Theorem ltmul1 8910

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 8909	. . 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 8898	. . . 4 |- ( ( C e. RR /\ 0 < C ) -> E. x e. RR ( 0 < x /\ ( C x. x ) = 1 ) )
4	3	3ad2ant3 1051	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> E. x e. RR ( 0 < x /\ ( C x. x ) = 1 ) )
5		simpl1 1031	. . . . . . . . . 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 1083	. . . . . . . . . 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 8346	. . . . . . . . 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 1032	. . . . . . . . . 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 8346	. . . . . . . . 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 535	. . . . . . . . . 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 545	. . . . . . . . . 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 1208	. . . . . . . 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 8909	. . . . . . . 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 8344	. . . . . . . . 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 8344	. . . . . . . . 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 8344	. . . . . . . . 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 8339	. . . . . . 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 8344	. . . . . . . . 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 8339	. . . . . . 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 4156	. . . . . 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 546	. . . . . . . 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 6091	. . . . . 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 6091	. . . . . 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 4156	. . . . 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 8333	. . . . 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 8333	. . . . 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 4156	. . . 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 2673	. 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 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   x. cmul 8174   < clt 8350

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-sub 8489  df-neg 8490

This theorem is referenced by:  lemul1  8911  reapmul1lem  8912  ltmul2  9176  ltdiv1  9188  ltdiv23  9212  recp1lt1  9219  ltmul1i  9240  ltmul1d  10118  mertenslemi1  12280  flodddiv4t2lthalf  12684  qnumgt0  12954  4sqlem12  13159  tangtx  15862  pellexlem2  16006  lgsquadlem1  16110  lgsquadlem2  16111