Theorem ltmul1t 5737

Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20.

Assertion

Ref	Expression
ltmul1t	|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A < B <-> (A x. C) < (B x. C)))

Proof of Theorem ltmul1t

Step	Hyp	Ref	Expression
1		breq1 2590	. . . . 5 |- (A = if(A e. RR, A, 0) -> (A < B <-> if(A e. RR, A, 0) < B))
2		opreq1 3907	. . . . . 6 |- (A = if(A e. RR, A, 0) -> (A x. C) = (if(A e. RR, A, 0) x. C))
3	2	breq1d 2597	. . . . 5 |- (A = if(A e. RR, A, 0) -> ((A x. C) < (B x. C) <-> (if(A e. RR, A, 0) x. C) < (B x. C)))
4	1, 3	bibi12d 627	. . . 4 |- (A = if(A e. RR, A, 0) -> ((A < B <-> (A x. C) < (B x. C)) <-> (if(A e. RR, A, 0) < B <-> (if(A e. RR, A, 0) x. C) < (B x. C))))
5	4	imbi2d 610	. . 3 |- (A = if(A e. RR, A, 0) -> ((0 < C -> (A < B <-> (A x. C) < (B x. C))) <-> (0 < C -> (if(A e. RR, A, 0) < B <-> (if(A e. RR, A, 0) x. C) < (B x. C)))))
6		breq2 2591	. . . . 5 |- (B = if(B e. RR, B, 0) -> (if(A e. RR, A, 0) < B <-> if(A e. RR, A, 0) < if(B e. RR, B, 0)))
7		opreq1 3907	. . . . . 6 |- (B = if(B e. RR, B, 0) -> (B x. C) = (if(B e. RR, B, 0) x. C))
8	7	breq2d 2598	. . . . 5 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) x. C) < (B x. C) <-> (if(A e. RR, A, 0) x. C) < (if(B e. RR, B, 0) x. C)))
9	6, 8	bibi12d 627	. . . 4 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) < B <-> (if(A e. RR, A, 0) x. C) < (B x. C)) <-> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) x. C) < (if(B e. RR, B, 0) x. C))))
10	9	imbi2d 610	. . 3 |- (B = if(B e. RR, B, 0) -> ((0 < C -> (if(A e. RR, A, 0) < B <-> (if(A e. RR, A, 0) x. C) < (B x. C))) <-> (0 < C -> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) x. C) < (if(B e. RR, B, 0) x. C)))))
11		breq2 2591	. . . 4 |- (C = if(C e. RR, C, 0) -> (0 < C <-> 0 < if(C e. RR, C, 0)))
12		opreq2 3908	. . . . . 6 |- (C = if(C e. RR, C, 0) -> (if(A e. RR, A, 0) x. C) = (if(A e. RR, A, 0) x. if(C e. RR, C, 0)))
13		opreq2 3908	. . . . . 6 |- (C = if(C e. RR, C, 0) -> (if(B e. RR, B, 0) x. C) = (if(B e. RR, B, 0) x. if(C e. RR, C, 0)))
14	12, 13	breq12d 2599	. . . . 5 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) x. C) < (if(B e. RR, B, 0) x. C) <-> (if(A e. RR, A, 0) x. if(C e. RR, C, 0)) < (if(B e. RR, B, 0) x. if(C e. RR, C, 0))))
15	14	bibi2d 616	. . . 4 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) x. C) < (if(B e. RR, B, 0) x. C)) <-> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) x. if(C e. RR, C, 0)) < (if(B e. RR, B, 0) x. if(C e. RR, C, 0)))))
16	11, 15	imbi12d 624	. . 3 |- (C = if(C e. RR, C, 0) -> ((0 < C -> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) x. C) < (if(B e. RR, B, 0) x. C))) <-> (0 < if(C e. RR, C, 0) -> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) x. if(C e. RR, C, 0)) < (if(B e. RR, B, 0) x. if(C e. RR, C, 0))))))
17		0re 5363	. . . . 5 |- 0 e. RR
18	17	elimel 2365	. . . 4 |- if(A e. RR, A, 0) e. RR
19	17	elimel 2365	. . . 4 |- if(B e. RR, B, 0) e. RR
20	17	elimel 2365	. . . 4 |- if(C e. RR, C, 0) e. RR
21	18, 19, 20	ltmul1 5729	. . 3 |- (0 < if(C e. RR, C, 0) -> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) x. if(C e. RR, C, 0)) < (if(B e. RR, B, 0) x. if(C e. RR, C, 0))))
22	5, 10, 16, 21	dedth3h 2359	. 2 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (0 < C -> (A < B <-> (A x. C) < (B x. C))))
23	22	imp 350	1 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A < B <-> (A x. C) < (B x. C)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 772   = wceq 1099   e. wcel 1105  ifcif 2332   class class class wbr 2587  (class class class)co 3902  RRcr 5156  0cc0 5157   x. cmul 5162   < clt 5409

This theorem is referenced by:  ltmul2t 5738  lemul1t 5739  ltdiv23t 5791  recp1lt1 5800  ltdivp1 5806  expordit 6482  climmullem4 7010  efcltlem1 7197

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-inf2 4549

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-nel 1564  df-ral 1625  df-rex 1626  df-reu 1627  df-rab 1628  df-v 1787  df-sbc 1913  df-csb 1973  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-pss 2026  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-int 2502  df-iun 2536  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-id 2797  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-lim 2916  df-suc 2917  df-om 3095  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-f1 3158  df-fo 3159  df-f1o 3160  df-fv 3161  df-rdg 3871  df-opr 3904  df-oprab 3905  df-1st 4017  df-2nd 4018  df-1o 4071  df-oadd 4073  df-omul 4074  df-er 4199  df-ec 4201  df-qs 4204  df-en 4305  df-dom 4306  df-sdom 4307  df-ni 4923  df-pli 4924  df-mi 4925  df-lti 4926  df-plpq 4958  df-mpq 4959  df-enq 4960  df-nq 4961  df-plq 4962  df-mq 4963  df-rq 4964  df-ltq 4965  df-1q 4966  df-np 5009  df-1p 5010  df-plp 5011  df-mp 5012  df-ltp 5013  df-plpr 5087  df-mpr 5088  df-enr 5089  df-nr 5090  df-plr 5091  df-mr 5092  df-ltr 5093  df-0r 5094  df-1r 5095  df-m1r 5096  df-c 5163  df-0 5164  df-1 5165  df-i 5166  df-r 5167  df-plus 5168  df-mul 5169  df-lt 5170  df-sub 5279  df-neg 5281  df-pnf 5410  df-mnf 5411  df-xr 5412  df-ltxr 5413  df-le 5414

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