Theorem xaddass2 10104

Description: Associativity of extended real addition. See xaddass 10103 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 1047	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> A e. RR* )
2		xnegcl 10066	. . . . . 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 1048	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> A =/= +oo )
5		pnfxr 8231	. . . . . . . . 9 |- +oo e. RR*
6		xneg11 10068	. . . . . . . . 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 2443	. . . . . . 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 10062	. . . . . . 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 2430	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e A =/= -oo )
13		simp2l 1049	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> B e. RR* )
14		xnegcl 10066	. . . . . 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 1050	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> B =/= +oo )
17		xneg11 10068	. . . . . . . . 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 2443	. . . . . . 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 2430	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e B =/= -oo )
22		simp3l 1051	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> C e. RR* )
23		xnegcl 10066	. . . . . 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 1052	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> C =/= +oo )
26		xneg11 10068	. . . . . . . . 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 2443	. . . . . . 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 2430	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\ ( C e. RR* /\ C =/= +oo ) ) -> -e C =/= -oo )
31		xaddass 10103	. . . . 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 1289	. . . 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 10102	. . . . . 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 6032	. . . 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 10102	. . . . . 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 6033	. . . 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 2274	. . 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 10094	. . . . 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 10102	. . . 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 10094	. . . . 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 10102	. . . 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 2274	. 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 10094	. . . 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 10094	. . . 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 10068	. . 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 1004   = wceq 1397   e. wcel 2202   =/= wne 2402  (class class class)co 6017   +oocpnf 8210   -oocmnf 8211   RR*cxr 8212   -ecxne 10003   +ecxad 10004

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-xr 8217  df-sub 8351  df-neg 8352  df-xneg 10006  df-xadd 10007

This theorem is referenced by: (None)