Theorem fmul01lt1 39818

Description: Given a finite multiplication of values betweeen 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
fmul01lt1.1	|- F/_ i B
fmul01lt1.2	|- F/ i ph
fmul01lt1.3	|- F/_ j A
fmul01lt1.4	|- A = seq 1 ( x. , B )
fmul01lt1.5	|- ( ph -> M e. NN )
fmul01lt1.6	|- ( ph -> B : ( 1 ... M ) --> RR )
fmul01lt1.7	|- ( ( ph /\ i e. ( 1 ... M ) ) -> 0 <_ ( B ` i ) )
fmul01lt1.8	|- ( ( ph /\ i e. ( 1 ... M ) ) -> ( B ` i ) <_ 1 )
fmul01lt1.9	|- ( ph -> E e. RR+ )
fmul01lt1.10	|- ( ph -> E. j e. ( 1 ... M ) ( B ` j ) < E )

Assertion

Ref	Expression
fmul01lt1	|- ( ph -> ( A ` M ) < E )

Distinct variable groups:   i, j, E   i, M, j   ph, j

Allowed substitution hints:   ph( i)   A( i, j)   B( i, j)

Proof of Theorem fmul01lt1

Step	Hyp	Ref	Expression
1		fmul01lt1.10	. 2 |- ( ph -> E. j e. ( 1 ... M ) ( B ` j ) < E )
2		nfv 1843	. . 3 |- F/ j ph
3		fmul01lt1.3	. . . . 5 |- F/_ j A
4		nfcv 2764	. . . . 5 |- F/_ j M
5	3, 4	nffv 6198	. . . 4 |- F/_ j ( A ` M )
6		nfcv 2764	. . . 4 |- F/_ j <
7		nfcv 2764	. . . 4 |- F/_ j E
8	5, 6, 7	nfbr 4699	. . 3 |- F/ j ( A ` M ) < E
9		fmul01lt1.1	. . . . 5 |- F/_ i B
10		fmul01lt1.2	. . . . . 6 |- F/ i ph
11		nfv 1843	. . . . . 6 |- F/ i j e. ( 1 ... M )
12		nfcv 2764	. . . . . . . 8 |- F/_ i j
13	9, 12	nffv 6198	. . . . . . 7 |- F/_ i ( B ` j )
14		nfcv 2764	. . . . . . 7 |- F/_ i <
15		nfcv 2764	. . . . . . 7 |- F/_ i E
16	13, 14, 15	nfbr 4699	. . . . . 6 |- F/ i ( B ` j ) < E
17	10, 11, 16	nf3an 1831	. . . . 5 |- F/ i ( ph /\ j e. ( 1 ... M ) /\ ( B ` j ) < E )
18		fmul01lt1.4	. . . . 5 |- A = seq 1 ( x. , B )
19		1zzd 11408	. . . . 5 |- ( ( ph /\ j e. ( 1 ... M ) /\ ( B ` j ) < E ) -> 1 e. ZZ )
20		fmul01lt1.5	. . . . . . 7 |- ( ph -> M e. NN )
21		elnnuz 11724	. . . . . . 7 |- ( M e. NN <-> M e. ( ZZ>= ` 1 ) )
22	20, 21	sylib 208	. . . . . 6 |- ( ph -> M e. ( ZZ>= ` 1 ) )
23	22	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ j e. ( 1 ... M ) /\ ( B ` j ) < E ) -> M e. ( ZZ>= ` 1 ) )
24		fmul01lt1.6	. . . . . . 7 |- ( ph -> B : ( 1 ... M ) --> RR )
25	24	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( B ` i ) e. RR )
26	25	3ad2antl1 1223	. . . . 5 |- ( ( ( ph /\ j e. ( 1 ... M ) /\ ( B ` j ) < E ) /\ i e. ( 1 ... M ) ) -> ( B ` i ) e. RR )
27		fmul01lt1.7	. . . . . 6 |- ( ( ph /\ i e. ( 1 ... M ) ) -> 0 <_ ( B ` i ) )
28	27	3ad2antl1 1223	. . . . 5 |- ( ( ( ph /\ j e. ( 1 ... M ) /\ ( B ` j ) < E ) /\ i e. ( 1 ... M ) ) -> 0 <_ ( B ` i ) )
29		fmul01lt1.8	. . . . . 6 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( B ` i ) <_ 1 )
30	29	3ad2antl1 1223	. . . . 5 |- ( ( ( ph /\ j e. ( 1 ... M ) /\ ( B ` j ) < E ) /\ i e. ( 1 ... M ) ) -> ( B ` i ) <_ 1 )
31		fmul01lt1.9	. . . . . 6 |- ( ph -> E e. RR+ )
32	31	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ j e. ( 1 ... M ) /\ ( B ` j ) < E ) -> E e. RR+ )
33		simp2 1062	. . . . 5 |- ( ( ph /\ j e. ( 1 ... M ) /\ ( B ` j ) < E ) -> j e. ( 1 ... M ) )
34		simp3 1063	. . . . 5 |- ( ( ph /\ j e. ( 1 ... M ) /\ ( B ` j ) < E ) -> ( B ` j ) < E )
35	9, 17, 18, 19, 23, 26, 28, 30, 32, 33, 34	fmul01lt1lem2 39817	. . . 4 |- ( ( ph /\ j e. ( 1 ... M ) /\ ( B ` j ) < E ) -> ( A ` M ) < E )
36	35	3exp 1264	. . 3 |- ( ph -> ( j e. ( 1 ... M ) -> ( ( B ` j ) < E -> ( A ` M ) < E ) ) )
37	2, 8, 36	rexlimd 3026	. 2 |- ( ph -> ( E. j e. ( 1 ... M ) ( B ` j ) < E -> ( A ` M ) < E ) )
38	1, 37	mpd 15	1 |- ( ph -> ( A ` M ) < E )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   E.wrex 2913   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   NNcn 11020   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326   seqcseq 12801

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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802

This theorem is referenced by:  stoweidlem48  40265