Theorem fprodge1 11646

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 8015	. 2 |- 1 e. RR*
2		pnfxr 8009	. 2 |- +oo e. RR*
3		fprodge1.ph	. . 3 |- F/ k ph
4		1re 7955	. . . . . 6 |- 1 e. RR
5		icossre 9953	. . . . . 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 7902	. . . . 5 |- RR C_ CC
8	6, 7	sstri 3164	. . . 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 3151	. . . . . . . 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 3151	. . . . . . . 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 7987	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR )
17	16	rexrd 8006	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR* )
18		1t1e1 9070	. . . . . 6 |- ( 1 x. 1 ) = 1
19	4	a1i 9	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 e. RR )
20		0le1 8437	. . . . . . . 8 |- 0 <_ 1
21	20	a1i 9	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 0 <_ 1 )
22		icogelb 10265	. . . . . . . . 9 |- ( ( 1 e. RR* /\ +oo e. RR* /\ x e. ( 1 [,) +oo ) ) -> 1 <_ x )
23	1, 2, 22	mp3an12 1327	. . . . . . . 8 |- ( x e. ( 1 [,) +oo ) -> 1 <_ x )
24	23	adantr 276	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ x )
25		icogelb 10265	. . . . . . . . 9 |- ( ( 1 e. RR* /\ +oo e. RR* /\ y e. ( 1 [,) +oo ) ) -> 1 <_ y )
26	1, 2, 25	mp3an12 1327	. . . . . . . 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 8898	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( 1 x. 1 ) <_ ( x x. y ) )
29	18, 28	eqbrtrrid 4039	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ ( x x. y ) )
30	16	ltpnfd 9780	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) < +oo )
31	10, 11, 17, 29, 30	elicod 10264	. . . 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 8006	. . . 4 |- ( ( ph /\ k e. A ) -> B e. RR* )
38		fprodge1.ge	. . . 4 |- ( ( ph /\ k e. A ) -> 1 <_ B )
39	36	ltpnfd 9780	. . . 4 |- ( ( ph /\ k e. A ) -> B < +oo )
40	34, 35, 37, 38, 39	elicod 10264	. . 3 |- ( ( ph /\ k e. A ) -> B e. ( 1 [,) +oo ) )
41		1le1 8528	. . . . 5 |- 1 <_ 1
42		ltpnf 9779	. . . . . 6 |- ( 1 e. RR -> 1 < +oo )
43	4, 42	ax-mp 5	. . . . 5 |- 1 < +oo
44		elico2 9936	. . . . . 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 1179	. . . 4 |- 1 e. ( 1 [,) +oo )
47	46	a1i 9	. . 3 |- ( ph -> 1 e. ( 1 [,) +oo ) )
48	3, 9, 32, 33, 40, 47	fprodcllemf 11620	. 2 |- ( ph -> prod_ k e. A B e. ( 1 [,) +oo ) )
49		icogelb 10265	. 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 1341	1 |- ( ph -> 1 <_ prod_ k e. A B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   F/wnf 1460   e. wcel 2148   C_ wss 3129   class class class wbr 4003  (class class class)co 5874   Fincfn 6739   CCcc 7808   RRcr 7809   0cc0 7810   1c1 7811   x. cmul 7815   +oocpnf 7988   RR*cxr 7990   < clt 7991   <_ cle 7992   [,)cico 9889   prod_cprod 11557

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-ico 9893  df-fz 10008  df-fzo 10142  df-seqfrec 10445  df-exp 10519  df-ihash 10755  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-clim 11286  df-proddc 11558

This theorem is referenced by: (None)