Theorem fprodge1 11809

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 8090	. 2 |- 1 e. RR*
2		pnfxr 8084	. 2 |- +oo e. RR*
3		fprodge1.ph	. . 3 |- F/ k ph
4		1re 8030	. . . . . 6 |- 1 e. RR
5		icossre 10034	. . . . . 6 |- ( ( 1 e. RR /\ +oo e. RR* ) -> ( 1 [,) +oo ) C_ RR )
6	4, 2, 5	mp2an 426	. . . . 5 |- ( 1 [,) +oo ) C_ RR
7		ax-resscn 7976	. . . . 5 |- RR C_ CC
8	6, 7	sstri 3193	. . . 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 3180	. . . . . . . 8 |- ( x e. ( 1 [,) +oo ) -> x e. RR )
13	12	adantr 276	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> x e. RR )
14	6	sseli 3180	. . . . . . . 8 |- ( y e. ( 1 [,) +oo ) -> y e. RR )
15	14	adantl 277	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> y e. RR )
16	13, 15	remulcld 8062	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR )
17	16	rexrd 8081	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR* )
18		1t1e1 9148	. . . . . 6 |- ( 1 x. 1 ) = 1
19	4	a1i 9	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 e. RR )
20		0le1 8513	. . . . . . . 8 |- 0 <_ 1
21	20	a1i 9	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 0 <_ 1 )
22		icogelb 10360	. . . . . . . . 9 |- ( ( 1 e. RR* /\ +oo e. RR* /\ x e. ( 1 [,) +oo ) ) -> 1 <_ x )
23	1, 2, 22	mp3an12 1338	. . . . . . . 8 |- ( x e. ( 1 [,) +oo ) -> 1 <_ x )
24	23	adantr 276	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ x )
25		icogelb 10360	. . . . . . . . 9 |- ( ( 1 e. RR* /\ +oo e. RR* /\ y e. ( 1 [,) +oo ) ) -> 1 <_ y )
26	1, 2, 25	mp3an12 1338	. . . . . . . 8 |- ( y e. ( 1 [,) +oo ) -> 1 <_ y )
27	26	adantl 277	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ y )
28	19, 13, 19, 15, 21, 21, 24, 27	lemul12ad 8974	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( 1 x. 1 ) <_ ( x x. y ) )
29	18, 28	eqbrtrrid 4070	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ ( x x. y ) )
30	16	ltpnfd 9861	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) < +oo )
31	10, 11, 17, 29, 30	elicod 10359	. . . 4 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. ( 1 [,) +oo ) )
32	31	adantl 277	. . 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 8081	. . . 4 |- ( ( ph /\ k e. A ) -> B e. RR* )
38		fprodge1.ge	. . . 4 |- ( ( ph /\ k e. A ) -> 1 <_ B )
39	36	ltpnfd 9861	. . . 4 |- ( ( ph /\ k e. A ) -> B < +oo )
40	34, 35, 37, 38, 39	elicod 10359	. . 3 |- ( ( ph /\ k e. A ) -> B e. ( 1 [,) +oo ) )
41		1le1 8604	. . . . 5 |- 1 <_ 1
42		ltpnf 9860	. . . . . 6 |- ( 1 e. RR -> 1 < +oo )
43	4, 42	ax-mp 5	. . . . 5 |- 1 < +oo
44		elico2 10017	. . . . . 6 |- ( ( 1 e. RR /\ +oo e. RR* ) -> ( 1 e. ( 1 [,) +oo ) <-> ( 1 e. RR /\ 1 <_ 1 /\ 1 < +oo ) ) )
45	4, 2, 44	mp2an 426	. . . . 5 |- ( 1 e. ( 1 [,) +oo ) <-> ( 1 e. RR /\ 1 <_ 1 /\ 1 < +oo ) )
46	4, 41, 43, 45	mpbir3an 1181	. . . 4 |- 1 e. ( 1 [,) +oo )
47	46	a1i 9	. . 3 |- ( ph -> 1 e. ( 1 [,) +oo ) )
48	3, 9, 32, 33, 40, 47	fprodcllemf 11783	. 2 |- ( ph -> prod_ k e. A B e. ( 1 [,) +oo ) )
49		icogelb 10360	. 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 1352	1 |- ( ph -> 1 <_ prod_ k e. A B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   F/wnf 1474   e. wcel 2167   C_ wss 3157   class class class wbr 4034  (class class class)co 5925   Fincfn 6803   CCcc 7882   RRcr 7883   0cc0 7884   1c1 7885   x. cmul 7889   +oocpnf 8063   RR*cxr 8065   < clt 8066   <_ cle 8067   [,)cico 9970   prod_cprod 11720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7975  ax-resscn 7976  ax-1cn 7977  ax-1re 7978  ax-icn 7979  ax-addcl 7980  ax-addrcl 7981  ax-mulcl 7982  ax-mulrcl 7983  ax-addcom 7984  ax-mulcom 7985  ax-addass 7986  ax-mulass 7987  ax-distr 7988  ax-i2m1 7989  ax-0lt1 7990  ax-1rid 7991  ax-0id 7992  ax-rnegex 7993  ax-precex 7994  ax-cnre 7995  ax-pre-ltirr 7996  ax-pre-ltwlin 7997  ax-pre-lttrn 7998  ax-pre-apti 7999  ax-pre-ltadd 8000  ax-pre-mulgt0 8001  ax-pre-mulext 8002  ax-arch 8003  ax-caucvg 8004

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6202  df-2nd 6203  df-recs 6367  df-irdg 6432  df-frec 6453  df-1o 6478  df-oadd 6482  df-er 6596  df-en 6804  df-dom 6805  df-fin 6806  df-pnf 8068  df-mnf 8069  df-xr 8070  df-ltxr 8071  df-le 8072  df-sub 8204  df-neg 8205  df-reap 8607  df-ap 8614  df-div 8705  df-inn 8996  df-2 9054  df-3 9055  df-4 9056  df-n0 9255  df-z 9332  df-uz 9607  df-q 9699  df-rp 9734  df-ico 9974  df-fz 10089  df-fzo 10223  df-seqfrec 10545  df-exp 10636  df-ihash 10873  df-cj 11012  df-re 11013  df-im 11014  df-rsqrt 11168  df-abs 11169  df-clim 11449  df-proddc 11721

This theorem is referenced by: (None)