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Theorem xaddass2 11300
Description: Associativity of extended real addition. See xaddass 11299 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddass2  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  ( ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )

Proof of Theorem xaddass2
StepHypRef Expression
1 simp1l 1012 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  A  e.  RR* )
2 xnegcl 11270 . . . . . 6  |-  ( A  e.  RR*  ->  -e
A  e.  RR* )
31, 2syl 16 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e A  e.  RR* )
4 simp1r 1013 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  A  =/= +oo )
5 pnfxr 11179 . . . . . . . . 9  |- +oo  e.  RR*
6 xneg11 11272 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  (  -e A  =  -e +oo  <->  A  = +oo ) )
71, 5, 6sylancl 662 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
A  =  -e +oo 
<->  A  = +oo )
)
87necon3bid 2703 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
A  =/=  -e +oo 
<->  A  =/= +oo )
)
94, 8mpbird 232 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e A  =/=  -e +oo )
10 xnegpnf 11266 . . . . . . 7  |-  -e +oo  = -oo
1110a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e +oo  = -oo )
129, 11neeqtrd 2740 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e A  =/= -oo )
13 simp2l 1014 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  B  e.  RR* )
14 xnegcl 11270 . . . . . 6  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
1513, 14syl 16 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e B  e.  RR* )
16 simp2r 1015 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  B  =/= +oo )
17 xneg11 11272 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\ +oo  e.  RR* )  ->  (  -e B  =  -e +oo  <->  B  = +oo ) )
1813, 5, 17sylancl 662 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
B  =  -e +oo 
<->  B  = +oo )
)
1918necon3bid 2703 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
B  =/=  -e +oo 
<->  B  =/= +oo )
)
2016, 19mpbird 232 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e B  =/=  -e +oo )
2120, 11neeqtrd 2740 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e B  =/= -oo )
22 simp3l 1016 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  C  e.  RR* )
23 xnegcl 11270 . . . . . 6  |-  ( C  e.  RR*  ->  -e
C  e.  RR* )
2422, 23syl 16 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e C  e.  RR* )
25 simp3r 1017 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  C  =/= +oo )
26 xneg11 11272 . . . . . . . . 9  |-  ( ( C  e.  RR*  /\ +oo  e.  RR* )  ->  (  -e C  =  -e +oo  <->  C  = +oo ) )
2722, 5, 26sylancl 662 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
C  =  -e +oo 
<->  C  = +oo )
)
2827necon3bid 2703 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
C  =/=  -e +oo 
<->  C  =/= +oo )
)
2925, 28mpbird 232 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e C  =/=  -e +oo )
3029, 11neeqtrd 2740 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e C  =/= -oo )
31 xaddass 11299 . . . . 5  |-  ( ( (  -e A  e.  RR*  /\  -e
A  =/= -oo )  /\  (  -e B  e.  RR*  /\  -e
B  =/= -oo )  /\  (  -e C  e.  RR*  /\  -e
C  =/= -oo )
)  ->  ( (  -e A +e  -e B ) +e  -e C )  =  (  -e A +e
(  -e B +e  -e C ) ) )
323, 12, 15, 21, 24, 30, 31syl222anc 1235 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  ( (  -e A +e  -e B ) +e  -e C )  =  (  -e A +e
(  -e B +e  -e C ) ) )
33 xnegdi 11298 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -e
( A +e
B )  =  ( 
-e A +e  -e B ) )
341, 13, 33syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e ( A +e B )  =  (  -e A +e  -e B ) )
3534oveq1d 6191 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
( A +e
B ) +e  -e C )  =  ( (  -e
A +e  -e B ) +e  -e C ) )
36 xnegdi 11298 . . . . . 6  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  -e
( B +e
C )  =  ( 
-e B +e  -e C ) )
3713, 22, 36syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e ( B +e C )  =  (  -e B +e  -e C ) )
3837oveq2d 6192 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
A +e  -e ( B +e C ) )  =  (  -e
A +e ( 
-e B +e  -e C ) ) )
3932, 35, 383eqtr4d 2500 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
( A +e
B ) +e  -e C )  =  (  -e A +e  -e
( B +e
C ) ) )
40 xaddcl 11294 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  e.  RR* )
411, 13, 40syl2anc 661 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  ( A +e B )  e.  RR* )
42 xnegdi 11298 . . . 4  |-  ( ( ( A +e
B )  e.  RR*  /\  C  e.  RR* )  -> 
-e ( ( A +e B ) +e C )  =  (  -e ( A +e B ) +e  -e C ) )
4341, 22, 42syl2anc 661 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e ( ( A +e
B ) +e
C )  =  ( 
-e ( A +e B ) +e  -e
C ) )
44 xaddcl 11294 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B +e C )  e.  RR* )
4513, 22, 44syl2anc 661 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  ( B +e C )  e.  RR* )
46 xnegdi 11298 . . . 4  |-  ( ( A  e.  RR*  /\  ( B +e C )  e.  RR* )  ->  -e
( A +e
( B +e
C ) )  =  (  -e A +e  -e
( B +e
C ) ) )
471, 45, 46syl2anc 661 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e ( A +e ( B +e C ) )  =  ( 
-e A +e  -e ( B +e C ) ) )
4839, 43, 473eqtr4d 2500 . 2  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e ( ( A +e
B ) +e
C )  =  -e ( A +e ( B +e C ) ) )
49 xaddcl 11294 . . . 4  |-  ( ( ( A +e
B )  e.  RR*  /\  C  e.  RR* )  ->  ( ( A +e B ) +e C )  e. 
RR* )
5041, 22, 49syl2anc 661 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  ( ( A +e B ) +e C )  e.  RR* )
51 xaddcl 11294 . . . 4  |-  ( ( A  e.  RR*  /\  ( B +e C )  e.  RR* )  ->  ( A +e ( B +e C ) )  e.  RR* )
521, 45, 51syl2anc 661 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  ( A +e ( B +e C ) )  e.  RR* )
53 xneg11 11272 . . 3  |-  ( ( ( ( A +e B ) +e C )  e. 
RR*  /\  ( A +e ( B +e C ) )  e.  RR* )  ->  (  -e ( ( A +e
B ) +e
C )  =  -e ( A +e ( B +e C ) )  <-> 
( ( A +e B ) +e C )  =  ( A +e
( B +e
C ) ) ) )
5450, 52, 53syl2anc 661 . 2  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
( ( A +e B ) +e C )  = 
-e ( A +e ( B +e C ) )  <->  ( ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) ) )
5548, 54mpbid 210 1  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  ( ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    =/= wne 2641  (class class class)co 6176   +oocpnf 9502   -oocmnf 9503   RR*cxr 9504    -ecxne 11173   +ecxad 11174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-po 4725  df-so 4726  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-sub 9684  df-neg 9685  df-xneg 11176  df-xadd 11177
This theorem is referenced by: (None)
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