Theorem mulgt1 9137

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 8400	. . . . . . . . 9 |- 0 < 1
4		0re 8274	. . . . . . . . . 10 |- 0 e. RR
5		1re 8273	. . . . . . . . . 10 |- 1 e. RR
6		lttr 8347	. . . . . . . . . 10 |- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
7	4, 5, 6	mp3an12 1364	. . . . . . . . 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 9130	. . . . . . . . . . 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 1361	. . . . . . . . 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 8234	. . . . . . 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 4119	. . . . 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 8255	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
23		lttr 8347	. . . . 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 1361	. . . 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 1005   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   RRcr 8126   0cc0 8127   1c1 8128   x. cmul 8132   < clt 8308

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-lttrn 8241  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-sub 8446  df-neg 8447

This theorem is referenced by:  mulgt1d  9210  addltmul  9475  uz2mulcl  9940