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Theorem xaddass2 9646
Description: Associativity of extended real addition. See xaddass 9645 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 1005 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> A e. RR* )
2 xnegcl 9608 . . . . . 6 |- ( A e. RR* -> -e A e. RR* )
31, 2syl 14 . . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e A e. RR* )
4 simp1r 1006 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> A =/= +oo )
5 pnfxr 7811 . . . . . . . . 9 |- +oo e. RR*
6 xneg11 9610 . . . . . . . . 9 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( -e A = -e +oo <-> A = +oo ) )
71, 5, 6sylancl 409 . . . . . . . 8 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e A = -e +oo <-> A = +oo ) )
87necon3bid 2347 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e A =/= -e +oo <-> A =/= +oo ) )
94, 8mpbird 166 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e A =/= -e +oo )
10 xnegpnf 9604 . . . . . . 7 |- -e +oo = -oo
1110a1i 9 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e +oo = -oo )
129, 11neeqtrd 2334 . . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e A =/= -oo )
13 simp2l 1007 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> B e. RR* )
14 xnegcl 9608 . . . . . 6 |- ( B e. RR* -> -e B e. RR* )
1513, 14syl 14 . . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e B e. RR* )
16 simp2r 1008 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> B =/= +oo )
17 xneg11 9610 . . . . . . . . 9 |- ( ( B e. RR* /\ +oo e. RR* ) -> ( -e B = -e +oo <-> B = +oo ) )
1813, 5, 17sylancl 409 . . . . . . . 8 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e B = -e +oo <-> B = +oo ) )
1918necon3bid 2347 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e B =/= -e +oo <-> B =/= +oo ) )
2016, 19mpbird 166 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e B =/= -e +oo )
2120, 11neeqtrd 2334 . . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e B =/= -oo )
22 simp3l 1009 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> C e. RR* )
23 xnegcl 9608 . . . . . 6 |- ( C e. RR* -> -e C e. RR* )
2422, 23syl 14 . . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e C e. RR* )
25 simp3r 1010 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> C =/= +oo )
26 xneg11 9610 . . . . . . . . 9 |- ( ( C e. RR* /\ +oo e. RR* ) -> ( -e C = -e +oo <-> C = +oo ) )
2722, 5, 26sylancl 409 . . . . . . . 8 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e C = -e +oo <-> C = +oo ) )
2827necon3bid 2347 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e C =/= -e +oo <-> C =/= +oo ) )
2925, 28mpbird 166 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e C =/= -e +oo )
3029, 11neeqtrd 2334 . . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e C =/= -oo )
31 xaddass 9645 . . . . 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 1232 . . . 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 9644 . . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
341, 13, 33syl2anc 408 . . . . 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 5782 . . . 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 9644 . . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> -e ( B +e C ) = ( -e B +e -e C ) )
3713, 22, 36syl2anc 408 . . . . 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 5783 . . . 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 2180 . . 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 9636 . . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* )
411, 13, 40syl2anc 408 . . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( A +e B ) e. RR* )
42 xnegdi 9644 . . . 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 408 . . 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 9636 . . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B +e C ) e. RR* )
4513, 22, 44syl2anc 408 . . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( B +e C ) e. RR* )
46 xnegdi 9644 . . . 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 408 . . 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 2180 . 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 9636 . . . 4 |- ( ( ( A +e B ) e. RR* /\ C e. RR* ) -> ( ( A +e B ) +e C ) e. RR* )
5041, 22, 49syl2anc 408 . . 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 9636 . . . 4 |- ( ( A e. RR* /\ ( B +e C ) e. RR* ) -> ( A +e ( B +e C ) ) e. RR* )
521, 45, 51syl2anc 408 . . 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 9610 . . 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 408 . 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 146 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 set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   =/= wne 2306  (class class class)co 5767   +oocpnf 7790   -oocmnf 7791   RR*cxr 7792   -ecxne 9549   +ecxad 9550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-pnf 7795  df-mnf 7796  df-xr 7797  df-sub 7928  df-neg 7929  df-xneg 9552  df-xadd 9553
This theorem is referenced by: (None)
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