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Theorem xadddi 11408
Description: Distributive property for extended real addition and multiplication. Like xaddass 11362, this has an unusual domain of correctness due to counterexamples like  ( +oo  x.  (
2  -  1 ) )  = -oo  =/=  ( ( +oo  x.  2 )  -  ( +oo  x.  1 ) )  =  ( +oo  - +oo )  =  0. In this theorem we show that if the multiplier is real then everything works as expected. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xadddi  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )

Proof of Theorem xadddi
StepHypRef Expression
1 xadddilem 11407 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
2 simpl2 998 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  B  e.  RR* )
3 simpl3 999 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  C  e.  RR* )
4 xaddcl 11357 . . . . . 6  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B +e C )  e.  RR* )
52, 3, 4syl2anc 659 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  ( B +e C )  e.  RR* )
6 xmul02 11381 . . . . 5  |-  ( ( B +e C )  e.  RR*  ->  ( 0 xe ( B +e C ) )  =  0 )
75, 6syl 16 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe ( B +e C ) )  =  0 )
8 0xr 9551 . . . . 5  |-  0  e.  RR*
9 xaddid1 11359 . . . . 5  |-  ( 0  e.  RR*  ->  ( 0 +e 0 )  =  0 )
108, 9ax-mp 5 . . . 4  |-  ( 0 +e 0 )  =  0
117, 10syl6eqr 2441 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe ( B +e C ) )  =  ( 0 +e 0 ) )
12 simpr 459 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  0  =  A )
1312oveq1d 6211 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe ( B +e C ) )  =  ( A xe ( B +e C ) ) )
14 xmul02 11381 . . . . . 6  |-  ( B  e.  RR*  ->  ( 0 xe B )  =  0 )
152, 14syl 16 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe B )  =  0 )
1612oveq1d 6211 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe B )  =  ( A xe B ) )
1715, 16eqtr3d 2425 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  0  =  ( A xe B ) )
18 xmul02 11381 . . . . . 6  |-  ( C  e.  RR*  ->  ( 0 xe C )  =  0 )
193, 18syl 16 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe C )  =  0 )
2012oveq1d 6211 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe C )  =  ( A xe C ) )
2119, 20eqtr3d 2425 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  0  =  ( A xe C ) )
2217, 21oveq12d 6214 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 +e 0 )  =  ( ( A xe B ) +e ( A xe C ) ) )
2311, 13, 223eqtr3d 2431 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
24 simp1 994 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  A  e.  RR )
2524adantr 463 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  A  e.  RR )
26 rexneg 11331 . . . . . . 7  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
27 renegcl 9795 . . . . . . 7  |-  ( A  e.  RR  ->  -u A  e.  RR )
2826, 27eqeltrd 2470 . . . . . 6  |-  ( A  e.  RR  ->  -e
A  e.  RR )
2925, 28syl 16 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  -e
A  e.  RR )
30 simpl2 998 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  B  e.  RR* )
31 simpl3 999 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  C  e.  RR* )
3224rexrd 9554 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  A  e.  RR* )
33 xlt0neg1 11339 . . . . . . 7  |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
3432, 33syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( A  <  0  <->  0  <  -e A ) )
3534biimpa 482 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  0  <  -e A )
36 xadddilem 11407 . . . . 5  |-  ( ( (  -e A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  -e A )  ->  (  -e
A xe ( B +e C ) )  =  ( (  -e A xe B ) +e (  -e A xe C ) ) )
3729, 30, 31, 35, 36syl31anc 1229 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (  -e A xe ( B +e
C ) )  =  ( (  -e
A xe B ) +e ( 
-e A xe C ) ) )
3832adantr 463 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  A  e.  RR* )
3930, 31, 4syl2anc 659 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  ( B +e C )  e.  RR* )
40 xmulneg1 11382 . . . . 5  |-  ( ( A  e.  RR*  /\  ( B +e C )  e.  RR* )  ->  (  -e A xe ( B +e
C ) )  = 
-e ( A xe ( B +e C ) ) )
4138, 39, 40syl2anc 659 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (  -e A xe ( B +e
C ) )  = 
-e ( A xe ( B +e C ) ) )
42 xmulneg1 11382 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A xe B )  =  -e ( A xe B ) )
4338, 30, 42syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (  -e A xe B )  =  -e ( A xe B ) )
44 xmulneg1 11382 . . . . . . 7  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  (  -e A xe C )  =  -e ( A xe C ) )
4538, 31, 44syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (  -e A xe C )  =  -e ( A xe C ) )
4643, 45oveq12d 6214 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (
(  -e A xe B ) +e (  -e
A xe C ) )  =  ( 
-e ( A xe B ) +e  -e
( A xe C ) ) )
47 xmulcl 11386 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  e.  RR* )
4838, 30, 47syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  ( A xe B )  e.  RR* )
49 xmulcl 11386 . . . . . . 7  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A xe C )  e.  RR* )
5038, 31, 49syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  ( A xe C )  e.  RR* )
51 xnegdi 11361 . . . . . 6  |-  ( ( ( A xe B )  e.  RR*  /\  ( A xe C )  e.  RR* )  ->  -e ( ( A xe B ) +e ( A xe C ) )  =  ( 
-e ( A xe B ) +e  -e
( A xe C ) ) )
5248, 50, 51syl2anc 659 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  -e
( ( A xe B ) +e ( A xe C ) )  =  (  -e
( A xe B ) +e  -e ( A xe C ) ) )
5346, 52eqtr4d 2426 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (
(  -e A xe B ) +e (  -e
A xe C ) )  =  -e ( ( A xe B ) +e ( A xe C ) ) )
5437, 41, 533eqtr3d 2431 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  -e
( A xe ( B +e
C ) )  = 
-e ( ( A xe B ) +e ( A xe C ) ) )
55 xmulcl 11386 . . . . 5  |-  ( ( A  e.  RR*  /\  ( B +e C )  e.  RR* )  ->  ( A xe ( B +e C ) )  e.  RR* )
5638, 39, 55syl2anc 659 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  ( A xe ( B +e C ) )  e.  RR* )
57 xaddcl 11357 . . . . 5  |-  ( ( ( A xe B )  e.  RR*  /\  ( A xe C )  e.  RR* )  ->  ( ( A xe B ) +e ( A xe C ) )  e.  RR* )
5848, 50, 57syl2anc 659 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (
( A xe B ) +e
( A xe C ) )  e. 
RR* )
59 xneg11 11335 . . . 4  |-  ( ( ( A xe ( B +e
C ) )  e. 
RR*  /\  ( ( A xe B ) +e ( A xe C ) )  e.  RR* )  ->  (  -e ( A xe ( B +e C ) )  =  -e ( ( A xe B ) +e ( A xe C ) )  <->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) ) )
6056, 58, 59syl2anc 659 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (  -e ( A xe ( B +e C ) )  =  -e ( ( A xe B ) +e
( A xe C ) )  <->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) ) )
6154, 60mpbid 210 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
62 0re 9507 . . 3  |-  0  e.  RR
63 lttri4 9580 . . 3  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  \/  0  =  A  \/  A  <  0
) )
6462, 24, 63sylancr 661 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
0  <  A  \/  0  =  A  \/  A  <  0 ) )
651, 23, 61, 64mpjao3dan 1293 1  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    \/ w3o 970    /\ w3a 971    = wceq 1399    e. wcel 1826   class class class wbr 4367  (class class class)co 6196   RRcr 9402   0cc0 9403   RR*cxr 9538    < clt 9539   -ucneg 9719    -ecxne 11236   +ecxad 11237   xecxmu 11238
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
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-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-xneg 11239  df-xadd 11240  df-xmul 11241
This theorem is referenced by:  xadddir  11409  xadddi2  11410
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