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Theorem rexdiv 27775
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 10180 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  /  B )  e.  RR )
2 recn 9493 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
3 recn 9493 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
4 id 22 . . . . . 6  |-  ( B  =/=  0  ->  B  =/=  0 )
52, 3, 43anim123i 1179 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) )
6 divcan2 10132 . . . . 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 6204 . . . . . 6  |-  ( x  =  ( A  /  B )  ->  ( B  x.  x )  =  ( B  x.  ( A  /  B
) ) )
98eqeq1d 2384 . . . . 5  |-  ( x  =  ( A  /  B )  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  ( A  /  B
) )  =  A ) )
109rspcev 3135 . . . 4  |-  ( ( ( A  /  B
)  e.  RR  /\  ( B  x.  ( A  /  B ) )  =  A )  ->  E. x  e.  RR  ( B  x.  x
)  =  A )
111, 7, 10syl2anc 659 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  E. x  e.  RR  ( B  x.  x )  =  A )
12 receu 10111 . . . 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 9460 . . . 4  |-  RR  C_  CC
15 id 22 . . . . 5  |-  ( ( B  x.  x )  =  A  ->  ( B  x.  x )  =  A )
1615rgenw 2743 . . . 4  |-  A. x  e.  RR  ( ( B  x.  x )  =  A  ->  ( B  x.  x )  =  A )
17 riotass2 6184 . . . 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 680 . . 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 659 . 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 9550 . . . 4  |-  ( A  e.  RR  ->  A  e.  RR* )
21 xdivval 27768 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
iota_ x  e.  RR*  ( B xe x )  =  A ) )
2220, 21syl3an1 1259 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
iota_ x  e.  RR*  ( B xe x )  =  A ) )
23 ressxr 9548 . . . . 5  |-  RR  C_  RR*
2423a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  RR  C_ 
RR* )
25 rexmul 11384 . . . . . . . 8  |-  ( ( B  e.  RR  /\  x  e.  RR )  ->  ( B xe x )  =  ( B  x.  x ) )
2625eqeq1d 2384 . . . . . . 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 2796 . . . . 5  |-  ( B  e.  RR  ->  A. x  e.  RR  ( ( B  x.  x )  =  A  ->  ( B xe x )  =  A ) )
29283ad2ant2 1016 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  A. x  e.  RR  ( ( B  x.  x )  =  A  ->  ( B xe x )  =  A ) )
30 xreceu 27771 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B xe x )  =  A )
3120, 30syl3an1 1259 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B xe x )  =  A )
32 riotass2 6184 . . . 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 1227 . . 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 2426 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
iota_ x  e.  RR  ( B  x.  x
)  =  A ) )
35 divval 10126 . . 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 2433 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733   E!wreu 2734    C_ wss 3389   iota_crio 6157  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403    x. cmul 9408   RR*cxr 9538    / cdiv 10123   xecxmu 11238   /𝑒 cxdiv 27766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-xneg 11239  df-xmul 11241  df-xdiv 27767
This theorem is referenced by:  xdivid  27777  xdiv0  27778  rpxdivcld  27783  esumdivc  28231  probmeasb  28552  coinfliplem  28600
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