Theorem fprodrev 14707

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		fzfid 12772	. 2 |- ( ph -> ( ( K - N ) ... ( K - M ) ) e. Fin )
3		ovex 6678	. . . . 5 |- ( K - j ) e. _V
4		eqid 2622	. . . . 5 |- ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) = ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) )
5	3, 4	fnmpti 6022	. . . 4 |- ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) Fn ( ( K - N ) ... ( K - M ) )
6	5	a1i 11	. . 3 |- ( ph -> ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) Fn ( ( K - N ) ... ( K - M ) ) )
7		ovex 6678	. . . . 5 |- ( K - k ) e. _V
8		eqid 2622	. . . . 5 |- ( k e. ( M ... N ) |-> ( K - k ) ) = ( k e. ( M ... N ) |-> ( K - k ) )
9	7, 8	fnmpti 6022	. . . 4 |- ( k e. ( M ... N ) |-> ( K - k ) ) Fn ( M ... N )
10		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> k = ( K - j ) )
11		simprl 794	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. ( ( K - N ) ... ( K - M ) ) )
12		fprodshft.2	. . . . . . . . . . . 12 |- ( ph -> M e. ZZ )
13	12	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> M e. ZZ )
14		fprodshft.3	. . . . . . . . . . . 12 |- ( ph -> N e. ZZ )
15	14	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> N e. ZZ )
16		fprodshft.1	. . . . . . . . . . . 12 |- ( ph -> K e. ZZ )
17	16	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> K e. ZZ )
18		elfzelz 12342	. . . . . . . . . . . 12 |- ( j e. ( ( K - N ) ... ( K - M ) ) -> j e. ZZ )
19	18	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. ZZ )
20		fzrev 12403	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ j e. ZZ ) ) -> ( j e. ( ( K - N ) ... ( K - M ) ) <-> ( K - j ) e. ( M ... N ) ) )
21	13, 15, 17, 19, 20	syl22anc 1327	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( j e. ( ( K - N ) ... ( K - M ) ) <-> ( K - j ) e. ( M ... N ) ) )
22	11, 21	mpbid 222	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - j ) e. ( M ... N ) )
23	10, 22	eqeltrd 2701	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> k e. ( M ... N ) )
24		oveq2 6658	. . . . . . . . . 10 |- ( k = ( K - j ) -> ( K - k ) = ( K - ( K - j ) ) )
25	24	ad2antll 765	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - k ) = ( K - ( K - j ) ) )
26	16	zcnd 11483	. . . . . . . . . . 11 |- ( ph -> K e. CC )
27	26	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> K e. CC )
28	18	zcnd 11483	. . . . . . . . . . 11 |- ( j e. ( ( K - N ) ... ( K - M ) ) -> j e. CC )
29	28	ad2antrl 764	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. CC )
30	27, 29	nncand 10397	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - ( K - j ) ) = j )
31	25, 30	eqtr2d 2657	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j = ( K - k ) )
32	23, 31	jca 554	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( k e. ( M ... N ) /\ j = ( K - k ) ) )
33		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> j = ( K - k ) )
34		simprl 794	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. ( M ... N ) )
35	12	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> M e. ZZ )
36	14	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> N e. ZZ )
37	16	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> K e. ZZ )
38		elfzelz 12342	. . . . . . . . . . . 12 |- ( k e. ( M ... N ) -> k e. ZZ )
39	38	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. ZZ )
40		fzrev2 12404	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ k e. ZZ ) ) -> ( k e. ( M ... N ) <-> ( K - k ) e. ( ( K - N ) ... ( K - M ) ) ) )
41	35, 36, 37, 39, 40	syl22anc 1327	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( k e. ( M ... N ) <-> ( K - k ) e. ( ( K - N ) ... ( K - M ) ) ) )
42	34, 41	mpbid 222	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - k ) e. ( ( K - N ) ... ( K - M ) ) )
43	33, 42	eqeltrd 2701	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> j e. ( ( K - N ) ... ( K - M ) ) )
44		oveq2 6658	. . . . . . . . . 10 |- ( j = ( K - k ) -> ( K - j ) = ( K - ( K - k ) ) )
45	44	ad2antll 765	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - j ) = ( K - ( K - k ) ) )
46	26	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> K e. CC )
47	38	zcnd 11483	. . . . . . . . . . 11 |- ( k e. ( M ... N ) -> k e. CC )
48	47	ad2antrl 764	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. CC )
49	46, 48	nncand 10397	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - ( K - k ) ) = k )
50	45, 49	eqtr2d 2657	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k = ( K - j ) )
51	43, 50	jca 554	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) )
52	32, 51	impbida 877	. . . . . 6 |- ( ph -> ( ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) <-> ( k e. ( M ... N ) /\ j = ( K - k ) ) ) )
53	52	mptcnv 5534	. . . . 5 |- ( ph -> `' ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) = ( k e. ( M ... N ) |-> ( K - k ) ) )
54	53	fneq1d 5981	. . . 4 |- ( ph -> ( `' ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) Fn ( M ... N ) <-> ( k e. ( M ... N ) |-> ( K - k ) ) Fn ( M ... N ) ) )
55	9, 54	mpbiri 248	. . 3 |- ( ph -> `' ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) Fn ( M ... N ) )
56		dff1o4 6145	. . 3 |- ( ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) : ( ( K - N ) ... ( K - M ) ) -1-1-onto-> ( M ... N ) <-> ( ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) Fn ( ( K - N ) ... ( K - M ) ) /\ `' ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) Fn ( M ... N ) ) )
57	6, 55, 56	sylanbrc 698	. 2 |- ( ph -> ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) : ( ( K - N ) ... ( K - M ) ) -1-1-onto-> ( M ... N ) )
58		oveq2 6658	. . . 4 |- ( j = k -> ( K - j ) = ( K - k ) )
59	58, 4, 7	fvmpt 6282	. . 3 |- ( k e. ( ( K - N ) ... ( K - M ) ) -> ( ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) ` k ) = ( K - k ) )
60	59	adantl 482	. 2 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> ( ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) ` k ) = ( K - k ) )
61		fprodshft.4	. 2 |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
62	1, 2, 57, 60, 61	fprodf1o 14676	1 |- ( ph -> prod_ j e. ( M ... N ) A = prod_ k e. ( ( K - N ) ... ( K - M ) ) B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   `'ccnv 5113   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   - cmin 10266   ZZcz 11377   ...cfz 12326   prod_cprod 14635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636

This theorem is referenced by:  fallfacval3  14743  bcprod  31624