Theorem fprodge1 14726

Description: If all of the terms of a finite product are larger 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		1re 10039	. . . 4 |- 1 e. RR
2	1	rexri 10097	. . 3 |- 1 e. RR*
3	2	a1i 11	. 2 |- ( ph -> 1 e. RR* )
4		pnfxr 10092	. . 3 |- +oo e. RR*
5	4	a1i 11	. 2 |- ( ph -> +oo e. RR* )
6		fprodge1.ph	. . 3 |- F/ k ph
7		icossre 12254	. . . . . 6 |- ( ( 1 e. RR /\ +oo e. RR* ) -> ( 1 [,) +oo ) C_ RR )
8	1, 4, 7	mp2an 708	. . . . 5 |- ( 1 [,) +oo ) C_ RR
9		ax-resscn 9993	. . . . 5 |- RR C_ CC
10	8, 9	sstri 3612	. . . 4 |- ( 1 [,) +oo ) C_ CC
11	10	a1i 11	. . 3 |- ( ph -> ( 1 [,) +oo ) C_ CC )
12	2	a1i 11	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 e. RR* )
13	4	a1i 11	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> +oo e. RR* )
14	8	sseli 3599	. . . . . . . 8 |- ( x e. ( 1 [,) +oo ) -> x e. RR )
15	14	adantr 481	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> x e. RR )
16	8	sseli 3599	. . . . . . . 8 |- ( y e. ( 1 [,) +oo ) -> y e. RR )
17	16	adantl 482	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> y e. RR )
18	15, 17	remulcld 10070	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR )
19	18	rexrd 10089	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR* )
20		1t1e1 11175	. . . . . . . 8 |- ( 1 x. 1 ) = 1
21	20	eqcomi 2631	. . . . . . 7 |- 1 = ( 1 x. 1 )
22	21	a1i 11	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 = ( 1 x. 1 ) )
23	1	a1i 11	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 e. RR )
24		0le1 10551	. . . . . . . 8 |- 0 <_ 1
25	24	a1i 11	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 0 <_ 1 )
26	2	a1i 11	. . . . . . . . 9 |- ( x e. ( 1 [,) +oo ) -> 1 e. RR* )
27	4	a1i 11	. . . . . . . . 9 |- ( x e. ( 1 [,) +oo ) -> +oo e. RR* )
28		id 22	. . . . . . . . 9 |- ( x e. ( 1 [,) +oo ) -> x e. ( 1 [,) +oo ) )
29		icogelb 12225	. . . . . . . . 9 |- ( ( 1 e. RR* /\ +oo e. RR* /\ x e. ( 1 [,) +oo ) ) -> 1 <_ x )
30	26, 27, 28, 29	syl3anc 1326	. . . . . . . 8 |- ( x e. ( 1 [,) +oo ) -> 1 <_ x )
31	30	adantr 481	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ x )
32	2	a1i 11	. . . . . . . . 9 |- ( y e. ( 1 [,) +oo ) -> 1 e. RR* )
33	4	a1i 11	. . . . . . . . 9 |- ( y e. ( 1 [,) +oo ) -> +oo e. RR* )
34		id 22	. . . . . . . . 9 |- ( y e. ( 1 [,) +oo ) -> y e. ( 1 [,) +oo ) )
35		icogelb 12225	. . . . . . . . 9 |- ( ( 1 e. RR* /\ +oo e. RR* /\ y e. ( 1 [,) +oo ) ) -> 1 <_ y )
36	32, 33, 34, 35	syl3anc 1326	. . . . . . . 8 |- ( y e. ( 1 [,) +oo ) -> 1 <_ y )
37	36	adantl 482	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ y )
38	23, 15, 23, 17, 25, 25, 31, 37	lemul12ad 10966	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( 1 x. 1 ) <_ ( x x. y ) )
39	22, 38	eqbrtrd 4675	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ ( x x. y ) )
40	18	ltpnfd 11955	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) < +oo )
41	12, 13, 19, 39, 40	elicod 12224	. . . 4 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. ( 1 [,) +oo ) )
42	41	adantl 482	. . 3 |- ( ( ph /\ ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) ) -> ( x x. y ) e. ( 1 [,) +oo ) )
43		fprodge1.a	. . 3 |- ( ph -> A e. Fin )
44	2	a1i 11	. . . 4 |- ( ( ph /\ k e. A ) -> 1 e. RR* )
45	4	a1i 11	. . . 4 |- ( ( ph /\ k e. A ) -> +oo e. RR* )
46		fprodge1.b	. . . . 5 |- ( ( ph /\ k e. A ) -> B e. RR )
47	46	rexrd 10089	. . . 4 |- ( ( ph /\ k e. A ) -> B e. RR* )
48		fprodge1.ge	. . . 4 |- ( ( ph /\ k e. A ) -> 1 <_ B )
49	46	ltpnfd 11955	. . . 4 |- ( ( ph /\ k e. A ) -> B < +oo )
50	44, 45, 47, 48, 49	elicod 12224	. . 3 |- ( ( ph /\ k e. A ) -> B e. ( 1 [,) +oo ) )
51		1le1 10655	. . . . . 6 |- 1 <_ 1
52		ltpnf 11954	. . . . . . 7 |- ( 1 e. RR -> 1 < +oo )
53	1, 52	ax-mp 5	. . . . . 6 |- 1 < +oo
54	1, 51, 53	3pm3.2i 1239	. . . . 5 |- ( 1 e. RR /\ 1 <_ 1 /\ 1 < +oo )
55		elico2 12237	. . . . . 6 |- ( ( 1 e. RR /\ +oo e. RR* ) -> ( 1 e. ( 1 [,) +oo ) <-> ( 1 e. RR /\ 1 <_ 1 /\ 1 < +oo ) ) )
56	1, 4, 55	mp2an 708	. . . . 5 |- ( 1 e. ( 1 [,) +oo ) <-> ( 1 e. RR /\ 1 <_ 1 /\ 1 < +oo ) )
57	54, 56	mpbir 221	. . . 4 |- 1 e. ( 1 [,) +oo )
58	57	a1i 11	. . 3 |- ( ph -> 1 e. ( 1 [,) +oo ) )
59	6, 11, 42, 43, 50, 58	fprodcllemf 14688	. 2 |- ( ph -> prod_ k e. A B e. ( 1 [,) +oo ) )
60		icogelb 12225	. 2 |- ( ( 1 e. RR* /\ +oo e. RR* /\ prod_ k e. A B e. ( 1 [,) +oo ) ) -> 1 <_ prod_ k e. A B )
61	3, 5, 59, 60	syl3anc 1326	1 |- ( ph -> 1 <_ prod_ k e. A B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   [,)cico 12177   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-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-ico 12181  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:  fprodle  14727