ) -> j e. CC )
32	31	ad2antrl 482	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. CC )
33	30, 32	nncand 8214	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - ( K - j ) ) = j )
34	28, 33	eqtr2d 2199	. . . . 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 522	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> j = ( K - k ) )
37		simprl 521	. . . . . . 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 482	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. ZZ )
42		fzrev2 10020	. . . . . . . 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 1229	. . . . . . 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 2243	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> j e. ( ( K - N ) ... ( K - M ) ) )
46		oveq2 5850	. . . . . . 7 |- ( j = ( K - k ) -> ( K - j ) = ( K - ( K - k ) ) )
47	46	ad2antll 483	. . . . . 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 9314	. . . . . . . 8 |- ( k e. ( M ... N ) -> k e. CC )
50	49	ad2antrl 482	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. CC )
51	48, 50	nncand 8214	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - ( K - k ) ) = k )
52	47, 51	eqtr2d 2199	. . . . 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 586	. . 3 |- ( ph -> ( ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) <-> ( k e. ( M ... N ) /\ j = ( K - k ) ) ) )
55	8, 12, 16, 54	f1od 6041	. 2 |- ( ph -> ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) : ( ( K - N ) ... ( K - M ) ) -1-1-onto-> ( M ... N ) )
56		oveq2 5850	. . 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 9960	. . . . 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 9318	. . 3 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> ( K - k ) e. ZZ )
62	8, 56, 57, 61	fvmptd3 5579	. 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 11529	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 1343   e. wcel 2136   |-> cmpt 4043  (class class class)co 5842   CCcc 7751   - cmin 8069   ZZcz 9191   ...cfz 9944   prod_cprod 11491

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-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  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-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  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-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  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-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-proddc 11492

This theorem is referenced by: (None)