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Theorem xmulasslem3 12116
Description: Lemma for xmulass 12117. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmulasslem3 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
Proof of Theorem xmulasslem3
StepHypRef Expression
1 recn 10026 . . . . . . . . . 10 |- ( A e. RR -> A e. CC )
2 recn 10026 . . . . . . . . . 10 |- ( B e. RR -> B e. CC )
3 recn 10026 . . . . . . . . . 10 |- ( C e. RR -> C e. CC )
4 mulass 10024 . . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
51, 2, 3, 4syl3an 1368 . . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
653expa 1265 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
7 remulcl 10021 . . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
8 rexmul 12101 . . . . . . . . 9 |- ( ( ( A x. B ) e. RR /\ C e. RR ) -> ( ( A x. B ) xe C ) = ( ( A x. B ) x. C ) )
97, 8sylan 488 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A x. B ) xe C ) = ( ( A x. B ) x. C ) )
10 remulcl 10021 . . . . . . . . . 10 |- ( ( B e. RR /\ C e. RR ) -> ( B x. C ) e. RR )
11 rexmul 12101 . . . . . . . . . 10 |- ( ( A e. RR /\ ( B x. C ) e. RR ) -> ( A xe ( B x. C ) ) = ( A x. ( B x. C ) ) )
1210, 11sylan2 491 . . . . . . . . 9 |- ( ( A e. RR /\ ( B e. RR /\ C e. RR ) ) -> ( A xe ( B x. C ) ) = ( A x. ( B x. C ) ) )
1312anassrs 680 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A xe ( B x. C ) ) = ( A x. ( B x. C ) ) )
146, 9, 133eqtr4d 2666 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A x. B ) xe C ) = ( A xe ( B x. C ) ) )
15 rexmul 12101 . . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A xe B ) = ( A x. B ) )
1615adantr 481 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A xe B ) = ( A x. B ) )
1716oveq1d 6665 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A xe B ) xe C ) = ( ( A x. B ) xe C ) )
18 rexmul 12101 . . . . . . . . 9 |- ( ( B e. RR /\ C e. RR ) -> ( B xe C ) = ( B x. C ) )
1918adantll 750 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( B xe C ) = ( B x. C ) )
2019oveq2d 6666 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A xe ( B xe C ) ) = ( A xe ( B x. C ) ) )
2114, 17, 203eqtr4d 2666 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
2221adantll 750 . . . . 5 |- ( ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ ( A e. RR /\ B e. RR ) ) /\ C e. RR ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
23 oveq2 6658 . . . . . . . . 9 |- ( C = +oo -> ( ( A xe B ) xe C ) = ( ( A xe B ) xe +oo ) )
24 simp1l 1085 . . . . . . . . . . 11 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> A e. RR* )
25 simp2l 1087 . . . . . . . . . . 11 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> B e. RR* )
26 xmulcl 12103 . . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> ( A xe B ) e. RR* )
2724, 25, 26syl2anc 693 . . . . . . . . . 10 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( A xe B ) e. RR* )
28 xmulgt0 12113 . . . . . . . . . . 11 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) -> 0 < ( A xe B ) )
29283adant3 1081 . . . . . . . . . 10 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> 0 < ( A xe B ) )
30 xmulpnf1 12104 . . . . . . . . . 10 |- ( ( ( A xe B ) e. RR* /\ 0 < ( A xe B ) ) -> ( ( A xe B ) xe +oo ) = +oo )
3127, 29, 30syl2anc 693 . . . . . . . . 9 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( ( A xe B ) xe +oo ) = +oo )
3223, 31sylan9eqr 2678 . . . . . . . 8 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ C = +oo ) -> ( ( A xe B ) xe C ) = +oo )
33 simpl1 1064 . . . . . . . . 9 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ C = +oo ) -> ( A e. RR* /\ 0 < A ) )
34 xmulpnf1 12104 . . . . . . . . 9 |- ( ( A e. RR* /\ 0 < A ) -> ( A xe +oo ) = +oo )
3533, 34syl 17 . . . . . . . 8 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ C = +oo ) -> ( A xe +oo ) = +oo )
3632, 35eqtr4d 2659 . . . . . . 7 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ C = +oo ) -> ( ( A xe B ) xe C ) = ( A xe +oo ) )
37 oveq2 6658 . . . . . . . . 9 |- ( C = +oo -> ( B xe C ) = ( B xe +oo ) )
38 xmulpnf1 12104 . . . . . . . . . 10 |- ( ( B e. RR* /\ 0 < B ) -> ( B xe +oo ) = +oo )
39383ad2ant2 1083 . . . . . . . . 9 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( B xe +oo ) = +oo )
4037, 39sylan9eqr 2678 . . . . . . . 8 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ C = +oo ) -> ( B xe C ) = +oo )
4140oveq2d 6666 . . . . . . 7 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ C = +oo ) -> ( A xe ( B xe C ) ) = ( A xe +oo ) )
4236, 41eqtr4d 2659 . . . . . 6 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ C = +oo ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
4342adantlr 751 . . . . 5 |- ( ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ ( A e. RR /\ B e. RR ) ) /\ C = +oo ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
44 simpl3r 1117 . . . . . 6 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ ( A e. RR /\ B e. RR ) ) -> 0 < C )
45 xmulasslem2 12112 . . . . . 6 |- ( ( 0 < C /\ C = -oo ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
4644, 45sylan 488 . . . . 5 |- ( ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ ( A e. RR /\ B e. RR ) ) /\ C = -oo ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
47 simp3l 1089 . . . . . . 7 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> C e. RR* )
48 elxr 11950 . . . . . . 7 |- ( C e. RR* <-> ( C e. RR \/ C = +oo \/ C = -oo ) )
4947, 48sylib 208 . . . . . 6 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( C e. RR \/ C = +oo \/ C = -oo ) )
5049adantr 481 . . . . 5 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ ( A e. RR /\ B e. RR ) ) -> ( C e. RR \/ C = +oo \/ C = -oo ) )
5122, 43, 46, 50mpjao3dan 1395 . . . 4 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ ( A e. RR /\ B e. RR ) ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
5251anassrs 680 . . 3 |- ( ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A e. RR ) /\ B e. RR ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
53 xmulpnf2 12105 . . . . . . . 8 |- ( ( C e. RR* /\ 0 < C ) -> ( +oo xe C ) = +oo )
54533ad2ant3 1084 . . . . . . 7 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( +oo xe C ) = +oo )
55343ad2ant1 1082 . . . . . . 7 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( A xe +oo ) = +oo )
5654, 55eqtr4d 2659 . . . . . 6 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( +oo xe C ) = ( A xe +oo ) )
5756adantr 481 . . . . 5 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ B = +oo ) -> ( +oo xe C ) = ( A xe +oo ) )
58 oveq2 6658 . . . . . . 7 |- ( B = +oo -> ( A xe B ) = ( A xe +oo ) )
5958, 55sylan9eqr 2678 . . . . . 6 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ B = +oo ) -> ( A xe B ) = +oo )
6059oveq1d 6665 . . . . 5 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ B = +oo ) -> ( ( A xe B ) xe C ) = ( +oo xe C ) )
61 oveq1 6657 . . . . . . 7 |- ( B = +oo -> ( B xe C ) = ( +oo xe C ) )
6261, 54sylan9eqr 2678 . . . . . 6 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ B = +oo ) -> ( B xe C ) = +oo )
6362oveq2d 6666 . . . . 5 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ B = +oo ) -> ( A xe ( B xe C ) ) = ( A xe +oo ) )
6457, 60, 633eqtr4d 2666 . . . 4 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ B = +oo ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
6564adantlr 751 . . 3 |- ( ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A e. RR ) /\ B = +oo ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
66 simpl2r 1115 . . . 4 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A e. RR ) -> 0 < B )
67 xmulasslem2 12112 . . . 4 |- ( ( 0 < B /\ B = -oo ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
6866, 67sylan 488 . . 3 |- ( ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A e. RR ) /\ B = -oo ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
69 elxr 11950 . . . . 5 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
7025, 69sylib 208 . . . 4 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
7170adantr 481 . . 3 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A e. RR ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
7252, 65, 68, 71mpjao3dan 1395 . 2 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A e. RR ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
73 simpl3 1066 . . . 4 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A = +oo ) -> ( C e. RR* /\ 0 < C ) )
7473, 53syl 17 . . 3 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A = +oo ) -> ( +oo xe C ) = +oo )
75 oveq1 6657 . . . . 5 |- ( A = +oo -> ( A xe B ) = ( +oo xe B ) )
76 xmulpnf2 12105 . . . . . 6 |- ( ( B e. RR* /\ 0 < B ) -> ( +oo xe B ) = +oo )
77763ad2ant2 1083 . . . . 5 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( +oo xe B ) = +oo )
7875, 77sylan9eqr 2678 . . . 4 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A = +oo ) -> ( A xe B ) = +oo )
7978oveq1d 6665 . . 3 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A = +oo ) -> ( ( A xe B ) xe C ) = ( +oo xe C ) )
80 oveq1 6657 . . . 4 |- ( A = +oo -> ( A xe ( B xe C ) ) = ( +oo xe ( B xe C ) ) )
81 xmulcl 12103 . . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> ( B xe C ) e. RR* )
8225, 47, 81syl2anc 693 . . . . 5 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( B xe C ) e. RR* )
83 xmulgt0 12113 . . . . . 6 |- ( ( ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> 0 < ( B xe C ) )
84833adant1 1079 . . . . 5 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> 0 < ( B xe C ) )
85 xmulpnf2 12105 . . . . 5 |- ( ( ( B xe C ) e. RR* /\ 0 < ( B xe C ) ) -> ( +oo xe ( B xe C ) ) = +oo )
8682, 84, 85syl2anc 693 . . . 4 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( +oo xe ( B xe C ) ) = +oo )
8780, 86sylan9eqr 2678 . . 3 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A = +oo ) -> ( A xe ( B xe C ) ) = +oo )
8874, 79, 873eqtr4d 2666 . 2 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A = +oo ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
89 simp1r 1086 . . 3 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> 0 < A )
90 xmulasslem2 12112 . . 3 |- ( ( 0 < A /\ A = -oo ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
9189, 90sylan 488 . 2 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) /\ A = -oo ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
92 elxr 11950 . . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
9324, 92sylib 208 . 2 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
9472, 88, 91, 93mpjao3dan 1395 1 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) /\ ( C e. RR* /\ 0 < C ) ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   xecxmu 11945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-xmul 11948
This theorem is referenced by:  xmulass  12117
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