Theorem xaddass 9826

Description: Associativity of extended real addition. The correct condition here is "it is not the case that both +oo and -oo appear as one of A , B , C, i.e. -. { +oo , -oo } C_ { A , B , C }", but this condition is difficult to work with, so we break the theorem into two parts: this one, where -oo is not present in A , B , C, and xaddass2 9827, where +oo is not present. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xaddass	|- ( ( ( 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 xaddass

Step	Hyp	Ref	Expression
1		recn 7907	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
2		recn 7907	. . . . . . . . . 10 |- ( B e. RR -> B e. CC )
3		recn 7907	. . . . . . . . . 10 |- ( C e. RR -> C e. CC )
4		addass 7904	. . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5	1, 2, 3, 4	syl3an 1275	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
6	5	3expa 1198	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
7		readdcl 7900	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
8		rexadd 9809	. . . . . . . . 9 |- ( ( ( A + B ) e. RR /\ C e. RR ) -> ( ( A + B ) +e C ) = ( ( A + B ) + C ) )
9	7, 8	sylan 281	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A + B ) +e C ) = ( ( A + B ) + C ) )
10		readdcl 7900	. . . . . . . . . 10 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
11		rexadd 9809	. . . . . . . . . 10 |- ( ( A e. RR /\ ( B + C ) e. RR ) -> ( A +e ( B + C ) ) = ( A + ( B + C ) ) )
12	10, 11	sylan2 284	. . . . . . . . 9 |- ( ( A e. RR /\ ( B e. RR /\ C e. RR ) ) -> ( A +e ( B + C ) ) = ( A + ( B + C ) ) )
13	12	anassrs 398	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A +e ( B + C ) ) = ( A + ( B + C ) ) )
14	6, 9, 13	3eqtr4d 2213	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A + B ) +e C ) = ( A +e ( B + C ) ) )
15		rexadd 9809	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
16	15	adantr 274	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A +e B ) = ( A + B ) )
17	16	oveq1d 5868	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A +e B ) +e C ) = ( ( A + B ) +e C ) )
18		rexadd 9809	. . . . . . . . 9 |- ( ( B e. RR /\ C e. RR ) -> ( B +e C ) = ( B + C ) )
19	18	adantll 473	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( B +e C ) = ( B + C ) )
20	19	oveq2d 5869	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A +e ( B +e C ) ) = ( A +e ( B + C ) ) )
21	14, 17, 20	3eqtr4d 2213	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
22	21	adantll 473	. . . . 5 |- ( ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ ( A e. RR /\ B e. RR ) ) /\ C e. RR ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
23		oveq2 5861	. . . . . . . . 9 |- ( C = +oo -> ( ( A +e B ) +e C ) = ( ( A +e B ) +e +oo ) )
24		simp1l 1016	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> A e. RR* )
25		simp2l 1018	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> B e. RR* )
26		xaddcl 9817	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* )
27	24, 25, 26	syl2anc 409	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A +e B ) e. RR* )
28		xaddnemnf 9814	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) ) -> ( A +e B ) =/= -oo )
29	28	3adant3 1012	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A +e B ) =/= -oo )
30		xaddpnf1 9803	. . . . . . . . . 10 |- ( ( ( A +e B ) e. RR* /\ ( A +e B ) =/= -oo ) -> ( ( A +e B ) +e +oo ) = +oo )
31	27, 29, 30	syl2anc 409	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( ( A +e B ) +e +oo ) = +oo )
32	23, 31	sylan9eqr 2225	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( ( A +e B ) +e C ) = +oo )
33		xaddpnf1 9803	. . . . . . . . . 10 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
34	33	3ad2ant1 1013	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A +e +oo ) = +oo )
35	34	adantr 274	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( A +e +oo ) = +oo )
36	32, 35	eqtr4d 2206	. . . . . . 7 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( ( A +e B ) +e C ) = ( A +e +oo ) )
37		oveq2 5861	. . . . . . . . 9 |- ( C = +oo -> ( B +e C ) = ( B +e +oo ) )
38		xaddpnf1 9803	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B +e +oo ) = +oo )
39	38	3ad2ant2 1014	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( B +e +oo ) = +oo )
40	37, 39	sylan9eqr 2225	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( B +e C ) = +oo )
41	40	oveq2d 5869	. . . . . . 7 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( A +e ( B +e C ) ) = ( A +e +oo ) )
42	36, 41	eqtr4d 2206	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
43	42	adantlr 474	. . . . 5 |- ( ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ ( A e. RR /\ B e. RR ) ) /\ C = +oo ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
44		simp3 994	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( C e. RR* /\ C =/= -oo ) )
45		xrnemnf 9734	. . . . . . 7 |- ( ( C e. RR* /\ C =/= -oo ) <-> ( C e. RR \/ C = +oo ) )
46	44, 45	sylib 121	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( C e. RR \/ C = +oo ) )
47	46	adantr 274	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ ( A e. RR /\ B e. RR ) ) -> ( C e. RR \/ C = +oo ) )
48	22, 43, 47	mpjaodan 793	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ ( A e. RR /\ B e. RR ) ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
49	48	anassrs 398	. . 3 |- ( ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A e. RR ) /\ B e. RR ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
50		xaddpnf2 9804	. . . . . . . 8 |- ( ( C e. RR* /\ C =/= -oo ) -> ( +oo +e C ) = +oo )
51	50	3ad2ant3 1015	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( +oo +e C ) = +oo )
52	51, 34	eqtr4d 2206	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( +oo +e C ) = ( A +e +oo ) )
53	52	adantr 274	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( +oo +e C ) = ( A +e +oo ) )
54		oveq2 5861	. . . . . . 7 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
55	54, 34	sylan9eqr 2225	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( A +e B ) = +oo )
56	55	oveq1d 5868	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( ( A +e B ) +e C ) = ( +oo +e C ) )
57		oveq1 5860	. . . . . . 7 |- ( B = +oo -> ( B +e C ) = ( +oo +e C ) )
58	57, 51	sylan9eqr 2225	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( B +e C ) = +oo )
59	58	oveq2d 5869	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( A +e ( B +e C ) ) = ( A +e +oo ) )
60	53, 56, 59	3eqtr4d 2213	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
61	60	adantlr 474	. . 3 |- ( ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A e. RR ) /\ B = +oo ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
62		simpl2 996	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A e. RR ) -> ( B e. RR* /\ B =/= -oo ) )
63		xrnemnf 9734	. . . 4 |- ( ( B e. RR* /\ B =/= -oo ) <-> ( B e. RR \/ B = +oo ) )
64	62, 63	sylib 121	. . 3 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A e. RR ) -> ( B e. RR \/ B = +oo ) )
65	49, 61, 64	mpjaodan 793	. 2 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A e. RR ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
66		simpl3 997	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( C e. RR* /\ C =/= -oo ) )
67	66, 50	syl 14	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( +oo +e C ) = +oo )
68		simpl2l 1045	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> B e. RR* )
69		simpl3l 1047	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> C e. RR* )
70		xaddcl 9817	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> ( B +e C ) e. RR* )
71	68, 69, 70	syl2anc 409	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( B +e C ) e. RR* )
72		simpl2 996	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( B e. RR* /\ B =/= -oo ) )
73		xaddnemnf 9814	. . . . . 6 |- ( ( ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( B +e C ) =/= -oo )
74	72, 66, 73	syl2anc 409	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( B +e C ) =/= -oo )
75		xaddpnf2 9804	. . . . 5 |- ( ( ( B +e C ) e. RR* /\ ( B +e C ) =/= -oo ) -> ( +oo +e ( B +e C ) ) = +oo )
76	71, 74, 75	syl2anc 409	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( +oo +e ( B +e C ) ) = +oo )
77	67, 76	eqtr4d 2206	. . 3 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( +oo +e C ) = ( +oo +e ( B +e C ) ) )
78		simpr 109	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> A = +oo )
79	78	oveq1d 5868	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( A +e B ) = ( +oo +e B ) )
80		xaddpnf2 9804	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
81	72, 80	syl 14	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( +oo +e B ) = +oo )
82	79, 81	eqtrd 2203	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( A +e B ) = +oo )
83	82	oveq1d 5868	. . 3 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( ( A +e B ) +e C ) = ( +oo +e C ) )
84	78	oveq1d 5868	. . 3 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( A +e ( B +e C ) ) = ( +oo +e ( B +e C ) ) )
85	77, 83, 84	3eqtr4d 2213	. 2 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
86		simp1 992	. . 3 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A e. RR* /\ A =/= -oo ) )
87		xrnemnf 9734	. . 3 |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) )
88	86, 87	sylib 121	. 2 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A e. RR \/ A = +oo ) )
89	65, 85, 88	mpjaodan 793	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 103   \/ wo 703   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340  (class class class)co 5853   CCcc 7772   RRcr 7773   + caddc 7777   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953   +ecxad 9727

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871  ax-addass 7876  ax-rnegex 7883

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-xadd 9730

This theorem is referenced by:  xaddass2  9827  xpncan  9828  xadd4d  9842