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Theorem xrge0addass 27838
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 11632 . . 3  |-  ( 0 [,] +oo )  C_  RR*
2 simp1 996 . . 3  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  A  e.  ( 0 [,] +oo )
)
31, 2sseldi 3497 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  A  e.  RR* )
4 0xr 9657 . . . . . . 7  |-  0  e.  RR*
54a1i 11 . . . . . 6  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  0  e.  RR* )
6 pnfxr 11346 . . . . . . 7  |- +oo  e.  RR*
76a1i 11 . . . . . 6  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  -> +oo  e.  RR* )
8 elicc4 11616 . . . . . 6  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  A  e. 
RR* )  ->  ( A  e.  ( 0 [,] +oo )  <->  ( 0  <_  A  /\  A  <_ +oo ) ) )
95, 7, 3, 8syl3anc 1228 . . . . 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 11399 . . 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 997 . . 3  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  B  e.  ( 0 [,] +oo )
)
151, 14sseldi 3497 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  B  e.  RR* )
16 elicc4 11616 . . . . . 6  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  B  e. 
RR* )  ->  ( B  e.  ( 0 [,] +oo )  <->  ( 0  <_  B  /\  B  <_ +oo ) ) )
175, 7, 15, 16syl3anc 1228 . . . . 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 11399 . . 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 998 . . 3  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  C  e.  ( 0 [,] +oo )
)
231, 22sseldi 3497 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  C  e.  RR* )
24 elicc4 11616 . . . . . 6  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( 0 [,] +oo )  <->  ( 0  <_  C  /\  C  <_ +oo ) ) )
255, 7, 23, 24syl3anc 1228 . . . . 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 11399 . . 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 11466 . 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 1244 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456  (class class class)co 6296   0cc0 9509   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    <_ cle 9646   +ecxad 11341   [,]cicc 11557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-addass 9574  ax-i2m1 9577  ax-1ne0 9578  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-xadd 11344  df-icc 11561
This theorem is referenced by:  inelcarsg  28454
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