Theorem xaddass2 9939

Description: Associativity of extended real addition. See xaddass 9938 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

Step	Hyp	Ref	Expression
1		simp1l 1023	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> A e. RR* )
2		xnegcl 9901	. . . . . 6 |- ( A e. RR* -> -e A e. RR* )
3	1, 2	syl 14	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e A e. RR* )
4		simp1r 1024	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> A =/= +oo )
5		pnfxr 8074	. . . . . . . . 9 |- +oo e. RR*
6		xneg11 9903	. . . . . . . . 9 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( -e A = -e +oo <-> A = +oo ) )
7	1, 5, 6	sylancl 413	. . . . . . . 8 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e A = -e +oo <-> A = +oo ) )
8	7	necon3bid 2405	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e A =/= -e +oo <-> A =/= +oo ) )
9	4, 8	mpbird 167	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e A =/= -e +oo )
10		xnegpnf 9897	. . . . . . 7 |- -e +oo = -oo
11	10	a1i 9	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e +oo = -oo )
12	9, 11	neeqtrd 2392	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e A =/= -oo )
13		simp2l 1025	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> B e. RR* )
14		xnegcl 9901	. . . . . 6 |- ( B e. RR* -> -e B e. RR* )
15	13, 14	syl 14	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e B e. RR* )
16		simp2r 1026	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> B =/= +oo )
17		xneg11 9903	. . . . . . . . 9 |- ( ( B e. RR* /\ +oo e. RR* ) -> ( -e B = -e +oo <-> B = +oo ) )
18	13, 5, 17	sylancl 413	. . . . . . . 8 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e B = -e +oo <-> B = +oo ) )
19	18	necon3bid 2405	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e B =/= -e +oo <-> B =/= +oo ) )
20	16, 19	mpbird 167	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e B =/= -e +oo )
21	20, 11	neeqtrd 2392	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e B =/= -oo )
22		simp3l 1027	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> C e. RR* )
23		xnegcl 9901	. . . . . 6 |- ( C e. RR* -> -e C e. RR* )
24	22, 23	syl 14	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e C e. RR* )
25		simp3r 1028	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> C =/= +oo )
26		xneg11 9903	. . . . . . . . 9 |- ( ( C e. RR* /\ +oo e. RR* ) -> ( -e C = -e +oo <-> C = +oo ) )
27	22, 5, 26	sylancl 413	. . . . . . . 8 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e C = -e +oo <-> C = +oo ) )
28	27	necon3bid 2405	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( -e C =/= -e +oo <-> C =/= +oo ) )
29	25, 28	mpbird 167	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e C =/= -e +oo )
30	29, 11	neeqtrd 2392	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e C =/= -oo )
31		xaddass 9938	. . . . 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 ) ) )
32	3, 12, 15, 21, 24, 30, 31	syl222anc 1265	. . . 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 9937	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
34	1, 13, 33	syl2anc 411	. . . . 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 ) )
35	34	oveq1d 5934	. . . 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 9937	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> -e ( B +e C ) = ( -e B +e -e C ) )
37	13, 22, 36	syl2anc 411	. . . . 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 ) )
38	37	oveq2d 5935	. . . 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 ) ) )
39	32, 35, 38	3eqtr4d 2236	. . 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 9929	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* )
41	1, 13, 40	syl2anc 411	. . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( A +e B ) e. RR* )
42		xnegdi 9937	. . . 4 |- ( ( ( A +e B ) e. RR* /\ C e. RR* ) -> -e ( ( A +e B ) +e C ) = ( -e ( A +e B ) +e -e C ) )
43	41, 22, 42	syl2anc 411	. . 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 9929	. . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B +e C ) e. RR* )
45	13, 22, 44	syl2anc 411	. . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> ( B +e C ) e. RR* )
46		xnegdi 9937	. . . 4 |- ( ( A e. RR* /\ ( B +e C ) e. RR* ) -> -e ( A +e ( B +e C ) ) = ( -e A +e -e ( B +e C ) ) )
47	1, 45, 46	syl2anc 411	. . 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 ) ) )
48	39, 43, 47	3eqtr4d 2236	. 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 9929	. . . 4 |- ( ( ( A +e B ) e. RR* /\ C e. RR* ) -> ( ( A +e B ) +e C ) e. RR* )
50	41, 22, 49	syl2anc 411	. . 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 9929	. . . 4 |- ( ( A e. RR* /\ ( B +e C ) e. RR* ) -> ( A +e ( B +e C ) ) e. RR* )
52	1, 45, 51	syl2anc 411	. . 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 9903	. . 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 ) ) ) )
54	50, 52, 53	syl2anc 411	. 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 ) ) ) )
55	48, 54	mpbid 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   =/= wne 2364  (class class class)co 5919   +oocpnf 8053   -oocmnf 8054   RR*cxr 8055   -ecxne 9838   +ecxad 9839

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-pnf 8058  df-mnf 8059  df-xr 8060  df-sub 8194  df-neg 8195  df-xneg 9841  df-xadd 9842

This theorem is referenced by: (None)