Theorem fprodge1 11529

Description: If all of the terms of a finite product are greater than or equal to 1, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
fprodge1.ph	|- F/ k ph
fprodge1.a	|- ( ph -> A e. Fin )
fprodge1.b	|- ( ( ph /\ k e. A ) -> B e. RR )
fprodge1.ge	|- ( ( ph /\ k e. A ) -> 1 <_ B )

Assertion

Ref	Expression
fprodge1	|- ( ph -> 1 <_ prod_ k e. A B )

Distinct variable group:   A, k

Allowed substitution hints:   ph( k)   B( k)

Proof of Theorem fprodge1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1xr 7930	. 2 |- 1 e. RR*
2		pnfxr 7924	. 2 |- +oo e. RR*
3		fprodge1.ph	. . 3 |- F/ k ph
4		1re 7871	. . . . . 6 |- 1 e. RR
5		icossre 9851	. . . . . 6 |- ( ( 1 e. RR /\ +oo e. RR* ) -> ( 1 [,) +oo ) C_ RR )
6	4, 2, 5	mp2an 423	. . . . 5 |- ( 1 [,) +oo ) C_ RR
7		ax-resscn 7818	. . . . 5 |- RR C_ CC
8	6, 7	sstri 3137	. . . 4 |- ( 1 [,) +oo ) C_ CC
9	8	a1i 9	. . 3 |- ( ph -> ( 1 [,) +oo ) C_ CC )
10	1	a1i 9	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 e. RR* )
11	2	a1i 9	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> +oo e. RR* )
12	6	sseli 3124	. . . . . . . 8 |- ( x e. ( 1 [,) +oo ) -> x e. RR )
13	12	adantr 274	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> x e. RR )
14	6	sseli 3124	. . . . . . . 8 |- ( y e. ( 1 [,) +oo ) -> y e. RR )
15	14	adantl 275	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> y e. RR )
16	13, 15	remulcld 7902	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR )
17	16	rexrd 7921	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR* )
18		1t1e1 8979	. . . . . 6 |- ( 1 x. 1 ) = 1
19	4	a1i 9	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 e. RR )
20		0le1 8350	. . . . . . . 8 |- 0 <_ 1
21	20	a1i 9	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 0 <_ 1 )
22		icogelb 10158	. . . . . . . . 9 |- ( ( 1 e. RR* /\ +oo e. RR* /\ x e. ( 1 [,) +oo ) ) -> 1 <_ x )
23	1, 2, 22	mp3an12 1309	. . . . . . . 8 |- ( x e. ( 1 [,) +oo ) -> 1 <_ x )
24	23	adantr 274	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ x )
25		icogelb 10158	. . . . . . . . 9 |- ( ( 1 e. RR* /\ +oo e. RR* /\ y e. ( 1 [,) +oo ) ) -> 1 <_ y )
26	1, 2, 25	mp3an12 1309	. . . . . . . 8 |- ( y e. ( 1 [,) +oo ) -> 1 <_ y )
27	26	adantl 275	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ y )
28	19, 13, 19, 15, 21, 21, 24, 27	lemul12ad 8807	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( 1 x. 1 ) <_ ( x x. y ) )
29	18, 28	eqbrtrrid 4000	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ ( x x. y ) )
30	16	ltpnfd 9681	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) < +oo )
31	10, 11, 17, 29, 30	elicod 10157	. . . 4 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. ( 1 [,) +oo ) )
32	31	adantl 275	. . 3 |- ( ( ph /\ ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) ) -> ( x x. y ) e. ( 1 [,) +oo ) )
33		fprodge1.a	. . 3 |- ( ph -> A e. Fin )
34	1	a1i 9	. . . 4 |- ( ( ph /\ k e. A ) -> 1 e. RR* )
35	2	a1i 9	. . . 4 |- ( ( ph /\ k e. A ) -> +oo e. RR* )
36		fprodge1.b	. . . . 5 |- ( ( ph /\ k e. A ) -> B e. RR )
37	36	rexrd 7921	. . . 4 |- ( ( ph /\ k e. A ) -> B e. RR* )
38		fprodge1.ge	. . . 4 |- ( ( ph /\ k e. A ) -> 1 <_ B )
39	36	ltpnfd 9681	. . . 4 |- ( ( ph /\ k e. A ) -> B < +oo )
40	34, 35, 37, 38, 39	elicod 10157	. . 3 |- ( ( ph /\ k e. A ) -> B e. ( 1 [,) +oo ) )
41		1le1 8441	. . . . 5 |- 1 <_ 1
42		ltpnf 9680	. . . . . 6 |- ( 1 e. RR -> 1 < +oo )
43	4, 42	ax-mp 5	. . . . 5 |- 1 < +oo
44		elico2 9834	. . . . . 6 |- ( ( 1 e. RR /\ +oo e. RR* ) -> ( 1 e. ( 1 [,) +oo ) <-> ( 1 e. RR /\ 1 <_ 1 /\ 1 < +oo ) ) )
45	4, 2, 44	mp2an 423	. . . . 5 |- ( 1 e. ( 1 [,) +oo ) <-> ( 1 e. RR /\ 1 <_ 1 /\ 1 < +oo ) )
46	4, 41, 43, 45	mpbir3an 1164	. . . 4 |- 1 e. ( 1 [,) +oo )
47	46	a1i 9	. . 3 |- ( ph -> 1 e. ( 1 [,) +oo ) )
48	3, 9, 32, 33, 40, 47	fprodcllemf 11503	. 2 |- ( ph -> prod_ k e. A B e. ( 1 [,) +oo ) )
49		icogelb 10158	. 2 |- ( ( 1 e. RR* /\ +oo e. RR* /\ prod_ k e. A B e. ( 1 [,) +oo ) ) -> 1 <_ prod_ k e. A B )
50	1, 2, 48, 49	mp3an12i 1323	1 |- ( ph -> 1 <_ prod_ k e. A B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   F/wnf 1440   e. wcel 2128   C_ wss 3102   class class class wbr 3965  (class class class)co 5821   Fincfn 6682   CCcc 7724   RRcr 7725   0cc0 7726   1c1 7727   x. cmul 7731   +oocpnf 7903   RR*cxr 7905   < clt 7906   <_ cle 7907   [,)cico 9787   prod_cprod 11440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-mulrcl 7825  ax-addcom 7826  ax-mulcom 7827  ax-addass 7828  ax-mulass 7829  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-1rid 7833  ax-0id 7834  ax-rnegex 7835  ax-precex 7836  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-apti 7841  ax-pre-ltadd 7842  ax-pre-mulgt0 7843  ax-pre-mulext 7844  ax-arch 7845  ax-caucvg 7846

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-isom 5178  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-frec 6335  df-1o 6360  df-oadd 6364  df-er 6477  df-en 6683  df-dom 6684  df-fin 6685  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-reap 8444  df-ap 8451  df-div 8540  df-inn 8828  df-2 8886  df-3 8887  df-4 8888  df-n0 9085  df-z 9162  df-uz 9434  df-q 9522  df-rp 9554  df-ico 9791  df-fz 9906  df-fzo 10035  df-seqfrec 10338  df-exp 10412  df-ihash 10643  df-cj 10735  df-re 10736  df-im 10737  df-rsqrt 10891  df-abs 10892  df-clim 11169  df-proddc 11441

This theorem is referenced by: (None)