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Theorem xadddi 11499
Description: Distributive property for extended real addition and multiplication. Like xaddass 11453, 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 9608 . . 3  |-  0  e.  RR
2 simp1 996 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  A  e.  RR )
3 lttri4 9681 . . 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 11498 . . 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 1000 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  B  e.  RR* )
7 simpl3 1001 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  C  e.  RR* )
8 xaddcl 11448 . . . . . . 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 11472 . . . . . 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 9652 . . . . . 6  |-  0  e.  RR*
13 xaddid1 11450 . . . . . 6  |-  ( 0  e.  RR*  ->  ( 0 +e 0 )  =  0 )
1412, 13ax-mp 5 . . . . 5  |-  ( 0 +e 0 )  =  0
1511, 14syl6eqr 2526 . . . 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 6310 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe ( B +e C ) )  =  ( A xe ( B +e C ) ) )
18 xmul02 11472 . . . . . . 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 6310 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe B )  =  ( A xe B ) )
2119, 20eqtr3d 2510 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  0  =  ( A xe B ) )
22 xmul02 11472 . . . . . . 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 6310 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  (
0 xe C )  =  ( A xe C ) )
2523, 24eqtr3d 2510 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  =  A )  ->  0  =  ( A xe C ) )
2621, 25oveq12d 6313 . . . 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 2516 . . 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 11422 . . . . . . . 8  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
30 renegcl 9894 . . . . . . . 8  |-  ( A  e.  RR  ->  -u A  e.  RR )
3129, 30eqeltrd 2555 . . . . . . 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 1000 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  B  e.  RR* )
34 simpl3 1001 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  0 )  ->  C  e.  RR* )
352rexrd 9655 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  A  e.  RR* )
36 xlt0neg1 11430 . . . . . . . 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 11498 . . . . . 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 1231 . . . . 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 11473 . . . . . 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 11473 . . . . . . . 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 11473 . . . . . . . 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 6313 . . . . . 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 11477 . . . . . . . 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 11477 . . . . . . . 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 11452 . . . . . . 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 2511 . . . . 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 2516 . . . 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 11477 . . . . . 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 11448 . . . . . 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 11426 . . . . 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 1294 . 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 972    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4453  (class class class)co 6295   RRcr 9503   0cc0 9504   RR*cxr 9639    < clt 9640   -ucneg 9818    -ecxne 11327   +ecxad 11328   xecxmu 11329
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-xneg 11330  df-xadd 11331  df-xmul 11332
This theorem is referenced by:  xadddir  11500  xadddi2  11501
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