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Theorem rexdiv 27318
Description: The extended real division operation when both arguments are real. (Contributed by Thierry Arnoux, 18-Dec-2016.)
Assertion
Ref Expression
rexdiv  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  ( A  /  B ) )

Proof of Theorem rexdiv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 redivcl 10263 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  /  B )  e.  RR )
2 recn 9582 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
3 recn 9582 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
4 id 22 . . . . . 6  |-  ( B  =/=  0  ->  B  =/=  0 )
52, 3, 43anim123i 1181 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) )
6 divcan2 10215 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  /  B ) )  =  A )
75, 6syl 16 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( B  x.  ( A  /  B ) )  =  A )
8 oveq2 6292 . . . . . 6  |-  ( x  =  ( A  /  B )  ->  ( B  x.  x )  =  ( B  x.  ( A  /  B
) ) )
98eqeq1d 2469 . . . . 5  |-  ( x  =  ( A  /  B )  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  ( A  /  B
) )  =  A ) )
109rspcev 3214 . . . 4  |-  ( ( ( A  /  B
)  e.  RR  /\  ( B  x.  ( A  /  B ) )  =  A )  ->  E. x  e.  RR  ( B  x.  x
)  =  A )
111, 7, 10syl2anc 661 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. x  e.  RR  ( B  x.  x )  =  A )
12 receu 10194 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  E! x  e.  CC  ( B  x.  x )  =  A )
135, 12syl 16 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  CC  ( B  x.  x )  =  A )
14 ax-resscn 9549 . . . 4  |-  RR  C_  CC
15 id 22 . . . . 5  |-  ( ( B  x.  x )  =  A  ->  ( B  x.  x )  =  A )
1615rgenw 2825 . . . 4  |-  A. x  e.  RR  ( ( B  x.  x )  =  A  ->  ( B  x.  x )  =  A )
17 riotass2 6272 . . . 4  |-  ( ( ( RR  C_  CC  /\ 
A. x  e.  RR  ( ( B  x.  x )  =  A  ->  ( B  x.  x )  =  A ) )  /\  ( E. x  e.  RR  ( B  x.  x
)  =  A  /\  E! x  e.  CC  ( B  x.  x
)  =  A ) )  ->  ( iota_ x  e.  RR  ( B  x.  x )  =  A )  =  (
iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
1814, 16, 17mpanl12 682 . . 3  |-  ( ( E. x  e.  RR  ( B  x.  x
)  =  A  /\  E! x  e.  CC  ( B  x.  x
)  =  A )  ->  ( iota_ x  e.  RR  ( B  x.  x )  =  A )  =  ( iota_ x  e.  CC  ( B  x.  x )  =  A ) )
1911, 13, 18syl2anc 661 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( iota_ x  e.  RR  ( B  x.  x )  =  A )  =  (
iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
20 rexr 9639 . . . 4  |-  ( A  e.  RR  ->  A  e.  RR* )
21 xdivval 27311 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
iota_ x  e.  RR*  ( B xe x )  =  A ) )
2220, 21syl3an1 1261 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
iota_ x  e.  RR*  ( B xe x )  =  A ) )
23 ressxr 9637 . . . . 5  |-  RR  C_  RR*
2423a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  RR  C_ 
RR* )
25 rexmul 11463 . . . . . . . 8  |-  ( ( B  e.  RR  /\  x  e.  RR )  ->  ( B xe x )  =  ( B  x.  x ) )
2625eqeq1d 2469 . . . . . . 7  |-  ( ( B  e.  RR  /\  x  e.  RR )  ->  ( ( B xe x )  =  A  <->  ( B  x.  x )  =  A ) )
2726biimprd 223 . . . . . 6  |-  ( ( B  e.  RR  /\  x  e.  RR )  ->  ( ( B  x.  x )  =  A  ->  ( B xe x )  =  A ) )
2827ralrimiva 2878 . . . . 5  |-  ( B  e.  RR  ->  A. x  e.  RR  ( ( B  x.  x )  =  A  ->  ( B xe x )  =  A ) )
29283ad2ant2 1018 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  A. x  e.  RR  ( ( B  x.  x )  =  A  ->  ( B xe x )  =  A ) )
30 xreceu 27314 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B xe x )  =  A )
3120, 30syl3an1 1261 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B xe x )  =  A )
32 riotass2 6272 . . . 4  |-  ( ( ( RR  C_  RR*  /\  A. x  e.  RR  (
( B  x.  x
)  =  A  -> 
( B xe x )  =  A ) )  /\  ( E. x  e.  RR  ( B  x.  x
)  =  A  /\  E! x  e.  RR*  ( B xe x )  =  A ) )  ->  ( iota_ x  e.  RR  ( B  x.  x )  =  A )  =  ( iota_ x  e.  RR*  ( B xe x )  =  A ) )
3324, 29, 11, 31, 32syl22anc 1229 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( iota_ x  e.  RR  ( B  x.  x )  =  A )  =  (
iota_ x  e.  RR*  ( B xe x )  =  A ) )
3422, 33eqtr4d 2511 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
iota_ x  e.  RR  ( B  x.  x
)  =  A ) )
35 divval 10209 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
365, 35syl 16 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
3719, 34, 363eqtr4d 2518 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  ( A  /  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   E!wreu 2816    C_ wss 3476   iota_crio 6244  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492    x. cmul 9497   RR*cxr 9627    / cdiv 10206   xecxmu 11317   /𝑒 cxdiv 27309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-xneg 11318  df-xmul 11320  df-xdiv 27310
This theorem is referenced by:  xdivid  27320  xdiv0  27321  rpxdivcld  27326  esumdivc  27757  probmeasb  28037  coinfliplem  28085
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