Theorem fprodcl2lem 14680

Description: Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)

Hypotheses

Ref	Expression
fprodcllem.1	|- ( ph -> S C_ CC )
fprodcllem.2	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
fprodcllem.3	|- ( ph -> A e. Fin )
fprodcllem.4	|- ( ( ph /\ k e. A ) -> B e. S )
fprodcl2lem.5	|- ( ph -> A =/= (/) )

Assertion

Ref	Expression
fprodcl2lem	|- ( ph -> prod_ k e. A B e. S )

Distinct variable groups:   A, k, x, y   x, B, y   ph, k, x, y   S, k, x, y

Allowed substitution hint:   B( k)

Proof of Theorem fprodcl2lem

Dummy variables f m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fprodcl2lem.5	. . . 4 |- ( ph -> A =/= (/) )
2	1	a1d 25	. . 3 |- ( ph -> ( -. prod_ k e. A B e. S -> A =/= (/) ) )
3	2	necon4bd 2814	. 2 |- ( ph -> ( A = (/) -> prod_ k e. A B e. S ) )
4		prodfc 14675	. . . . . . 7 |- prod_ m e. A ( ( k e. A |-> B ) ` m ) = prod_ k e. A B
5		fveq2 6191	. . . . . . . 8 |- ( m = ( f ` x ) -> ( ( k e. A |-> B ) ` m ) = ( ( k e. A |-> B ) ` ( f ` x ) ) )
6		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( # ` A ) e. NN )
7		simprr 796	. . . . . . . 8 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> f : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
8		fprodcllem.1	. . . . . . . . . . . . 13 |- ( ph -> S C_ CC )
9	8	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ k e. A ) -> S C_ CC )
10		fprodcllem.4	. . . . . . . . . . . 12 |- ( ( ph /\ k e. A ) -> B e. S )
11	9, 10	sseldd 3604	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> B e. CC )
12		eqid 2622	. . . . . . . . . . 11 |- ( k e. A |-> B ) = ( k e. A |-> B )
13	11, 12	fmptd 6385	. . . . . . . . . 10 |- ( ph -> ( k e. A |-> B ) : A --> CC )
14	13	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> ( ( k e. A |-> B ) ` m ) e. CC )
15	14	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) /\ m e. A ) -> ( ( k e. A |-> B ) ` m ) e. CC )
16		f1of 6137	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> f : ( 1 ... ( # ` A ) ) --> A )
17	16	ad2antll 765	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> f : ( 1 ... ( # ` A ) ) --> A )
18		fvco3 6275	. . . . . . . . 9 |- ( ( f : ( 1 ... ( # ` A ) ) --> A /\ x e. ( 1 ... ( # ` A ) ) ) -> ( ( ( k e. A |-> B ) o. f ) ` x ) = ( ( k e. A |-> B ) ` ( f ` x ) ) )
19	17, 18	sylan 488	. . . . . . . 8 |- ( ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) /\ x e. ( 1 ... ( # ` A ) ) ) -> ( ( ( k e. A |-> B ) o. f ) ` x ) = ( ( k e. A |-> B ) ` ( f ` x ) ) )
20	5, 6, 7, 15, 19	fprod 14671	. . . . . . 7 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> prod_ m e. A ( ( k e. A |-> B ) ` m ) = ( seq 1 ( x. , ( ( k e. A |-> B ) o. f ) ) ` ( # ` A ) ) )
21	4, 20	syl5eqr 2670	. . . . . 6 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> prod_ k e. A B = ( seq 1 ( x. , ( ( k e. A |-> B ) o. f ) ) ` ( # ` A ) ) )
22		nnuz 11723	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
23	6, 22	syl6eleq 2711	. . . . . . 7 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( # ` A ) e. ( ZZ>= ` 1 ) )
24	10, 12	fmptd 6385	. . . . . . . . . 10 |- ( ph -> ( k e. A |-> B ) : A --> S )
25	24	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( k e. A |-> B ) : A --> S )
26		fco 6058	. . . . . . . . 9 |- ( ( ( k e. A |-> B ) : A --> S /\ f : ( 1 ... ( # ` A ) ) --> A ) -> ( ( k e. A |-> B ) o. f ) : ( 1 ... ( # ` A ) ) --> S )
27	25, 17, 26	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( ( k e. A |-> B ) o. f ) : ( 1 ... ( # ` A ) ) --> S )
28	27	ffvelrnda 6359	. . . . . . 7 |- ( ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) /\ x e. ( 1 ... ( # ` A ) ) ) -> ( ( ( k e. A |-> B ) o. f ) ` x ) e. S )
29		fprodcllem.2	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
30	29	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
31	23, 28, 30	seqcl 12821	. . . . . 6 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( seq 1 ( x. , ( ( k e. A |-> B ) o. f ) ) ` ( # ` A ) ) e. S )
32	21, 31	eqeltrd 2701	. . . . 5 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> prod_ k e. A B e. S )
33	32	expr 643	. . . 4 |- ( ( ph /\ ( # ` A ) e. NN ) -> ( f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> prod_ k e. A B e. S ) )
34	33	exlimdv 1861	. . 3 |- ( ( ph /\ ( # ` A ) e. NN ) -> ( E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> prod_ k e. A B e. S ) )
35	34	expimpd 629	. 2 |- ( ph -> ( ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) -> prod_ k e. A B e. S ) )
36		fprodcllem.3	. . 3 |- ( ph -> A e. Fin )
37		fz1f1o 14441	. . 3 |- ( A e. Fin -> ( A = (/) \/ ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) )
38	36, 37	syl 17	. 2 |- ( ph -> ( A = (/) \/ ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) )
39	3, 35, 38	mpjaod 396	1 |- ( ph -> prod_ k e. A B e. S )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   1c1 9937   x. cmul 9941   NNcn 11020   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117   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:  fprodcllem  14681