Theorem ltmul1 8928

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 8927	. . 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 8916	. . . 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 8362	. . . . . . . . 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 8362	. . . . . . . . 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 8927	. . . . . . . 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 8360	. . . . . . . . 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 8360	. . . . . . . . 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 8360	. . . . . . . . 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 8355	. . . . . . 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 8360	. . . . . . . . 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 8355	. . . . . . 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 4162	. . . . . 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 6104	. . . . . 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 6104	. . . . . 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 4162	. . . . 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 8349	. . . . 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 8349	. . . . 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 4162	. . . 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4131  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   1c1 8186   x. cmul 8190   < clt 8366

This proof depends on 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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-sub 8506  df-neg 8507

This theorem is used by:  lemul1  8929  reapmul1lem  8930  ltmul2  9194  ltdiv1  9206  ltdiv23  9230  recp1lt1  9237  ltmul1i  9258  ltmul1d  10156  mertenslemi1  12327  flodddiv4t2lthalf  12731  qnumgt0  13003  4sqlem12  13210  tangtx  16065  pellexlem2  16230  lgsquadlem1  16401  lgsquadlem2  16402