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Theorem lemul1OLD 7012
Description: Multiplication of both sides of 'less than or equal to' by a positive number.
Assertion
Ref Expression
lemul1OLD |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A <_ B <-> (A x. C) <_ (B x. C)))
Proof of Theorem lemul1OLD
StepHypRef Expression
1 simpl1 879 . . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> A e. RR)
2 simpl2 880 . . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> B e. RR)
3 simp3 878 . . . . 5 |- ((A e. RR /\ B e. RR /\ C e. RR) -> C e. RR)
4 id 73 . . . . 5 |- (0 < C -> 0 < C)
53, 4anim12i 360 . . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (C e. RR /\ 0 < C))
6 ltmul1 7008 . . . 4 |- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> (A < B <-> (A x. C) < (B x. C)))
71, 2, 5, 6syl111anc 1100 . . 3 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A < B <-> (A x. C) < (B x. C)))
8 recn 6466 . . . . . . 7 |- (A e. RR -> A e. CC)
9 recn 6466 . . . . . . 7 |- (B e. RR -> B e. CC)
10 recn 6466 . . . . . . 7 |- (C e. RR -> C e. CC)
118, 9, 103anim123i 1053 . . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (A e. CC /\ B e. CC /\ C e. CC))
1211adantr 425 . . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A e. CC /\ B e. CC /\ C e. CC))
13 gt0ne0 6800 . . . . . 6 |- ((C e. RR /\ 0 < C) -> C =/= 0)
14133ad2antl3 1040 . . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> C =/= 0)
15 mulcan2 6881 . . . . . . . . 9 |- ((A e. CC /\ B e. CC /\ (C e. CC /\ C =/= 0)) -> ((A x. C) = (B x. C) <-> A = B))
16153exp 1066 . . . . . . . 8 |- (A e. CC -> (B e. CC -> ((C e. CC /\ C =/= 0) -> ((A x. C) = (B x. C) <-> A = B))))
1716exp4a 409 . . . . . . 7 |- (A e. CC -> (B e. CC -> (C e. CC -> (C =/= 0 -> ((A x. C) = (B x. C) <-> A = B)))))
18173imp 1061 . . . . . 6 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (C =/= 0 -> ((A x. C) = (B x. C) <-> A = B)))
1918imp 377 . . . . 5 |- (((A e. CC /\ B e. CC /\ C e. CC) /\ C =/= 0) -> ((A x. C) = (B x. C) <-> A = B))
2012, 14, 19syl11anc 524 . . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> ((A x. C) = (B x. C) <-> A = B))
2120bicomd 580 . . 3 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A = B <-> (A x. C) = (B x. C)))
227, 21orbi12d 689 . 2 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> ((A < B \/ A = B) <-> ((A x. C) < (B x. C) \/ (A x. C) = (B x. C))))
23 leloe 6688 . . . 4 |- ((A e. RR /\ B e. RR) -> (A <_ B <-> (A < B \/ A = B)))
24233adant3 896 . . 3 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (A <_ B <-> (A < B \/ A = B)))
2524adantr 425 . 2 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A <_ B <-> (A < B \/ A = B)))
26 remulcl 6457 . . . . 5 |- ((A e. RR /\ C e. RR) -> (A x. C) e. RR)
27263adant2 895 . . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (A x. C) e. RR)
28 remulcl 6457 . . . . 5 |- ((B e. RR /\ C e. RR) -> (B x. C) e. RR)
29283adant1 894 . . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (B x. C) e. RR)
30 leloe 6688 . . . 4 |- (((A x. C) e. RR /\ (B x. C) e. RR) -> ((A x. C) <_ (B x. C) <-> ((A x. C) < (B x. C) \/ (A x. C) = (B x. C))))
3127, 29, 30syl11anc 524 . . 3 |- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A x. C) <_ (B x. C) <-> ((A x. C) < (B x. C) \/ (A x. C) = (B x. C))))
3231adantr 425 . 2 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> ((A x. C) <_ (B x. C) <-> ((A x. C) < (B x. C) \/ (A x. C) = (B x. C))))
3322, 25, 323bitr4d 609 1 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A <_ B <-> (A x. C) <_ (B x. C)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   x. cmul 6391   <_ cle 6448   < clt 6653
This theorem is referenced by:  lemul1i 7017  lemul1a 7019  ledivp1i 7089  sqlecan 7887  sin01bndlem2 8734  cos01bndlem2 8736  cnlnadjlem7 11643
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658
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