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Theorem xaddass2 11445
Description: Associativity of extended real addition. See xaddass 11444 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 1018 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  A  e.  RR* )
2 xnegcl 11415 . . . . . 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 1019 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  A  =/= +oo )
5 pnfxr 11324 . . . . . . . . 9  |- +oo  e.  RR*
6 xneg11 11417 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  (  -e A  =  -e +oo  <->  A  = +oo ) )
71, 5, 6sylancl 660 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
A  =  -e +oo 
<->  A  = +oo )
)
87necon3bid 2712 . . . . . . 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 11411 . . . . . . 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 2749 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e A  =/= -oo )
13 simp2l 1020 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  B  e.  RR* )
14 xnegcl 11415 . . . . . 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 1021 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  B  =/= +oo )
17 xneg11 11417 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\ +oo  e.  RR* )  ->  (  -e B  =  -e +oo  <->  B  = +oo ) )
1813, 5, 17sylancl 660 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
B  =  -e +oo 
<->  B  = +oo )
)
1918necon3bid 2712 . . . . . . 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 2749 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e B  =/= -oo )
22 simp3l 1022 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  C  e.  RR* )
23 xnegcl 11415 . . . . . 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 1023 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  C  =/= +oo )
26 xneg11 11417 . . . . . . . . 9  |-  ( ( C  e.  RR*  /\ +oo  e.  RR* )  ->  (  -e C  =  -e +oo  <->  C  = +oo ) )
2722, 5, 26sylancl 660 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  (  -e
C  =  -e +oo 
<->  C  = +oo )
)
2827necon3bid 2712 . . . . . . 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 2749 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  -e C  =/= -oo )
31 xaddass 11444 . . . . 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 1242 . . . 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 11443 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -e
( A +e
B )  =  ( 
-e A +e  -e B ) )
341, 13, 33syl2anc 659 . . . . 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 6285 . . . 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 11443 . . . . . 6  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  -e
( B +e
C )  =  ( 
-e B +e  -e C ) )
3713, 22, 36syl2anc 659 . . . . 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 6286 . . . 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 2505 . . 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 11439 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  e.  RR* )
411, 13, 40syl2anc 659 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  ( A +e B )  e.  RR* )
42 xnegdi 11443 . . . 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 659 . . 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 11439 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B +e C )  e.  RR* )
4513, 22, 44syl2anc 659 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo )
)  ->  ( B +e C )  e.  RR* )
46 xnegdi 11443 . . . 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 659 . . 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 2505 . 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 11439 . . . 4  |-  ( ( ( A +e
B )  e.  RR*  /\  C  e.  RR* )  ->  ( ( A +e B ) +e C )  e. 
RR* )
5041, 22, 49syl2anc 659 . . 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 11439 . . . 4  |-  ( ( A  e.  RR*  /\  ( B +e C )  e.  RR* )  ->  ( A +e ( B +e C ) )  e.  RR* )
521, 45, 51syl2anc 659 . . 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 11417 . . 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 659 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649  (class class class)co 6270   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616    -ecxne 11318   +ecxad 11319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-sub 9798  df-neg 9799  df-xneg 11321  df-xadd 11322
This theorem is referenced by: (None)
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