Theorem prod1 14674

Description: Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.)

Assertion

Ref	Expression
prod1	|- ( ( A C_ ( ZZ>= ` M ) \/ A e. Fin ) -> prod_ k e. A 1 = 1 )

Distinct variable groups:   A, k   k, M

Proof of Theorem prod1

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		simpr 477	. . . 4 |- ( ( A C_ ( ZZ>= ` M ) /\ M e. ZZ ) -> M e. ZZ )
3		ax-1ne0 10005	. . . . 5 |- 1 =/= 0
4	3	a1i 11	. . . 4 |- ( ( A C_ ( ZZ>= ` M ) /\ M e. ZZ ) -> 1 =/= 0 )
5	1	prodfclim1 14625	. . . . 5 |- ( M e. ZZ -> seq M ( x. , ( ( ZZ>= ` M ) X. { 1 } ) ) ~~> 1 )
6	5	adantl 482	. . . 4 |- ( ( A C_ ( ZZ>= ` M ) /\ M e. ZZ ) -> seq M ( x. , ( ( ZZ>= ` M ) X. { 1 } ) ) ~~> 1 )
7		simpl 473	. . . 4 |- ( ( A C_ ( ZZ>= ` M ) /\ M e. ZZ ) -> A C_ ( ZZ>= ` M ) )
8		1ex 10035	. . . . . . 7 |- 1 e. _V
9	8	fvconst2 6469	. . . . . 6 |- ( k e. ( ZZ>= ` M ) -> ( ( ( ZZ>= ` M ) X. { 1 } ) ` k ) = 1 )
10		ifid 4125	. . . . . 6 |- if ( k e. A , 1 , 1 ) = 1
11	9, 10	syl6eqr 2674	. . . . 5 |- ( k e. ( ZZ>= ` M ) -> ( ( ( ZZ>= ` M ) X. { 1 } ) ` k ) = if ( k e. A , 1 , 1 ) )
12	11	adantl 482	. . . 4 |- ( ( ( A C_ ( ZZ>= ` M ) /\ M e. ZZ ) /\ k e. ( ZZ>= ` M ) ) -> ( ( ( ZZ>= ` M ) X. { 1 } ) ` k ) = if ( k e. A , 1 , 1 ) )
13		1cnd 10056	. . . 4 |- ( ( ( A C_ ( ZZ>= ` M ) /\ M e. ZZ ) /\ k e. A ) -> 1 e. CC )
14	1, 2, 4, 6, 7, 12, 13	zprodn0 14669	. . 3 |- ( ( A C_ ( ZZ>= ` M ) /\ M e. ZZ ) -> prod_ k e. A 1 = 1 )
15		uzf 11690	. . . . . . . . 9 |- ZZ>= : ZZ --> ~P ZZ
16	15	fdmi 6052	. . . . . . . 8 |- dom ZZ>= = ZZ
17	16	eleq2i 2693	. . . . . . 7 |- ( M e. dom ZZ>= <-> M e. ZZ )
18		ndmfv 6218	. . . . . . 7 |- ( -. M e. dom ZZ>= -> ( ZZ>= ` M ) = (/) )
19	17, 18	sylnbir 321	. . . . . 6 |- ( -. M e. ZZ -> ( ZZ>= ` M ) = (/) )
20	19	sseq2d 3633	. . . . 5 |- ( -. M e. ZZ -> ( A C_ ( ZZ>= ` M ) <-> A C_ (/) ) )
21	20	biimpac 503	. . . 4 |- ( ( A C_ ( ZZ>= ` M ) /\ -. M e. ZZ ) -> A C_ (/) )
22		ss0 3974	. . . 4 |- ( A C_ (/) -> A = (/) )
23		prodeq1 14639	. . . . 5 |- ( A = (/) -> prod_ k e. A 1 = prod_ k e. (/) 1 )
24		prod0 14673	. . . . 5 |- prod_ k e. (/) 1 = 1
25	23, 24	syl6eq 2672	. . . 4 |- ( A = (/) -> prod_ k e. A 1 = 1 )
26	21, 22, 25	3syl 18	. . 3 |- ( ( A C_ ( ZZ>= ` M ) /\ -. M e. ZZ ) -> prod_ k e. A 1 = 1 )
27	14, 26	pm2.61dan 832	. 2 |- ( A C_ ( ZZ>= ` M ) -> prod_ k e. A 1 = 1 )
28		fz1f1o 14441	. . 3 |- ( A e. Fin -> ( A = (/) \/ ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) )
29		eqidd 2623	. . . . . . . . 9 |- ( k = ( f ` j ) -> 1 = 1 )
30		simpl 473	. . . . . . . . 9 |- ( ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) -> ( # ` A ) e. NN )
31		simpr 477	. . . . . . . . 9 |- ( ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) -> f : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
32		1cnd 10056	. . . . . . . . 9 |- ( ( ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) /\ k e. A ) -> 1 e. CC )
33		elfznn 12370	. . . . . . . . . . 11 |- ( j e. ( 1 ... ( # ` A ) ) -> j e. NN )
34	8	fvconst2 6469	. . . . . . . . . . 11 |- ( j e. NN -> ( ( NN X. { 1 } ) ` j ) = 1 )
35	33, 34	syl 17	. . . . . . . . . 10 |- ( j e. ( 1 ... ( # ` A ) ) -> ( ( NN X. { 1 } ) ` j ) = 1 )
36	35	adantl 482	. . . . . . . . 9 |- ( ( ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) /\ j e. ( 1 ... ( # ` A ) ) ) -> ( ( NN X. { 1 } ) ` j ) = 1 )
37	29, 30, 31, 32, 36	fprod 14671	. . . . . . . 8 |- ( ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) -> prod_ k e. A 1 = ( seq 1 ( x. , ( NN X. { 1 } ) ) ` ( # ` A ) ) )
38		nnuz 11723	. . . . . . . . . 10 |- NN = ( ZZ>= ` 1 )
39	38	prodf1 14623	. . . . . . . . 9 |- ( ( # ` A ) e. NN -> ( seq 1 ( x. , ( NN X. { 1 } ) ) ` ( # ` A ) ) = 1 )
40	39	adantr 481	. . . . . . . 8 |- ( ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) -> ( seq 1 ( x. , ( NN X. { 1 } ) ) ` ( # ` A ) ) = 1 )
41	37, 40	eqtrd 2656	. . . . . . 7 |- ( ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) -> prod_ k e. A 1 = 1 )
42	41	ex 450	. . . . . 6 |- ( ( # ` A ) e. NN -> ( f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> prod_ k e. A 1 = 1 ) )
43	42	exlimdv 1861	. . . . 5 |- ( ( # ` A ) e. NN -> ( E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> prod_ k e. A 1 = 1 ) )
44	43	imp 445	. . . 4 |- ( ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) -> prod_ k e. A 1 = 1 )
45	25, 44	jaoi 394	. . 3 |- ( ( A = (/) \/ ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> prod_ k e. A 1 = 1 )
46	28, 45	syl 17	. 2 |- ( A e. Fin -> prod_ k e. A 1 = 1 )
47	27, 46	jaoi 394	1 |- ( ( A C_ ( ZZ>= ` M ) \/ A e. Fin ) -> prod_ k e. A 1 = 1 )

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   ifcif 4086   ~Pcpw 4158   {csn 4177   class class class wbr 4653   X. cxp 5112   dom cdm 5114   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117   ~~> cli 14215   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-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:  fprodex01  29571  etransclem35  40486