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Theorem xaddass2 10254
Description: Associativity of extended real addition. See xaddass 10253 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 1052 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> A e. RR* )
2 xnegcl 10216 . . . . . 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 1053 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> A =/= +oo )
5 pnfxr 8371 . . . . . . . . 9 |- +oo e. RR*
6 xneg11 10218 . . . . . . . . 9 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( -e A = -e +oo <-> A = +oo ) )
71, 5, 6sylancl 417 . . . . . . . 8 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e A = -e +oo <-> A = +oo ) )
87necon3bid 2461 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e A =/= -e +oo <-> A =/= +oo ) )
94, 8mpbird 167 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e A =/= -e +oo )
10 xnegpnf 10212 . . . . . . 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 2448 . . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e A =/= -oo )
13 simp2l 1054 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> B e. RR* )
14 xnegcl 10216 . . . . . 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 1055 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> B =/= +oo )
17 xneg11 10218 . . . . . . . . 9 |- ( ( B e. RR* /\ +oo e. RR* ) -> ( -e B = -e +oo <-> B = +oo ) )
1813, 5, 17sylancl 417 . . . . . . . 8 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e B = -e +oo <-> B = +oo ) )
1918necon3bid 2461 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e B =/= -e +oo <-> B =/= +oo ) )
2016, 19mpbird 167 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e B =/= -e +oo )
2120, 11neeqtrd 2448 . . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e B =/= -oo )
22 simp3l 1056 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> C e. RR* )
23 xnegcl 10216 . . . . . 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 1057 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> C =/= +oo )
26 xneg11 10218 . . . . . . . . 9 |- ( ( C e. RR* /\ +oo e. RR* ) -> ( -e C = -e +oo <-> C = +oo ) )
2722, 5, 26sylancl 417 . . . . . . . 8 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e C = -e +oo <-> C = +oo ) )
2827necon3bid 2461 . . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e C =/= -e +oo <-> C =/= +oo ) )
2925, 28mpbird 167 . . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e C =/= -e +oo )
3029, 11neeqtrd 2448 . . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e C =/= -oo )
31 xaddass 10253 . . . . 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 1294 . . . 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 10252 . . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
341, 13, 33syl2anc 415 . . . . 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 6093 . . . 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 10252 . . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> -e ( B +e C ) = ( -e B +e -e C ) )
3713, 22, 36syl2anc 415 . . . . 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 6094 . . . 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 2281 . . 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 10244 . . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* )
411, 13, 40syl2anc 415 . . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( A +e B ) e. RR* )
42 xnegdi 10252 . . . 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 415 . . 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 10244 . . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B +e C ) e. RR* )
4513, 22, 44syl2anc 415 . . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( B +e C ) e. RR* )
46 xnegdi 10252 . . . 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 415 . . 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 2281 . 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 10244 . . . 4 |- ( ( ( A +e B ) e. RR* /\ C e. RR* ) -> ( ( A +e B ) +e C ) e. RR* )
5041, 22, 49syl2anc 415 . . 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 10244 . . . 4 |- ( ( A e. RR* /\ ( B +e C ) e. RR* ) -> ( A +e ( B +e C ) ) e. RR* )
521, 45, 51syl2anc 415 . . 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 10218 . . 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 415 . 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 147 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 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420  (class class class)co 6078   +oocpnf 8350   -oocmnf 8351   RR*cxr 8352   -ecxne 10153   +ecxad 10154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-addcom 8272  ax-addass 8274  ax-distr 8276  ax-i2m1 8277  ax-0id 8280  ax-rnegex 8281  ax-cnre 8283
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-pnf 8355  df-mnf 8356  df-xr 8357  df-sub 8492  df-neg 8493  df-xneg 10156  df-xadd 10157
This theorem is referenced by: (None)
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