Theorem xaddass 10147

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 10148, 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 8208	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
2		recn 8208	. . . . . . . . . 10 |- ( B e. RR -> B e. CC )
3		recn 8208	. . . . . . . . . 10 |- ( C e. RR -> C e. CC )
4		addass 8205	. . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5	1, 2, 3, 4	syl3an 1316	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
6	5	3expa 1230	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
7		readdcl 8201	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
8		rexadd 10130	. . . . . . . . 9 |- ( ( ( A + B ) e. RR /\ C e. RR ) -> ( ( A + B ) +e C ) = ( ( A + B ) + C ) )
9	7, 8	sylan 283	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A + B ) +e C ) = ( ( A + B ) + C ) )
10		readdcl 8201	. . . . . . . . . 10 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
11		rexadd 10130	. . . . . . . . . 10 |- ( ( A e. RR /\ ( B + C ) e. RR ) -> ( A +e ( B + C ) ) = ( A + ( B + C ) ) )
12	10, 11	sylan2 286	. . . . . . . . 9 |- ( ( A e. RR /\ ( B e. RR /\ C e. RR ) ) -> ( A +e ( B + C ) ) = ( A + ( B + C ) ) )
13	12	anassrs 400	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A +e ( B + C ) ) = ( A + ( B + C ) ) )
14	6, 9, 13	3eqtr4d 2274	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A + B ) +e C ) = ( A +e ( B + C ) ) )
15		rexadd 10130	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
16	15	adantr 276	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A +e B ) = ( A + B ) )
17	16	oveq1d 6043	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A +e B ) +e C ) = ( ( A + B ) +e C ) )
18		rexadd 10130	. . . . . . . . 9 |- ( ( B e. RR /\ C e. RR ) -> ( B +e C ) = ( B + C ) )
19	18	adantll 476	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( B +e C ) = ( B + C ) )
20	19	oveq2d 6044	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A +e ( B +e C ) ) = ( A +e ( B + C ) ) )
21	14, 17, 20	3eqtr4d 2274	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
22	21	adantll 476	. . . . 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 6036	. . . . . . . . 9 |- ( C = +oo -> ( ( A +e B ) +e C ) = ( ( A +e B ) +e +oo ) )
24		simp1l 1048	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> A e. RR* )
25		simp2l 1050	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> B e. RR* )
26		xaddcl 10138	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* )
27	24, 25, 26	syl2anc 411	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A +e B ) e. RR* )
28		xaddnemnf 10135	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) ) -> ( A +e B ) =/= -oo )
29	28	3adant3 1044	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A +e B ) =/= -oo )
30		xaddpnf1 10124	. . . . . . . . . 10 |- ( ( ( A +e B ) e. RR* /\ ( A +e B ) =/= -oo ) -> ( ( A +e B ) +e +oo ) = +oo )
31	27, 29, 30	syl2anc 411	. . . . . . . . 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 2286	. . . . . . . 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 10124	. . . . . . . . . 10 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
34	33	3ad2ant1 1045	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A +e +oo ) = +oo )
35	34	adantr 276	. . . . . . . 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 2267	. . . . . . 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 6036	. . . . . . . . 9 |- ( C = +oo -> ( B +e C ) = ( B +e +oo ) )
38		xaddpnf1 10124	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B +e +oo ) = +oo )
39	38	3ad2ant2 1046	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( B +e +oo ) = +oo )
40	37, 39	sylan9eqr 2286	. . . . . . . 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 6044	. . . . . . 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 2267	. . . . . 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 477	. . . . 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 1026	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( C e. RR* /\ C =/= -oo ) )
45		xrnemnf 10055	. . . . . . 7 |- ( ( C e. RR* /\ C =/= -oo ) <-> ( C e. RR \/ C = +oo ) )
46	44, 45	sylib 122	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( C e. RR \/ C = +oo ) )
47	46	adantr 276	. . . . 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 806	. . . 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 400	. . 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 10125	. . . . . . . 8 |- ( ( C e. RR* /\ C =/= -oo ) -> ( +oo +e C ) = +oo )
51	50	3ad2ant3 1047	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( +oo +e C ) = +oo )
52	51, 34	eqtr4d 2267	. . . . . 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 276	. . . . 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 6036	. . . . . . 7 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
55	54, 34	sylan9eqr 2286	. . . . . 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 6043	. . . . 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 6035	. . . . . . 7 |- ( B = +oo -> ( B +e C ) = ( +oo +e C ) )
58	57, 51	sylan9eqr 2286	. . . . . 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 6044	. . . . 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 2274	. . . 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 477	. . 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 1028	. . . 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 10055	. . . 4 |- ( ( B e. RR* /\ B =/= -oo ) <-> ( B e. RR \/ B = +oo ) )
64	62, 63	sylib 122	. . 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 806	. 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 1029	. . . . 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 1077	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> B e. RR* )
69		simpl3l 1079	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> C e. RR* )
70		xaddcl 10138	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> ( B +e C ) e. RR* )
71	68, 69, 70	syl2anc 411	. . . . 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 1028	. . . . . 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 10135	. . . . . 6 |- ( ( ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( B +e C ) =/= -oo )
74	72, 66, 73	syl2anc 411	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( B +e C ) =/= -oo )
75		xaddpnf2 10125	. . . . 5 |- ( ( ( B +e C ) e. RR* /\ ( B +e C ) =/= -oo ) -> ( +oo +e ( B +e C ) ) = +oo )
76	71, 74, 75	syl2anc 411	. . . 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 2267	. . 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 110	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> A = +oo )
79	78	oveq1d 6043	. . . . 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 10125	. . . . . 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 2264	. . . 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 6043	. . 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 6043	. . 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 2274	. 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 1024	. . 3 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A e. RR* /\ A =/= -oo ) )
87		xrnemnf 10055	. . 3 |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) )
88	86, 87	sylib 122	. 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 806	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   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   =/= wne 2403  (class class class)co 6028   CCcc 8073   RRcr 8074   + caddc 8078   +oocpnf 8254   -oocmnf 8255   RR*cxr 8256   +ecxad 10048

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172  ax-addass 8177  ax-rnegex 8184

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8259  df-mnf 8260  df-xr 8261  df-xadd 10051

This theorem is referenced by:  xaddass2  10148  xpncan  10149  xadd4d  10163