Theorem prodmolem3 14663

Description: Lemma for prodmo 14666. (Contributed by Scott Fenton, 4-Dec-2017.)

Hypotheses

Ref	Expression
prodmo.1	|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )
prodmo.2	|- ( ( ph /\ k e. A ) -> B e. CC )
prodmo.3	|- G = ( j e. NN |-> [_ ( f ` j ) / k ]_ B )
prodmolem3.4	|- H = ( j e. NN |-> [_ ( K ` j ) / k ]_ B )
prodmolem3.5	|- ( ph -> ( M e. NN /\ N e. NN ) )
prodmolem3.6	|- ( ph -> f : ( 1 ... M ) -1-1-onto-> A )
prodmolem3.7	|- ( ph -> K : ( 1 ... N ) -1-1-onto-> A )

Assertion

Ref	Expression
prodmolem3	|- ( ph -> ( seq 1 ( x. , G ) ` M ) = ( seq 1 ( x. , H ) ` N ) )

Distinct variable groups:   A, k   k, F   ph, k   B, j   f, j, k   j, G   j, k, ph   j, K   j, M

Allowed substitution hints:   ph( f)   A( f, j)   B( f, k)   F( f, j)   G( f, k)   H( f, j, k)   K( f, k)   M( f, k)   N( f, j, k)

Proof of Theorem prodmolem3

Dummy variables i m z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcl 10020	. . . 4 |- ( ( m e. CC /\ j e. CC ) -> ( m x. j ) e. CC )
2	1	adantl 482	. . 3 |- ( ( ph /\ ( m e. CC /\ j e. CC ) ) -> ( m x. j ) e. CC )
3		mulcom 10022	. . . 4 |- ( ( m e. CC /\ j e. CC ) -> ( m x. j ) = ( j x. m ) )
4	3	adantl 482	. . 3 |- ( ( ph /\ ( m e. CC /\ j e. CC ) ) -> ( m x. j ) = ( j x. m ) )
5		mulass 10024	. . . 4 |- ( ( m e. CC /\ j e. CC /\ z e. CC ) -> ( ( m x. j ) x. z ) = ( m x. ( j x. z ) ) )
6	5	adantl 482	. . 3 |- ( ( ph /\ ( m e. CC /\ j e. CC /\ z e. CC ) ) -> ( ( m x. j ) x. z ) = ( m x. ( j x. z ) ) )
7		prodmolem3.5	. . . . 5 |- ( ph -> ( M e. NN /\ N e. NN ) )
8	7	simpld 475	. . . 4 |- ( ph -> M e. NN )
9		nnuz 11723	. . . 4 |- NN = ( ZZ>= ` 1 )
10	8, 9	syl6eleq 2711	. . 3 |- ( ph -> M e. ( ZZ>= ` 1 ) )
11		ssid 3624	. . . 4 |- CC C_ CC
12	11	a1i 11	. . 3 |- ( ph -> CC C_ CC )
13		prodmolem3.6	. . . . . 6 |- ( ph -> f : ( 1 ... M ) -1-1-onto-> A )
14		f1ocnv 6149	. . . . . 6 |- ( f : ( 1 ... M ) -1-1-onto-> A -> `' f : A -1-1-onto-> ( 1 ... M ) )
15	13, 14	syl 17	. . . . 5 |- ( ph -> `' f : A -1-1-onto-> ( 1 ... M ) )
16		prodmolem3.7	. . . . 5 |- ( ph -> K : ( 1 ... N ) -1-1-onto-> A )
17		f1oco 6159	. . . . 5 |- ( ( `' f : A -1-1-onto-> ( 1 ... M ) /\ K : ( 1 ... N ) -1-1-onto-> A ) -> ( `' f o. K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) )
18	15, 16, 17	syl2anc 693	. . . 4 |- ( ph -> ( `' f o. K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) )
19		ovex 6678	. . . . . . . . . 10 |- ( 1 ... N ) e. _V
20	19	f1oen 7976	. . . . . . . . 9 |- ( ( `' f o. K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) -> ( 1 ... N ) ~~ ( 1 ... M ) )
21	18, 20	syl 17	. . . . . . . 8 |- ( ph -> ( 1 ... N ) ~~ ( 1 ... M ) )
22		fzfi 12771	. . . . . . . . 9 |- ( 1 ... N ) e. Fin
23		fzfi 12771	. . . . . . . . 9 |- ( 1 ... M ) e. Fin
24		hashen 13135	. . . . . . . . 9 |- ( ( ( 1 ... N ) e. Fin /\ ( 1 ... M ) e. Fin ) -> ( ( # ` ( 1 ... N ) ) = ( # ` ( 1 ... M ) ) <-> ( 1 ... N ) ~~ ( 1 ... M ) ) )
25	22, 23, 24	mp2an 708	. . . . . . . 8 |- ( ( # ` ( 1 ... N ) ) = ( # ` ( 1 ... M ) ) <-> ( 1 ... N ) ~~ ( 1 ... M ) )
26	21, 25	sylibr 224	. . . . . . 7 |- ( ph -> ( # ` ( 1 ... N ) ) = ( # ` ( 1 ... M ) ) )
27	7	simprd 479	. . . . . . . . 9 |- ( ph -> N e. NN )
28	27	nnnn0d 11351	. . . . . . . 8 |- ( ph -> N e. NN0 )
29		hashfz1 13134	. . . . . . . 8 |- ( N e. NN0 -> ( # ` ( 1 ... N ) ) = N )
30	28, 29	syl 17	. . . . . . 7 |- ( ph -> ( # ` ( 1 ... N ) ) = N )
31	8	nnnn0d 11351	. . . . . . . 8 |- ( ph -> M e. NN0 )
32		hashfz1 13134	. . . . . . . 8 |- ( M e. NN0 -> ( # ` ( 1 ... M ) ) = M )
33	31, 32	syl 17	. . . . . . 7 |- ( ph -> ( # ` ( 1 ... M ) ) = M )
34	26, 30, 33	3eqtr3rd 2665	. . . . . 6 |- ( ph -> M = N )
35	34	oveq2d 6666	. . . . 5 |- ( ph -> ( 1 ... M ) = ( 1 ... N ) )
36		f1oeq2 6128	. . . . 5 |- ( ( 1 ... M ) = ( 1 ... N ) -> ( ( `' f o. K ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M ) <-> ( `' f o. K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) ) )
37	35, 36	syl 17	. . . 4 |- ( ph -> ( ( `' f o. K ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M ) <-> ( `' f o. K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) ) )
38	18, 37	mpbird 247	. . 3 |- ( ph -> ( `' f o. K ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M ) )
39		elfznn 12370	. . . . . 6 |- ( m e. ( 1 ... M ) -> m e. NN )
40	39	adantl 482	. . . . 5 |- ( ( ph /\ m e. ( 1 ... M ) ) -> m e. NN )
41		f1of 6137	. . . . . . . 8 |- ( f : ( 1 ... M ) -1-1-onto-> A -> f : ( 1 ... M ) --> A )
42	13, 41	syl 17	. . . . . . 7 |- ( ph -> f : ( 1 ... M ) --> A )
43	42	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... M ) ) -> ( f ` m ) e. A )
44		prodmo.2	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> B e. CC )
45	44	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. k e. A B e. CC )
46	45	adantr 481	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... M ) ) -> A. k e. A B e. CC )
47		nfcsb1v 3549	. . . . . . . 8 |- F/_ k [_ ( f ` m ) / k ]_ B
48	47	nfel1 2779	. . . . . . 7 |- F/ k [_ ( f ` m ) / k ]_ B e. CC
49		csbeq1a 3542	. . . . . . . 8 |- ( k = ( f ` m ) -> B = [_ ( f ` m ) / k ]_ B )
50	49	eleq1d 2686	. . . . . . 7 |- ( k = ( f ` m ) -> ( B e. CC <-> [_ ( f ` m ) / k ]_ B e. CC ) )
51	48, 50	rspc 3303	. . . . . 6 |- ( ( f ` m ) e. A -> ( A. k e. A B e. CC -> [_ ( f ` m ) / k ]_ B e. CC ) )
52	43, 46, 51	sylc 65	. . . . 5 |- ( ( ph /\ m e. ( 1 ... M ) ) -> [_ ( f ` m ) / k ]_ B e. CC )
53		fveq2 6191	. . . . . . 7 |- ( j = m -> ( f ` j ) = ( f ` m ) )
54	53	csbeq1d 3540	. . . . . 6 |- ( j = m -> [_ ( f ` j ) / k ]_ B = [_ ( f ` m ) / k ]_ B )
55		prodmo.3	. . . . . 6 |- G = ( j e. NN |-> [_ ( f ` j ) / k ]_ B )
56	54, 55	fvmptg 6280	. . . . 5 |- ( ( m e. NN /\ [_ ( f ` m ) / k ]_ B e. CC ) -> ( G ` m ) = [_ ( f ` m ) / k ]_ B )
57	40, 52, 56	syl2anc 693	. . . 4 |- ( ( ph /\ m e. ( 1 ... M ) ) -> ( G ` m ) = [_ ( f ` m ) / k ]_ B )
58	57, 52	eqeltrd 2701	. . 3 |- ( ( ph /\ m e. ( 1 ... M ) ) -> ( G ` m ) e. CC )
59		f1oeq2 6128	. . . . . . . . . . . 12 |- ( ( 1 ... M ) = ( 1 ... N ) -> ( K : ( 1 ... M ) -1-1-onto-> A <-> K : ( 1 ... N ) -1-1-onto-> A ) )
60	35, 59	syl 17	. . . . . . . . . . 11 |- ( ph -> ( K : ( 1 ... M ) -1-1-onto-> A <-> K : ( 1 ... N ) -1-1-onto-> A ) )
61	16, 60	mpbird 247	. . . . . . . . . 10 |- ( ph -> K : ( 1 ... M ) -1-1-onto-> A )
62		f1of 6137	. . . . . . . . . 10 |- ( K : ( 1 ... M ) -1-1-onto-> A -> K : ( 1 ... M ) --> A )
63	61, 62	syl 17	. . . . . . . . 9 |- ( ph -> K : ( 1 ... M ) --> A )
64		fvco3 6275	. . . . . . . . 9 |- ( ( K : ( 1 ... M ) --> A /\ i e. ( 1 ... M ) ) -> ( ( `' f o. K ) ` i ) = ( `' f ` ( K ` i ) ) )
65	63, 64	sylan 488	. . . . . . . 8 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( `' f o. K ) ` i ) = ( `' f ` ( K ` i ) ) )
66	65	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( f ` ( ( `' f o. K ) ` i ) ) = ( f ` ( `' f ` ( K ` i ) ) ) )
67	13	adantr 481	. . . . . . . 8 |- ( ( ph /\ i e. ( 1 ... M ) ) -> f : ( 1 ... M ) -1-1-onto-> A )
68	63	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( K ` i ) e. A )
69		f1ocnvfv2 6533	. . . . . . . 8 |- ( ( f : ( 1 ... M ) -1-1-onto-> A /\ ( K ` i ) e. A ) -> ( f ` ( `' f ` ( K ` i ) ) ) = ( K ` i ) )
70	67, 68, 69	syl2anc 693	. . . . . . 7 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( f ` ( `' f ` ( K ` i ) ) ) = ( K ` i ) )
71	66, 70	eqtrd 2656	. . . . . 6 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( f ` ( ( `' f o. K ) ` i ) ) = ( K ` i ) )
72	71	csbeq1d 3540	. . . . 5 |- ( ( ph /\ i e. ( 1 ... M ) ) -> [_ ( f ` ( ( `' f o. K ) ` i ) ) / k ]_ B = [_ ( K ` i ) / k ]_ B )
73	72	fveq2d 6195	. . . 4 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( _I ` [_ ( f ` ( ( `' f o. K ) ` i ) ) / k ]_ B ) = ( _I ` [_ ( K ` i ) / k ]_ B ) )
74		f1of 6137	. . . . . . 7 |- ( ( `' f o. K ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M ) -> ( `' f o. K ) : ( 1 ... M ) --> ( 1 ... M ) )
75	38, 74	syl 17	. . . . . 6 |- ( ph -> ( `' f o. K ) : ( 1 ... M ) --> ( 1 ... M ) )
76	75	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( `' f o. K ) ` i ) e. ( 1 ... M ) )
77		elfznn 12370	. . . . 5 |- ( ( ( `' f o. K ) ` i ) e. ( 1 ... M ) -> ( ( `' f o. K ) ` i ) e. NN )
78		fveq2 6191	. . . . . . 7 |- ( j = ( ( `' f o. K ) ` i ) -> ( f ` j ) = ( f ` ( ( `' f o. K ) ` i ) ) )
79	78	csbeq1d 3540	. . . . . 6 |- ( j = ( ( `' f o. K ) ` i ) -> [_ ( f ` j ) / k ]_ B = [_ ( f ` ( ( `' f o. K ) ` i ) ) / k ]_ B )
80	79, 55	fvmpti 6281	. . . . 5 |- ( ( ( `' f o. K ) ` i ) e. NN -> ( G ` ( ( `' f o. K ) ` i ) ) = ( _I ` [_ ( f ` ( ( `' f o. K ) ` i ) ) / k ]_ B ) )
81	76, 77, 80	3syl 18	. . . 4 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` ( ( `' f o. K ) ` i ) ) = ( _I ` [_ ( f ` ( ( `' f o. K ) ` i ) ) / k ]_ B ) )
82		elfznn 12370	. . . . . 6 |- ( i e. ( 1 ... M ) -> i e. NN )
83	82	adantl 482	. . . . 5 |- ( ( ph /\ i e. ( 1 ... M ) ) -> i e. NN )
84		fveq2 6191	. . . . . . 7 |- ( j = i -> ( K ` j ) = ( K ` i ) )
85	84	csbeq1d 3540	. . . . . 6 |- ( j = i -> [_ ( K ` j ) / k ]_ B = [_ ( K ` i ) / k ]_ B )
86		prodmolem3.4	. . . . . 6 |- H = ( j e. NN |-> [_ ( K ` j ) / k ]_ B )
87	85, 86	fvmpti 6281	. . . . 5 |- ( i e. NN -> ( H ` i ) = ( _I ` [_ ( K ` i ) / k ]_ B ) )
88	83, 87	syl 17	. . . 4 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( H ` i ) = ( _I ` [_ ( K ` i ) / k ]_ B ) )
89	73, 81, 88	3eqtr4rd 2667	. . 3 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( H ` i ) = ( G ` ( ( `' f o. K ) ` i ) ) )
90	2, 4, 6, 10, 12, 38, 58, 89	seqf1o 12842	. 2 |- ( ph -> ( seq 1 ( x. , H ) ` M ) = ( seq 1 ( x. , G ) ` M ) )
91	34	fveq2d 6195	. 2 |- ( ph -> ( seq 1 ( x. , H ) ` M ) = ( seq 1 ( x. , H ) ` N ) )
92	90, 91	eqtr3d 2658	1 |- ( ph -> ( seq 1 ( x. , G ) ` M ) = ( seq 1 ( x. , H ) ` N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   [_csb 3533   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   Fincfn 7955   CCcc 9934   1c1 9937   x. cmul 9941   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-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-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118

This theorem is referenced by:  prodmolem2a  14664  prodmo  14666