Theorem prodindf 30085

Description: The product of indicators is one if and only if all values are in the set. (Contributed by Thierry Arnoux, 11-Dec-2021.)

Hypotheses

Ref	Expression
prodindf.1	|- ( ph -> O e. V )
prodindf.2	|- ( ph -> A e. Fin )
prodindf.3	|- ( ph -> B C_ O )
prodindf.4	|- ( ph -> F : A --> O )

Assertion

Ref	Expression
prodindf	|- ( ph -> prod_ k e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = if ( ran F C_ B , 1 , 0 ) )

Distinct variable groups:   A, k   B, k   k, F   k, O   ph, k

Allowed substitution hint:   V( k)

Proof of Theorem prodindf

Dummy variable l is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( k = l -> ( F ` k ) = ( F ` l ) )
2	1	fveq2d 6195	. . 3 |- ( k = l -> ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = ( ( (𝟭 ` O ) ` B ) ` ( F ` l ) ) )
3		prodindf.2	. . 3 |- ( ph -> A e. Fin )
4		prodindf.1	. . . . . 6 |- ( ph -> O e. V )
5		prodindf.3	. . . . . 6 |- ( ph -> B C_ O )
6		indf 30077	. . . . . 6 |- ( ( O e. V /\ B C_ O ) -> ( (𝟭 ` O ) ` B ) : O --> { 0 , 1 } )
7	4, 5, 6	syl2anc 693	. . . . 5 |- ( ph -> ( (𝟭 ` O ) ` B ) : O --> { 0 , 1 } )
8	7	adantr 481	. . . 4 |- ( ( ph /\ k e. A ) -> ( (𝟭 ` O ) ` B ) : O --> { 0 , 1 } )
9		prodindf.4	. . . . 5 |- ( ph -> F : A --> O )
10	9	ffvelrnda 6359	. . . 4 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. O )
11	8, 10	ffvelrnd 6360	. . 3 |- ( ( ph /\ k e. A ) -> ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) e. { 0 , 1 } )
12	2, 3, 11	fprodex01 29571	. 2 |- ( ph -> prod_ k e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = if ( A. l e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` l ) ) = 1 , 1 , 0 ) )
13		fveq2 6191	. . . . . . 7 |- ( l = k -> ( F ` l ) = ( F ` k ) )
14	13	fveq2d 6195	. . . . . 6 |- ( l = k -> ( ( (𝟭 ` O ) ` B ) ` ( F ` l ) ) = ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) )
15	14	eqeq1d 2624	. . . . 5 |- ( l = k -> ( ( ( (𝟭 ` O ) ` B ) ` ( F ` l ) ) = 1 <-> ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = 1 ) )
16	15	cbvralv 3171	. . . 4 |- ( A. l e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` l ) ) = 1 <-> A. k e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = 1 )
17	16	a1i 11	. . 3 |- ( ph -> ( A. l e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` l ) ) = 1 <-> A. k e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = 1 ) )
18	17	ifbid 4108	. 2 |- ( ph -> if ( A. l e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` l ) ) = 1 , 1 , 0 ) = if ( A. k e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = 1 , 1 , 0 ) )
19		eqid 2622	. . . . . 6 |- ( k e. A |-> ( F ` k ) ) = ( k e. A |-> ( F ` k ) )
20	19	rnmptss 6392	. . . . 5 |- ( A. k e. A ( F ` k ) e. B -> ran ( k e. A |-> ( F ` k ) ) C_ B )
21		nfv 1843	. . . . . . . 8 |- F/ k ph
22		nfmpt1 4747	. . . . . . . . . 10 |- F/_ k ( k e. A |-> ( F ` k ) )
23	22	nfrn 5368	. . . . . . . . 9 |- F/_ k ran ( k e. A |-> ( F ` k ) )
24		nfcv 2764	. . . . . . . . 9 |- F/_ k B
25	23, 24	nfss 3596	. . . . . . . 8 |- F/ k ran ( k e. A |-> ( F ` k ) ) C_ B
26	21, 25	nfan 1828	. . . . . . 7 |- F/ k ( ph /\ ran ( k e. A |-> ( F ` k ) ) C_ B )
27		simplr 792	. . . . . . . . 9 |- ( ( ( ph /\ ran ( k e. A |-> ( F ` k ) ) C_ B ) /\ k e. A ) -> ran ( k e. A |-> ( F ` k ) ) C_ B )
28	9	feqmptd 6249	. . . . . . . . . . . . . 14 |- ( ph -> F = ( k e. A |-> ( F ` k ) ) )
29		eqidd 2623	. . . . . . . . . . . . . 14 |- ( ph -> k = k )
30	28, 29	fveq12d 6197	. . . . . . . . . . . . 13 |- ( ph -> ( F ` k ) = ( ( k e. A |-> ( F ` k ) ) ` k ) )
31	30	ralrimivw 2967	. . . . . . . . . . . 12 |- ( ph -> A. k e. A ( F ` k ) = ( ( k e. A |-> ( F ` k ) ) ` k ) )
32	31	r19.21bi 2932	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( F ` k ) = ( ( k e. A |-> ( F ` k ) ) ` k ) )
33		ffn 6045	. . . . . . . . . . . . . . 15 |- ( F : A --> O -> F Fn A )
34	9, 33	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> F Fn A )
35	28	fneq1d 5981	. . . . . . . . . . . . . 14 |- ( ph -> ( F Fn A <-> ( k e. A |-> ( F ` k ) ) Fn A ) )
36	34, 35	mpbid 222	. . . . . . . . . . . . 13 |- ( ph -> ( k e. A |-> ( F ` k ) ) Fn A )
37	36	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ k e. A ) -> ( k e. A |-> ( F ` k ) ) Fn A )
38		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ k e. A ) -> k e. A )
39		fnfvelrn 6356	. . . . . . . . . . . 12 |- ( ( ( k e. A |-> ( F ` k ) ) Fn A /\ k e. A ) -> ( ( k e. A |-> ( F ` k ) ) ` k ) e. ran ( k e. A |-> ( F ` k ) ) )
40	37, 38, 39	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( ( k e. A |-> ( F ` k ) ) ` k ) e. ran ( k e. A |-> ( F ` k ) ) )
41	32, 40	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. ran ( k e. A |-> ( F ` k ) ) )
42	41	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ ran ( k e. A |-> ( F ` k ) ) C_ B ) /\ k e. A ) -> ( F ` k ) e. ran ( k e. A |-> ( F ` k ) ) )
43	27, 42	sseldd 3604	. . . . . . . 8 |- ( ( ( ph /\ ran ( k e. A |-> ( F ` k ) ) C_ B ) /\ k e. A ) -> ( F ` k ) e. B )
44	43	ex 450	. . . . . . 7 |- ( ( ph /\ ran ( k e. A |-> ( F ` k ) ) C_ B ) -> ( k e. A -> ( F ` k ) e. B ) )
45	26, 44	ralrimi 2957	. . . . . 6 |- ( ( ph /\ ran ( k e. A |-> ( F ` k ) ) C_ B ) -> A. k e. A ( F ` k ) e. B )
46	45	ex 450	. . . . 5 |- ( ph -> ( ran ( k e. A |-> ( F ` k ) ) C_ B -> A. k e. A ( F ` k ) e. B ) )
47	20, 46	impbid2 216	. . . 4 |- ( ph -> ( A. k e. A ( F ` k ) e. B <-> ran ( k e. A |-> ( F ` k ) ) C_ B ) )
48	4	adantr 481	. . . . . 6 |- ( ( ph /\ k e. A ) -> O e. V )
49	5	adantr 481	. . . . . 6 |- ( ( ph /\ k e. A ) -> B C_ O )
50		ind1a 30081	. . . . . 6 |- ( ( O e. V /\ B C_ O /\ ( F ` k ) e. O ) -> ( ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = 1 <-> ( F ` k ) e. B ) )
51	48, 49, 10, 50	syl3anc 1326	. . . . 5 |- ( ( ph /\ k e. A ) -> ( ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = 1 <-> ( F ` k ) e. B ) )
52	51	ralbidva 2985	. . . 4 |- ( ph -> ( A. k e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = 1 <-> A. k e. A ( F ` k ) e. B ) )
53	28	rneqd 5353	. . . . 5 |- ( ph -> ran F = ran ( k e. A |-> ( F ` k ) ) )
54	53	sseq1d 3632	. . . 4 |- ( ph -> ( ran F C_ B <-> ran ( k e. A |-> ( F ` k ) ) C_ B ) )
55	47, 52, 54	3bitr4d 300	. . 3 |- ( ph -> ( A. k e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = 1 <-> ran F C_ B ) )
56	55	ifbid 4108	. 2 |- ( ph -> if ( A. k e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = 1 , 1 , 0 ) = if ( ran F C_ B , 1 , 0 ) )
57	12, 18, 56	3eqtrd 2660	1 |- ( ph -> prod_ k e. A ( ( (𝟭 ` O ) ` B ) ` ( F ` k ) ) = if ( ran F C_ B , 1 , 0 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ifcif 4086   {cpr 4179   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888   Fincfn 7955   0cc0 9936   1c1 9937   prod_cprod 14635  𝟭cind 30072

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  df-ind 30073

This theorem is referenced by:  hashreprin  30698