Theorem fprodge1 11649

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 8018	. 2 |- 1 e. RR*
2		pnfxr 8012	. 2 |- +oo e. RR*
3		fprodge1.ph	. . 3 |- F/ k ph
4		1re 7958	. . . . . 6 |- 1 e. RR
5		icossre 9956	. . . . . 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 7905	. . . . 5 |- RR C_ CC
8	6, 7	sstri 3166	. . . 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 3153	. . . . . . . 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 3153	. . . . . . . 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 7990	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR )
17	16	rexrd 8009	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR* )
18		1t1e1 9073	. . . . . 6 |- ( 1 x. 1 ) = 1
19	4	a1i 9	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 e. RR )
20		0le1 8440	. . . . . . . 8 |- 0 <_ 1
21	20	a1i 9	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 0 <_ 1 )
22		icogelb 10268	. . . . . . . . 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 10268	. . . . . . . . 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 8901	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( 1 x. 1 ) <_ ( x x. y ) )
29	18, 28	eqbrtrrid 4041	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ ( x x. y ) )
30	16	ltpnfd 9783	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) < +oo )
31	10, 11, 17, 29, 30	elicod 10267	. . . 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 8009	. . . 4 |- ( ( ph /\ k e. A ) -> B e. RR* )
38		fprodge1.ge	. . . 4 |- ( ( ph /\ k e. A ) -> 1 <_ B )
39	36	ltpnfd 9783	. . . 4 |- ( ( ph /\ k e. A ) -> B < +oo )
40	34, 35, 37, 38, 39	elicod 10267	. . 3 |- ( ( ph /\ k e. A ) -> B e. ( 1 [,) +oo ) )
41		1le1 8531	. . . . 5 |- 1 <_ 1
42		ltpnf 9782	. . . . . 6 |- ( 1 e. RR -> 1 < +oo )
43	4, 42	ax-mp 5	. . . . 5 |- 1 < +oo
44		elico2 9939	. . . . . 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 11623	. 2 |- ( ph -> prod_ k e. A B e. ( 1 [,) +oo ) )
49		icogelb 10268	. 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 3131   class class class wbr 4005  (class class class)co 5877   Fincfn 6742   CCcc 7811   RRcr 7812   0cc0 7813   1c1 7814   x. cmul 7818   +oocpnf 7991   RR*cxr 7993   < clt 7994   <_ cle 7995   [,)cico 9892   prod_cprod 11560

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-ico 9896  df-fz 10011  df-fzo 10145  df-seqfrec 10448  df-exp 10522  df-ihash 10758  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-proddc 11561

This theorem is referenced by: (None)