Theorem fprodge1 11950

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 8131	. 2 |- 1 e. RR*
2		pnfxr 8125	. 2 |- +oo e. RR*
3		fprodge1.ph	. . 3 |- F/ k ph
4		1re 8071	. . . . . 6 |- 1 e. RR
5		icossre 10076	. . . . . 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 8017	. . . . 5 |- RR C_ CC
8	6, 7	sstri 3202	. . . 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 3189	. . . . . . . 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 3189	. . . . . . . 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 8103	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR )
17	16	rexrd 8122	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) e. RR* )
18		1t1e1 9189	. . . . . 6 |- ( 1 x. 1 ) = 1
19	4	a1i 9	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 e. RR )
20		0le1 8554	. . . . . . . 8 |- 0 <_ 1
21	20	a1i 9	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 0 <_ 1 )
22		icogelb 10408	. . . . . . . . 9 |- ( ( 1 e. RR* /\ +oo e. RR* /\ x e. ( 1 [,) +oo ) ) -> 1 <_ x )
23	1, 2, 22	mp3an12 1340	. . . . . . . 8 |- ( x e. ( 1 [,) +oo ) -> 1 <_ x )
24	23	adantr 276	. . . . . . 7 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ x )
25		icogelb 10408	. . . . . . . . 9 |- ( ( 1 e. RR* /\ +oo e. RR* /\ y e. ( 1 [,) +oo ) ) -> 1 <_ y )
26	1, 2, 25	mp3an12 1340	. . . . . . . 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 9015	. . . . . 6 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( 1 x. 1 ) <_ ( x x. y ) )
29	18, 28	eqbrtrrid 4080	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> 1 <_ ( x x. y ) )
30	16	ltpnfd 9903	. . . . 5 |- ( ( x e. ( 1 [,) +oo ) /\ y e. ( 1 [,) +oo ) ) -> ( x x. y ) < +oo )
31	10, 11, 17, 29, 30	elicod 10407	. . . 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 8122	. . . 4 |- ( ( ph /\ k e. A ) -> B e. RR* )
38		fprodge1.ge	. . . 4 |- ( ( ph /\ k e. A ) -> 1 <_ B )
39	36	ltpnfd 9903	. . . 4 |- ( ( ph /\ k e. A ) -> B < +oo )
40	34, 35, 37, 38, 39	elicod 10407	. . 3 |- ( ( ph /\ k e. A ) -> B e. ( 1 [,) +oo ) )
41		1le1 8645	. . . . 5 |- 1 <_ 1
42		ltpnf 9902	. . . . . 6 |- ( 1 e. RR -> 1 < +oo )
43	4, 42	ax-mp 5	. . . . 5 |- 1 < +oo
44		elico2 10059	. . . . . 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 1182	. . . 4 |- 1 e. ( 1 [,) +oo )
47	46	a1i 9	. . 3 |- ( ph -> 1 e. ( 1 [,) +oo ) )
48	3, 9, 32, 33, 40, 47	fprodcllemf 11924	. 2 |- ( ph -> prod_ k e. A B e. ( 1 [,) +oo ) )
49		icogelb 10408	. 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 1354	1 |- ( ph -> 1 <_ prod_ k e. A B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   F/wnf 1483   e. wcel 2176   C_ wss 3166   class class class wbr 4044  (class class class)co 5944   Fincfn 6827   CCcc 7923   RRcr 7924   0cc0 7925   1c1 7926   x. cmul 7930   +oocpnf 8104   RR*cxr 8106   < clt 8107   <_ cle 8108   [,)cico 10012   prod_cprod 11861

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-frec 6477  df-1o 6502  df-oadd 6506  df-er 6620  df-en 6828  df-dom 6829  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-ico 10016  df-fz 10131  df-fzo 10265  df-seqfrec 10593  df-exp 10684  df-ihash 10921  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590  df-proddc 11862

This theorem is referenced by: (None)