Theorem fprodrev 12260

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 9668	. . 3 |- ( ph -> ( K - N ) e. ZZ )
5		fprodshft.2	. . . 4 |- ( ph -> M e. ZZ )
6	2, 5	zsubcld 9668	. . 3 |- ( ph -> ( K - M ) e. ZZ )
7	4, 6	fzfigd 10756	. 2 |- ( ph -> ( ( K - N ) ... ( K - M ) ) e. Fin )
8		eqid 2231	. . 3 |- ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) = ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) )
9	2	adantr 276	. . . 4 |- ( ( ph /\ j e. ( ( K - N ) ... ( K - M ) ) ) -> K e. ZZ )
10		elfzelz 10322	. . . . 5 |- ( j e. ( ( K - N ) ... ( K - M ) ) -> j e. ZZ )
11	10	adantl 277	. . . 4 |- ( ( ph /\ j e. ( ( K - N ) ... ( K - M ) ) ) -> j e. ZZ )
12	9, 11	zsubcld 9668	. . 3 |- ( ( ph /\ j e. ( ( K - N ) ... ( K - M ) ) ) -> ( K - j ) e. ZZ )
13	2	adantr 276	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> K e. ZZ )
14		elfzelz 10322	. . . . 5 |- ( k e. ( M ... N ) -> k e. ZZ )
15	14	adantl 277	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> k e. ZZ )
16	13, 15	zsubcld 9668	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> ( K - k ) e. ZZ )
17		simprr 533	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> k = ( K - j ) )
18		simprl 531	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. ( ( K - N ) ... ( K - M ) ) )
19	5	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> M e. ZZ )
20	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> N e. ZZ )
21	2	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> K e. ZZ )
22	10	ad2antrl 490	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. ZZ )
23		fzrev 10381	. . . . . . . 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 1275	. . . . . . 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 147	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - j ) e. ( M ... N ) )
26	17, 25	eqeltrd 2308	. . . . 5 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> k e. ( M ... N ) )
27		oveq2 6036	. . . . . . 7 |- ( k = ( K - j ) -> ( K - k ) = ( K - ( K - j ) ) )
28	27	ad2antll 491	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - k ) = ( K - ( K - j ) ) )
29	2	zcnd 9664	. . . . . . . 8 |- ( ph -> K e. CC )
30	29	adantr 276	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> K e. CC )
31	10	zcnd 9664	. . . . . . . 8 |- ( j e. ( ( K - N ) ... ( K - M ) ) -> j e. CC )
32	31	ad2antrl 490	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. CC )
33	30, 32	nncand 8554	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - ( K - j ) ) = j )
34	28, 33	eqtr2d 2265	. . . . 5 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j = ( K - k ) )
35	26, 34	jca 306	. . . 4 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( k e. ( M ... N ) /\ j = ( K - k ) ) )
36		simprr 533	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> j = ( K - k ) )
37		simprl 531	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. ( M ... N ) )
38	5	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> M e. ZZ )
39	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> N e. ZZ )
40	2	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> K e. ZZ )
41	14	ad2antrl 490	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. ZZ )
42		fzrev2 10382	. . . . . . . 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 1275	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( k e. ( M ... N ) <-> ( K - k ) e. ( ( K - N ) ... ( K - M ) ) ) )
44	37, 43	mpbid 147	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - k ) e. ( ( K - N ) ... ( K - M ) ) )
45	36, 44	eqeltrd 2308	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> j e. ( ( K - N ) ... ( K - M ) ) )
46		oveq2 6036	. . . . . . 7 |- ( j = ( K - k ) -> ( K - j ) = ( K - ( K - k ) ) )
47	46	ad2antll 491	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - j ) = ( K - ( K - k ) ) )
48	29	adantr 276	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> K e. CC )
49	14	zcnd 9664	. . . . . . . 8 |- ( k e. ( M ... N ) -> k e. CC )
50	49	ad2antrl 490	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. CC )
51	48, 50	nncand 8554	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - ( K - k ) ) = k )
52	47, 51	eqtr2d 2265	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k = ( K - j ) )
53	45, 52	jca 306	. . . 4 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) )
54	35, 53	impbida 600	. . 3 |- ( ph -> ( ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) <-> ( k e. ( M ... N ) /\ j = ( K - k ) ) ) )
55	8, 12, 16, 54	f1od 6236	. 2 |- ( ph -> ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) : ( ( K - N ) ... ( K - M ) ) -1-1-onto-> ( M ... N ) )
56		oveq2 6036	. . 3 |- ( j = k -> ( K - j ) = ( K - k ) )
57		simpr 110	. . 3 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> k e. ( ( K - N ) ... ( K - M ) ) )
58	2	adantr 276	. . . 4 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> K e. ZZ )
59		elfzelz 10322	. . . . 5 |- ( k e. ( ( K - N ) ... ( K - M ) ) -> k e. ZZ )
60	59	adantl 277	. . . 4 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> k e. ZZ )
61	58, 60	zsubcld 9668	. . 3 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> ( K - k ) e. ZZ )
62	8, 56, 57, 61	fvmptd3 5749	. 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 12229	1 |- ( ph -> prod_ j e. ( M ... N ) A = prod_ k e. ( ( K - N ) ... ( K - M ) ) B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   |-> cmpt 4155  (class class class)co 6028   CCcc 8090   - cmin 8409   ZZcz 9540   ...cfz 10305   prod_cprod 12191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-ihash 11101  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-proddc 12192

This theorem is referenced by: (None)