Theorem ltmul1 8762

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 8761	. . 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 8750	. . . 4 |- ( ( C e. RR /\ 0 < C ) -> E. x e. RR ( 0 < x /\ ( C x. x ) = 1 ) )
4	3	3ad2ant3 1044	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> E. x e. RR ( 0 < x /\ ( C x. x ) = 1 ) )
5		simpl1 1024	. . . . . . . . . 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 1076	. . . . . . . . . 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 8200	. . . . . . . . 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 1025	. . . . . . . . . 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 8200	. . . . . . . . 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 1201	. . . . . . . 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 8761	. . . . . . . 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 8198	. . . . . . . . 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 8198	. . . . . . . . 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 8198	. . . . . . . . 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 8193	. . . . . . 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 8198	. . . . . . . . 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 8193	. . . . . . 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 4117	. . . . . 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 6029	. . . . . 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 6029	. . . . . 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 4117	. . . . 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 8186	. . . . 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 8186	. . . . 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 4117	. . . 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 2653	. 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 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4086  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   1c1 8023   x. cmul 8027   < clt 8204

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltadd 8138  ax-pre-mulgt0 8139

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-sub 8342  df-neg 8343

This theorem is referenced by:  lemul1  8763  reapmul1lem  8764  ltmul2  9026  ltdiv1  9038  ltdiv23  9062  recp1lt1  9069  ltmul1i  9090  ltmul1d  9963  mertenslemi1  12086  flodddiv4t2lthalf  12490  qnumgt0  12760  4sqlem12  12965  tangtx  15552  lgsquadlem1  15796  lgsquadlem2  15797