Theorem ltmul1 8622

Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
ltmul1	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )

Proof of Theorem ltmul1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltmul1a 8621	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ A < B ) -> ( A x. C ) < ( B x. C ) )
2	1	ex 115	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B -> ( A x. C ) < ( B x. C ) ) )
3		recexgt0 8610	. . . 4 |- ( ( C e. RR /\ 0 < C ) -> E. x e. RR ( 0 < x /\ ( C x. x ) = 1 ) )
4	3	3ad2ant3 1022	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> E. x e. RR ( 0 < x /\ ( C x. x ) = 1 ) )
5		simpl1 1002	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> A e. RR )
6		simpl3l 1054	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> C e. RR )
7	5, 6	remulcld 8060	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> ( A x. C ) e. RR )
8		simpl2 1003	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> B e. RR )
9	8, 6	remulcld 8060	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> ( B x. C ) e. RR )
10		simprl 529	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> x e. RR )
11		simprrl 539	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> 0 < x )
12	10, 11	jca 306	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> ( x e. RR /\ 0 < x ) )
13	7, 9, 12	3jca 1179	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> ( ( A x. C ) e. RR /\ ( B x. C ) e. RR /\ ( x e. RR /\ 0 < x ) ) )
14		ltmul1a 8621	. . . . . . . 8 |- ( ( ( ( A x. C ) e. RR /\ ( B x. C ) e. RR /\ ( x e. RR /\ 0 < x ) ) /\ ( A x. C ) < ( B x. C ) ) -> ( ( A x. C ) x. x ) < ( ( B x. C ) x. x ) )
15	13, 14	sylan 283	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> ( ( A x. C ) x. x ) < ( ( B x. C ) x. x ) )
16	5	recnd 8058	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> A e. CC )
17	16	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> A e. CC )
18	6	recnd 8058	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> C e. CC )
19	18	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> C e. CC )
20	10	recnd 8058	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> x e. CC )
21	20	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> x e. CC )
22	17, 19, 21	mulassd 8053	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> ( ( A x. C ) x. x ) = ( A x. ( C x. x ) ) )
23	8	recnd 8058	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> B e. CC )
24	23	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> B e. CC )
25	24, 19, 21	mulassd 8053	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> ( ( B x. C ) x. x ) = ( B x. ( C x. x ) ) )
26	15, 22, 25	3brtr3d 4065	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> ( A x. ( C x. x ) ) < ( B x. ( C x. x ) ) )
27		simprrr 540	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> ( C x. x ) = 1 )
28	27	adantr 276	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> ( C x. x ) = 1 )
29	28	oveq2d 5939	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> ( A x. ( C x. x ) ) = ( A x. 1 ) )
30	28	oveq2d 5939	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> ( B x. ( C x. x ) ) = ( B x. 1 ) )
31	26, 29, 30	3brtr3d 4065	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> ( A x. 1 ) < ( B x. 1 ) )
32	17	mulridd 8046	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> ( A x. 1 ) = A )
33	24	mulridd 8046	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> ( B x. 1 ) = B )
34	31, 32, 33	3brtr3d 4065	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) /\ ( A x. C ) < ( B x. C ) ) -> A < B )
35	34	ex 115	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ ( x e. RR /\ ( 0 < x /\ ( C x. x ) = 1 ) ) ) -> ( ( A x. C ) < ( B x. C ) -> A < B ) )
36	4, 35	rexlimddv 2619	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) < ( B x. C ) -> A < B ) )
37	2, 36	impbid 129	1 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   E.wrex 2476   class class class wbr 4034  (class class class)co 5923   CCcc 7880   RRcr 7881   0cc0 7882   1c1 7883   x. cmul 7887   < clt 8064

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7973  ax-resscn 7974  ax-1cn 7975  ax-1re 7976  ax-icn 7977  ax-addcl 7978  ax-addrcl 7979  ax-mulcl 7980  ax-mulrcl 7981  ax-addcom 7982  ax-mulcom 7983  ax-addass 7984  ax-mulass 7985  ax-distr 7986  ax-i2m1 7987  ax-1rid 7989  ax-0id 7990  ax-rnegex 7991  ax-precex 7992  ax-cnre 7993  ax-pre-ltadd 7998  ax-pre-mulgt0 7999

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-pnf 8066  df-mnf 8067  df-ltxr 8069  df-sub 8202  df-neg 8203

This theorem is referenced by:  lemul1  8623  reapmul1lem  8624  ltmul2  8886  ltdiv1  8898  ltdiv23  8922  recp1lt1  8929  ltmul1i  8950  ltmul1d  9816  mertenslemi1  11703  flodddiv4t2lthalf  12107  qnumgt0  12377  4sqlem12  12582  tangtx  15100  lgsquadlem1  15344  lgsquadlem2  15345