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Theorem xadddi 11258
Description: Distributive property for extended real addition and multiplication. Like xaddass 11212, 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 0re 9386 . . 3  |-  0  e.  RR
2 simp1 988 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  A  e.  RR )
3 lttri4 9459 . . 3  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  \/  0  =  A  \/  A  <  0
) )
41, 2, 3sylancr 663 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
0  <  A  \/  0  =  A  \/  A  <  0 ) )
5 xadddilem 11257 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
6 simpl2 992 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  B  e.  RR* )
7 simpl3 993 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  C  e.  RR* )
8 xaddcl 11207 . . . . . . 7  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B +e C )  e.  RR* )
96, 7, 8syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  ( B +e C )  e.  RR* )
10 xmul02 11231 . . . . . 6  |-  ( ( B +e C )  e.  RR*  ->  ( 0 xe ( B +e C ) )  =  0 )
119, 10syl 16 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe ( B +e C ) )  =  0 )
12 0xr 9430 . . . . . 6  |-  0  e.  RR*
13 xaddid1 11209 . . . . . 6  |-  ( 0  e.  RR*  ->  ( 0 +e 0 )  =  0 )
1412, 13ax-mp 5 . . . . 5  |-  ( 0 +e 0 )  =  0
1511, 14syl6eqr 2493 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe ( B +e C ) )  =  ( 0 +e 0 ) )
16 simpr 461 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  0  =  A )
1716oveq1d 6106 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe ( B +e C ) )  =  ( A xe ( B +e C ) ) )
18 xmul02 11231 . . . . . . 7  |-  ( B  e.  RR*  ->  ( 0 xe B )  =  0 )
196, 18syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe B )  =  0 )
2016oveq1d 6106 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe B )  =  ( A xe B ) )
2119, 20eqtr3d 2477 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  0  =  ( A xe B ) )
22 xmul02 11231 . . . . . . 7  |-  ( C  e.  RR*  ->  ( 0 xe C )  =  0 )
237, 22syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe C )  =  0 )
2416oveq1d 6106 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe C )  =  ( A xe C ) )
2523, 24eqtr3d 2477 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  0  =  ( A xe C ) )
2621, 25oveq12d 6109 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 +e 0 )  =  ( ( A xe B ) +e ( A xe C ) ) )
2715, 17, 263eqtr3d 2483 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
282adantr 465 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  A  e.  RR )
29 rexneg 11181 . . . . . . . 8  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
30 renegcl 9672 . . . . . . . 8  |-  ( A  e.  RR  ->  -u A  e.  RR )
3129, 30eqeltrd 2517 . . . . . . 7  |-  ( A  e.  RR  ->  -e
A  e.  RR )
3228, 31syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  -e
A  e.  RR )
33 simpl2 992 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  B  e.  RR* )
34 simpl3 993 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  C  e.  RR* )
352rexrd 9433 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  A  e.  RR* )
36 xlt0neg1 11189 . . . . . . . 8  |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
3735, 36syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( A  <  0  <->  0  <  -e A ) )
3837biimpa 484 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  0  <  -e A )
39 xadddilem 11257 . . . . . 6  |-  ( ( (  -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 ) ) )
4032, 33, 34, 38, 39syl31anc 1221 . . . . 5  |-  ( ( ( 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 ) ) )
4135adantr 465 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  A  e.  RR* )
4233, 34, 8syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  ( B +e C )  e.  RR* )
43 xmulneg1 11232 . . . . . 6  |-  ( ( A  e.  RR*  /\  ( B +e C )  e.  RR* )  ->  (  -e A xe ( B +e
C ) )  = 
-e ( A xe ( B +e C ) ) )
4441, 42, 43syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (  -e A xe ( B +e
C ) )  = 
-e ( A xe ( B +e C ) ) )
45 xmulneg1 11232 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A xe B )  =  -e ( A xe B ) )
4641, 33, 45syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (  -e A xe B )  =  -e ( A xe B ) )
47 xmulneg1 11232 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  (  -e A xe C )  =  -e ( A xe C ) )
4841, 34, 47syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (  -e A xe C )  =  -e ( A xe C ) )
4946, 48oveq12d 6109 . . . . . 6  |-  ( ( ( 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 ) ) )
50 xmulcl 11236 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  e.  RR* )
5141, 33, 50syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  ( A xe B )  e.  RR* )
52 xmulcl 11236 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A xe C )  e.  RR* )
5341, 34, 52syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  ( A xe C )  e.  RR* )
54 xnegdi 11211 . . . . . . 7  |-  ( ( ( 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 ) ) )
5551, 53, 54syl2anc 661 . . . . . 6  |-  ( ( ( 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 ) ) )
5649, 55eqtr4d 2478 . . . . 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 ( A xe C ) ) )
5740, 44, 563eqtr3d 2483 . . . 4  |-  ( ( ( 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 ) ) )
58 xmulcl 11236 . . . . . 6  |-  ( ( A  e.  RR*  /\  ( B +e C )  e.  RR* )  ->  ( A xe ( B +e C ) )  e.  RR* )
5941, 42, 58syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  ( A xe ( B +e C ) )  e.  RR* )
60 xaddcl 11207 . . . . . 6  |-  ( ( ( A xe B )  e.  RR*  /\  ( A xe C )  e.  RR* )  ->  ( ( A xe B ) +e ( A xe C ) )  e.  RR* )
6151, 53, 60syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  (
( A xe B ) +e
( A xe C ) )  e. 
RR* )
62 xneg11 11185 . . . . 5  |-  ( ( ( 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 ) ) ) )
6359, 61, 62syl2anc 661 . . . 4  |-  ( ( ( 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 ) ) ) )
6457, 63mpbid 210 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
655, 27, 643jaodan 1284 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  (
0  <  A  \/  0  =  A  \/  A  <  0 ) )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
664, 65mpdan 668 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 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4292  (class class class)co 6091   RRcr 9281   0cc0 9282   RR*cxr 9417    < clt 9418   -ucneg 9596    -ecxne 11086   +ecxad 11087   xecxmu 11088
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 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-xneg 11089  df-xadd 11090  df-xmul 11091
This theorem is referenced by:  xadddir  11259  xadddi2  11260
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