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Theorem xrge0addass 26163
Description: Associativity of extended nonnegative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)
Assertion
Ref Expression
xrge0addass  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )

Proof of Theorem xrge0addass
StepHypRef Expression
1 iccssxr 11390 . . 3  |-  ( 0 [,] +oo )  C_  RR*
2 simp1 988 . . 3  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  A  e.  ( 0 [,] +oo )
)
31, 2sseldi 3366 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  A  e.  RR* )
4 0xr 9442 . . . . . . 7  |-  0  e.  RR*
54a1i 11 . . . . . 6  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  0  e.  RR* )
6 pnfxr 11104 . . . . . . 7  |- +oo  e.  RR*
76a1i 11 . . . . . 6  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  -> +oo  e.  RR* )
8 elicc4 11374 . . . . . 6  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  A  e. 
RR* )  ->  ( A  e.  ( 0 [,] +oo )  <->  ( 0  <_  A  /\  A  <_ +oo ) ) )
95, 7, 3, 8syl3anc 1218 . . . . 5  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( A  e.  ( 0 [,] +oo ) 
<->  ( 0  <_  A  /\  A  <_ +oo )
) )
102, 9mpbid 210 . . . 4  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( 0  <_  A  /\  A  <_ +oo )
)
1110simpld 459 . . 3  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  0  <_  A
)
12 ge0nemnf 11157 . . 3  |-  ( ( A  e.  RR*  /\  0  <_  A )  ->  A  =/= -oo )
133, 11, 12syl2anc 661 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  A  =/= -oo )
14 simp2 989 . . 3  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  B  e.  ( 0 [,] +oo )
)
151, 14sseldi 3366 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  B  e.  RR* )
16 elicc4 11374 . . . . . 6  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  B  e. 
RR* )  ->  ( B  e.  ( 0 [,] +oo )  <->  ( 0  <_  B  /\  B  <_ +oo ) ) )
175, 7, 15, 16syl3anc 1218 . . . . 5  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( B  e.  ( 0 [,] +oo ) 
<->  ( 0  <_  B  /\  B  <_ +oo )
) )
1814, 17mpbid 210 . . . 4  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( 0  <_  B  /\  B  <_ +oo )
)
1918simpld 459 . . 3  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  0  <_  B
)
20 ge0nemnf 11157 . . 3  |-  ( ( B  e.  RR*  /\  0  <_  B )  ->  B  =/= -oo )
2115, 19, 20syl2anc 661 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  B  =/= -oo )
22 simp3 990 . . 3  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  C  e.  ( 0 [,] +oo )
)
231, 22sseldi 3366 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  C  e.  RR* )
24 elicc4 11374 . . . . . 6  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( 0 [,] +oo )  <->  ( 0  <_  C  /\  C  <_ +oo ) ) )
255, 7, 23, 24syl3anc 1218 . . . . 5  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( C  e.  ( 0 [,] +oo ) 
<->  ( 0  <_  C  /\  C  <_ +oo )
) )
2622, 25mpbid 210 . . . 4  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( 0  <_  C  /\  C  <_ +oo )
)
2726simpld 459 . . 3  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  0  <_  C
)
28 ge0nemnf 11157 . . 3  |-  ( ( C  e.  RR*  /\  0  <_  C )  ->  C  =/= -oo )
2923, 27, 28syl2anc 661 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  C  =/= -oo )
30 xaddass 11224 . 2  |-  ( ( ( 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 ) ) )
313, 13, 15, 21, 23, 29, 30syl222anc 1234 1  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +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 1369    e. wcel 1756    =/= wne 2618   class class class wbr 4304  (class class class)co 6103   0cc0 9294   +oocpnf 9427   -oocmnf 9428   RR*cxr 9429    <_ cle 9431   +ecxad 11099   [,]cicc 11315
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 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-addass 9359  ax-i2m1 9362  ax-1ne0 9363  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-xadd 11102  df-icc 11319
This theorem is referenced by: (None)
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