Theorem ltmul1 8866

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 8865	. . 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 8854	. . . 4 |- ( ( C e. RR /\ 0 < C ) -> E. x e. RR ( 0 < x /\ ( C x. x ) = 1 ) )
4	3	3ad2ant3 1047	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> E. x e. RR ( 0 < x /\ ( C x. x ) = 1 ) )
5		simpl1 1027	. . . . . . . . . 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 1079	. . . . . . . . . 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 8304	. . . . . . . . 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 1028	. . . . . . . . . 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 8304	. . . . . . . . 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 531	. . . . . . . . . 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 541	. . . . . . . . . 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 1204	. . . . . . . 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 8865	. . . . . . . 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 8302	. . . . . . . . 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 8302	. . . . . . . . 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 8302	. . . . . . . . 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 8297	. . . . . . 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 8302	. . . . . . . . 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 8297	. . . . . . 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 4140	. . . . . 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 542	. . . . . . . 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 6066	. . . . . 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 6066	. . . . . 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 4140	. . . . 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 8291	. . . . 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 8291	. . . . 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 4140	. . . 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 2665	. 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 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4109  (class class class)co 6050   CCcc 8125   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-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  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:  lemul1  8867  reapmul1lem  8868  ltmul2  9130  ltdiv1  9142  ltdiv23  9166  recp1lt1  9173  ltmul1i  9194  ltmul1d  10071  mertenslemi1  12221  flodddiv4t2lthalf  12625  qnumgt0  12895  4sqlem12  13100  tangtx  15703  pellexlem2  15846  lgsquadlem1  15950  lgsquadlem2  15951