Theorem mulgt1 8734

Description: The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)

Assertion

Ref	Expression
mulgt1	|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < ( A x. B ) )

Proof of Theorem mulgt1

Step	Hyp	Ref	Expression
1		simpl 108	. . . . 5 |- ( ( 1 < A /\ 1 < B ) -> 1 < A )
2	1	a1i 9	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> 1 < A ) )
3		0lt1 8002	. . . . . . . . 9 |- 0 < 1
4		0re 7878	. . . . . . . . . 10 |- 0 e. RR
5		1re 7877	. . . . . . . . . 10 |- 1 e. RR
6		lttr 7951	. . . . . . . . . 10 |- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
7	4, 5, 6	mp3an12 1309	. . . . . . . . 9 |- ( A e. RR -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
8	3, 7	mpani 427	. . . . . . . 8 |- ( A e. RR -> ( 1 < A -> 0 < A ) )
9	8	adantr 274	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < A -> 0 < A ) )
10		ltmul2 8727	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B <-> ( A x. 1 ) < ( A x. B ) ) )
11	10	biimpd 143	. . . . . . . . . 10 |- ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) )
12	5, 11	mp3an1 1306	. . . . . . . . 9 |- ( ( B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) )
13	12	exp32 363	. . . . . . . 8 |- ( B e. RR -> ( A e. RR -> ( 0 < A -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) ) ) )
14	13	impcom 124	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) ) )
15	9, 14	syld 45	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < A -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) ) )
16	15	impd 252	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> ( A x. 1 ) < ( A x. B ) ) )
17		ax-1rid 7839	. . . . . . 7 |- ( A e. RR -> ( A x. 1 ) = A )
18	17	adantr 274	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A x. 1 ) = A )
19	18	breq1d 3975	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) < ( A x. B ) <-> A < ( A x. B ) ) )
20	16, 19	sylibd 148	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> A < ( A x. B ) ) )
21	2, 20	jcad 305	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> ( 1 < A /\ A < ( A x. B ) ) ) )
22		remulcl 7860	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
23		lttr 7951	. . . . 5 |- ( ( 1 e. RR /\ A e. RR /\ ( A x. B ) e. RR ) -> ( ( 1 < A /\ A < ( A x. B ) ) -> 1 < ( A x. B ) ) )
24	5, 23	mp3an1 1306	. . . 4 |- ( ( A e. RR /\ ( A x. B ) e. RR ) -> ( ( 1 < A /\ A < ( A x. B ) ) -> 1 < ( A x. B ) ) )
25	22, 24	syldan 280	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ A < ( A x. B ) ) -> 1 < ( A x. B ) ) )
26	21, 25	syld 45	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> 1 < ( A x. B ) ) )
27	26	imp 123	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < ( A x. B ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1335   e. wcel 2128   class class class wbr 3965  (class class class)co 5824   RRcr 7731   0cc0 7732   1c1 7733   x. cmul 7737   < clt 7912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-lttrn 7846  ax-pre-ltadd 7848  ax-pre-mulgt0 7849

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-ltxr 7917  df-sub 8048  df-neg 8049

This theorem is referenced by:  mulgt1d  8807  addltmul  9069  uz2mulcl  9519