Theorem mulgt1 8779

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 8046	. . . . . . . . 9 |- 0 < 1
4		0re 7920	. . . . . . . . . 10 |- 0 e. RR
5		1re 7919	. . . . . . . . . 10 |- 1 e. RR
6		lttr 7993	. . . . . . . . . 10 |- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
7	4, 5, 6	mp3an12 1322	. . . . . . . . 9 |- ( A e. RR -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
8	3, 7	mpani 428	. . . . . . . 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 8772	. . . . . . . . . . 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 1319	. . . . . . . . 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 7881	. . . . . . 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 3999	. . . . 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 7902	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
23		lttr 7993	. . . . 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 1319	. . . 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 973   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775   x. cmul 7779   < clt 7954

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-lttrn 7888  ax-pre-ltadd 7890  ax-pre-mulgt0 7891

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093

This theorem is referenced by:  mulgt1d  8852  addltmul  9114  uz2mulcl  9567