Theorem mulgt1 8823

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 109	. . . . 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 8087	. . . . . . . . 9 |- 0 < 1
4		0re 7960	. . . . . . . . . 10 |- 0 e. RR
5		1re 7959	. . . . . . . . . 10 |- 1 e. RR
6		lttr 8034	. . . . . . . . . 10 |- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
7	4, 5, 6	mp3an12 1327	. . . . . . . . 9 |- ( A e. RR -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
8	3, 7	mpani 430	. . . . . . . 8 |- ( A e. RR -> ( 1 < A -> 0 < A ) )
9	8	adantr 276	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < A -> 0 < A ) )
10		ltmul2 8816	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B <-> ( A x. 1 ) < ( A x. B ) ) )
11	10	biimpd 144	. . . . . . . . . 10 |- ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) )
12	5, 11	mp3an1 1324	. . . . . . . . 9 |- ( ( B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) )
13	12	exp32 365	. . . . . . . 8 |- ( B e. RR -> ( A e. RR -> ( 0 < A -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) ) ) )
14	13	impcom 125	. . . . . . 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 254	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> ( A x. 1 ) < ( A x. B ) ) )
17		ax-1rid 7921	. . . . . . 7 |- ( A e. RR -> ( A x. 1 ) = A )
18	17	adantr 276	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A x. 1 ) = A )
19	18	breq1d 4015	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) < ( A x. B ) <-> A < ( A x. B ) ) )
20	16, 19	sylibd 149	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> A < ( A x. B ) ) )
21	2, 20	jcad 307	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> ( 1 < A /\ A < ( A x. B ) ) ) )
22		remulcl 7942	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
23		lttr 8034	. . . . 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 1324	. . . 4 |- ( ( A e. RR /\ ( A x. B ) e. RR ) -> ( ( 1 < A /\ A < ( A x. B ) ) -> 1 < ( A x. B ) ) )
25	22, 24	syldan 282	. . 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 124	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < ( A x. B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5878   RRcr 7813   0cc0 7814   1c1 7815   x. cmul 7819   < clt 7995

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-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-lttrn 7928  ax-pre-ltadd 7930  ax-pre-mulgt0 7931

This theorem depends on definitions:  df-bi 117  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-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-ltxr 8000  df-sub 8133  df-neg 8134

This theorem is referenced by:  mulgt1d  8896  addltmul  9158  uz2mulcl  9611