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Theorem xdivrec 27857
Description: Relationship between division and reciprocal. (Contributed by Thierry Arnoux, 5-Jul-2017.)
Assertion
Ref Expression
xdivrec  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  ( A xe ( 1 /𝑒  B ) ) )

Proof of Theorem xdivrec
StepHypRef Expression
1 simp2 995 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  B  e.  RR )
21rexrd 9632 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  B  e.  RR* )
3 simp1 994 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  A  e.  RR* )
4 1re 9584 . . . . . . . . 9  |-  1  e.  RR
54rexri 9635 . . . . . . . 8  |-  1  e.  RR*
65a1i 11 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  1  e.  RR* )
7 simp3 996 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  B  =/=  0 )
86, 1, 7xdivcld 27853 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
1 /𝑒 
B )  e.  RR* )
93, 8xmulcld 11497 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A xe ( 1 /𝑒  B ) )  e.  RR* )
10 xmulcom 11461 . . . . 5  |-  ( ( B  e.  RR*  /\  ( A xe ( 1 /𝑒  B ) )  e.  RR* )  ->  ( B xe ( A xe ( 1 /𝑒  B ) ) )  =  ( ( A xe ( 1 /𝑒  B ) ) xe B ) )
112, 9, 10syl2anc 659 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( B xe ( A xe ( 1 /𝑒  B ) ) )  =  ( ( A xe ( 1 /𝑒  B ) ) xe B ) )
12 xmulass 11482 . . . . 5  |-  ( ( A  e.  RR*  /\  (
1 /𝑒 
B )  e.  RR*  /\  B  e.  RR* )  ->  ( ( A xe ( 1 /𝑒  B ) ) xe B )  =  ( A xe ( ( 1 /𝑒  B ) xe B ) ) )
133, 8, 2, 12syl3anc 1226 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( A xe ( 1 /𝑒  B ) ) xe B )  =  ( A xe ( ( 1 /𝑒  B ) xe B ) ) )
14 xmulcom 11461 . . . . . . 7  |-  ( ( ( 1 /𝑒  B )  e.  RR*  /\  B  e.  RR* )  ->  ( ( 1 /𝑒  B ) xe B )  =  ( B xe ( 1 /𝑒  B ) ) )
158, 2, 14syl2anc 659 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( 1 /𝑒  B ) xe B )  =  ( B xe ( 1 /𝑒  B ) ) )
16 eqid 2454 . . . . . . 7  |-  ( 1 /𝑒  B )  =  ( 1 /𝑒  B )
17 xdivmul 27855 . . . . . . . 8  |-  ( ( 1  e.  RR*  /\  (
1 /𝑒 
B )  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( (
1 /𝑒 
B )  =  ( 1 /𝑒  B )  <->  ( B xe ( 1 /𝑒  B ) )  =  1 ) )
186, 8, 1, 7, 17syl112anc 1230 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( 1 /𝑒  B )  =  ( 1 /𝑒  B )  <->  ( B xe ( 1 /𝑒  B ) )  =  1 ) )
1916, 18mpbii 211 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( B xe ( 1 /𝑒  B ) )  =  1 )
2015, 19eqtrd 2495 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( 1 /𝑒  B ) xe B )  =  1 )
2120oveq2d 6286 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A xe ( ( 1 /𝑒  B ) xe B ) )  =  ( A xe 1 ) )
2211, 13, 213eqtrd 2499 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( B xe ( A xe ( 1 /𝑒  B ) ) )  =  ( A xe 1 ) )
23 xmulid1 11474 . . . 4  |-  ( A  e.  RR*  ->  ( A xe 1 )  =  A )
243, 23syl 16 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A xe 1 )  =  A )
2522, 24eqtrd 2495 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( B xe ( A xe ( 1 /𝑒  B ) ) )  =  A )
26 xdivmul 27855 . . 3  |-  ( ( A  e.  RR*  /\  ( A xe ( 1 /𝑒  B ) )  e.  RR*  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( ( A /𝑒  B )  =  ( A xe ( 1 /𝑒  B ) )  <->  ( B xe ( A xe ( 1 /𝑒  B ) ) )  =  A ) )
273, 9, 1, 7, 26syl112anc 1230 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( A /𝑒  B )  =  ( A xe ( 1 /𝑒  B ) )  <->  ( B xe ( A xe ( 1 /𝑒  B ) ) )  =  A ) )
2825, 27mpbird 232 1  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  ( A xe ( 1 /𝑒  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482   RR*cxr 9616   xecxmu 11320   /𝑒 cxdiv 27847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-xneg 11321  df-xmul 11323  df-xdiv 27848
This theorem is referenced by:  esumdivc  28312
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