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Theorem rexdiv 26106
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 10055 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  /  B )  e.  RR )
2 recn 9377 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
3 recn 9377 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
4 id 22 . . . . . 6  |-  ( B  =/=  0  ->  B  =/=  0 )
52, 3, 43anim123i 1173 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) )
6 divcan2 10007 . . . . 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 6104 . . . . . 6  |-  ( x  =  ( A  /  B )  ->  ( B  x.  x )  =  ( B  x.  ( A  /  B
) ) )
98eqeq1d 2451 . . . . 5  |-  ( x  =  ( A  /  B )  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  ( A  /  B
) )  =  A ) )
109rspcev 3078 . . . 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 9986 . . . 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 9344 . . . 4  |-  RR  C_  CC
15 id 22 . . . . 5  |-  ( ( B  x.  x )  =  A  ->  ( B  x.  x )  =  A )
1615rgenw 2788 . . . 4  |-  A. x  e.  RR  ( ( B  x.  x )  =  A  ->  ( B  x.  x )  =  A )
17 riotass2 6084 . . . 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 9434 . . . 4  |-  ( A  e.  RR  ->  A  e.  RR* )
21 xdivval 26099 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
iota_ x  e.  RR*  ( B xe x )  =  A ) )
2220, 21syl3an1 1251 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
iota_ x  e.  RR*  ( B xe x )  =  A ) )
23 ressxr 9432 . . . . 5  |-  RR  C_  RR*
2423a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  RR  C_ 
RR* )
25 rexmul 11239 . . . . . . . 8  |-  ( ( B  e.  RR  /\  x  e.  RR )  ->  ( B xe x )  =  ( B  x.  x ) )
2625eqeq1d 2451 . . . . . . 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 2804 . . . . 5  |-  ( B  e.  RR  ->  A. x  e.  RR  ( ( B  x.  x )  =  A  ->  ( B xe x )  =  A ) )
29283ad2ant2 1010 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  A. x  e.  RR  ( ( B  x.  x )  =  A  ->  ( B xe x )  =  A ) )
30 xreceu 26102 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B xe x )  =  A )
3120, 30syl3an1 1251 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B xe x )  =  A )
32 riotass2 6084 . . . 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 1219 . . 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 2478 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
iota_ x  e.  RR  ( B  x.  x
)  =  A ) )
35 divval 10001 . . 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 2485 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   E!wreu 2722    C_ wss 3333   iota_crio 6056  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287    x. cmul 9292   RR*cxr 9422    / cdiv 9998   xecxmu 11093   /𝑒 cxdiv 26097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-xneg 11094  df-xmul 11096  df-xdiv 26098
This theorem is referenced by:  xdivid  26108  xdiv0  26109  rpxdivcld  26114  esumdivc  26537  probmeasb  26818  coinfliplem  26866
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