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Theorem reapmul1 7379
Description: Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7546. (Contributed by Jim Kingdon, 8-Feb-2020.)
Assertion
Ref Expression
reapmul1  RR  RR  C  RR  C #  0 #  x.  C #  x.  C

Proof of Theorem reapmul1
StepHypRef Expression
1 0re 6825 . . . . 5  0  RR
2 reaplt 7372 . . . . 5  C  RR  0  RR  C #  0  C  <  0  0  <  C
31, 2mpan2 401 . . . 4  C  RR  C #  0  C  <  0  0  < 
C
43pm5.32i 427 . . 3  C  RR  C #  0  C  RR  C  <  0  0  <  C
5 simp1 903 . . . . . . . . . . 11  RR  RR  C  RR  C  <  0  RR
65recnd 6851 . . . . . . . . . 10  RR  RR  C  RR  C  <  0  CC
7 simp3l 931 . . . . . . . . . . 11  RR  RR  C  RR  C  <  0 
C  RR
87recnd 6851 . . . . . . . . . 10  RR  RR  C  RR  C  <  0 
C  CC
96, 8mulneg2d 7205 . . . . . . . . 9  RR  RR  C  RR  C  <  0  x.  -u C  -u  x.  C
10 simp2 904 . . . . . . . . . . 11  RR  RR  C  RR  C  <  0  RR
1110recnd 6851 . . . . . . . . . 10  RR  RR  C  RR  C  <  0  CC
1211, 8mulneg2d 7205 . . . . . . . . 9  RR  RR  C  RR  C  <  0  x.  -u C  -u  x.  C
139, 12breq12d 3768 . . . . . . . 8  RR  RR  C  RR  C  <  0  x.  -u C #  x.  -u C 
-u  x.  C #  -u  x.  C
147renegcld 7174 . . . . . . . . 9  RR  RR  C  RR  C  <  0  -u C  RR
15 simp3r 932 . . . . . . . . . 10  RR  RR  C  RR  C  <  0 
C  <  0
167lt0neg1d 7302 . . . . . . . . . 10  RR  RR  C  RR  C  <  0  C  <  0 
0  <  -u C
1715, 16mpbid 135 . . . . . . . . 9  RR  RR  C  RR  C  <  0  0  <  -u C
18 reapmul1lem 7378 . . . . . . . . 9  RR  RR  -u C  RR  0  <  -u C #  x.  -u C #  x.  -u C
195, 10, 14, 17, 18syl112anc 1138 . . . . . . . 8  RR  RR  C  RR  C  <  0 #  x.  -u C #  x.  -u C
205, 7remulcld 6853 . . . . . . . . . . 11  RR  RR  C  RR  C  <  0  x.  C  RR
2110, 7remulcld 6853 . . . . . . . . . . 11  RR  RR  C  RR  C  <  0  x.  C  RR
2220, 21ltnegd 7309 . . . . . . . . . 10  RR  RR  C  RR  C  <  0  x.  C  <  x.  C  -u  x.  C  <  -u  x.  C
2321, 20ltnegd 7309 . . . . . . . . . 10  RR  RR  C  RR  C  <  0  x.  C  <  x.  C  -u  x.  C  <  -u  x.  C
2422, 23orbi12d 706 . . . . . . . . 9  RR  RR  C  RR  C  <  0  x.  C  <  x.  C  x.  C  <  x.  C  -u  x.  C  <  -u  x.  C  -u  x.  C  <  -u  x.  C
25 reaplt 7372 . . . . . . . . . 10  x.  C  RR  x.  C  RR  x.  C #  x.  C  x.  C  <  x.  C  x.  C  <  x.  C
2620, 21, 25syl2anc 391 . . . . . . . . 9  RR  RR  C  RR  C  <  0  x.  C #  x.  C  x.  C  <  x.  C  x.  C  <  x.  C
2720renegcld 7174 . . . . . . . . . . 11  RR  RR  C  RR  C  <  0  -u  x.  C  RR
2821renegcld 7174 . . . . . . . . . . 11  RR  RR  C  RR  C  <  0  -u  x.  C  RR
29 reaplt 7372 . . . . . . . . . . 11 
-u  x.  C  RR  -u  x.  C  RR  -u  x.  C #  -u  x.  C  -u  x.  C  <  -u  x.  C  -u  x.  C  <  -u  x.  C
3027, 28, 29syl2anc 391 . . . . . . . . . 10  RR  RR  C  RR  C  <  0  -u  x.  C #  -u  x.  C  -u  x.  C  <  -u  x.  C  -u  x.  C  <  -u  x.  C
31 orcom 646 . . . . . . . . . 10 
-u  x.  C  <  -u  x.  C  -u  x.  C  <  -u  x.  C  -u  x.  C  <  -u  x.  C  -u  x.  C  <  -u  x.  C
3230, 31syl6bb 185 . . . . . . . . 9  RR  RR  C  RR  C  <  0  -u  x.  C #  -u  x.  C  -u  x.  C  <  -u  x.  C  -u  x.  C  <  -u  x.  C
3324, 26, 323bitr4d 209 . . . . . . . 8  RR  RR  C  RR  C  <  0  x.  C #  x.  C 
-u  x.  C #  -u  x.  C
3413, 19, 333bitr4d 209 . . . . . . 7  RR  RR  C  RR  C  <  0 #  x.  C #  x.  C
35343expa 1103 . . . . . 6  RR  RR  C  RR  C  <  0 #  x.  C #  x.  C
3635anassrs 380 . . . . 5  RR  RR  C  RR  C  <  0 #  x.  C #  x.  C
37 reapmul1lem 7378 . . . . . . 7  RR  RR  C  RR  0  <  C #  x.  C #  x.  C
38373expa 1103 . . . . . 6  RR  RR  C  RR  0  <  C #  x.  C #  x.  C
3938anassrs 380 . . . . 5  RR  RR  C  RR  0  <  C #  x.  C #  x.  C
4036, 39jaodan 709 . . . 4  RR  RR  C  RR  C  <  0  0  < 
C #  x.  C #  x.  C
4140anasss 379 . . 3  RR  RR  C  RR  C  <  0  0  <  C #  x.  C #  x.  C
424, 41sylan2b 271 . 2  RR  RR  C  RR  C #  0 #  x.  C #  x.  C
43423impa 1098 1  RR  RR  C  RR  C #  0 #  x.  C #  x.  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wo 628   w3a 884   wcel 1390   class class class wbr 3755  (class class class)co 5455   RRcr 6710   0cc0 6711    x. cmul 6716    < clt 6857   -ucneg 6980   # cap 7365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-mulrcl 6782  ax-addcom 6783  ax-mulcom 6784  ax-addass 6785  ax-mulass 6786  ax-distr 6787  ax-i2m1 6788  ax-1rid 6790  ax-0id 6791  ax-rnegex 6792  ax-precex 6793  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-lttrn 6797  ax-pre-apti 6798  ax-pre-ltadd 6799  ax-pre-mulgt0 6800
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-ltxr 6862  df-sub 6981  df-neg 6982  df-reap 7359  df-ap 7366
This theorem is referenced by: (None)
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