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Theorem outsidele 32239
Description: Relate OutsideOf to Seg<_. Theorem 6.13 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
outsidele |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( POutsideOf <. A , B >. -> ( <. P , A >. Seg<_ <. P , B >. <-> A Btwn <. P , B >. ) ) )
Proof of Theorem outsidele
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 simpl 473 . . . . . 6 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> N e. NN )
2 simpr1 1067 . . . . . 6 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> P e. ( EE ` N ) )
3 simpr2 1068 . . . . . 6 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
4 simpr3 1069 . . . . . 6 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
5 brsegle2 32216 . . . . . 6 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( P e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( <. P , A >. Seg<_ <. P , B >. <-> E. y e. ( EE ` N ) ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) )
61, 2, 3, 2, 4, 5syl122anc 1335 . . . . 5 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( <. P , A >. Seg<_ <. P , B >. <-> E. y e. ( EE ` N ) ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) )
76adantr 481 . . . 4 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ POutsideOf <. A , B >. ) -> ( <. P , A >. Seg<_ <. P , B >. <-> E. y e. ( EE ` N ) ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) )
8 simprl 794 . . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> POutsideOf <. A , B >. )
9 outsideofcom 32235 . . . . . . . . . . 11 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( POutsideOf <. A , B >. <-> POutsideOf <. B , A >. ) )
109ad2antrr 762 . . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> ( POutsideOf <. A , B >. <-> POutsideOf <. B , A >. ) )
118, 10mpbid 222 . . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> POutsideOf <. B , A >. )
12 simpll 790 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> N e. NN )
13 simplr1 1103 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> P e. ( EE ` N ) )
14 simplr3 1105 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> B e. ( EE ` N ) )
1512, 13, 14cgrrflxd 32095 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> <. P , B >.Cgr <. P , B >. )
1615adantr 481 . . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> <. P , B >.Cgr <. P , B >. )
1711, 16jca 554 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> ( POutsideOf <. B , A >. /\ <. P , B >.Cgr <. P , B >. ) )
18 simprrl 804 . . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> A Btwn <. P , y >. )
19 simpr 477 . . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> y e. ( EE ` N ) )
20 simplr2 1104 . . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> A e. ( EE ` N ) )
21 btwncolinear1 32176 . . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ y e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( A Btwn <. P , y >. -> P Colinear <. y , A >. ) )
2212, 13, 19, 20, 21syl13anc 1328 . . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> ( A Btwn <. P , y >. -> P Colinear <. y , A >. ) )
2322adantr 481 . . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> ( A Btwn <. P , y >. -> P Colinear <. y , A >. ) )
2418, 23mpd 15 . . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> P Colinear <. y , A >. )
25 outsidene1 32230 . . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( POutsideOf <. A , B >. -> A =/= P ) )
2625ad2antrr 762 . . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> ( POutsideOf <. A , B >. -> A =/= P ) )
278, 26mpd 15 . . . . . . . . . . . 12 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> A =/= P )
2827neneqd 2799 . . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> -. A = P )
29 df-3an 1039 . . . . . . . . . . . . 13 |- ( ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) /\ P Btwn <. y , A >. ) <-> ( ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) /\ P Btwn <. y , A >. ) )
30 simpr2l 1120 . . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) /\ P Btwn <. y , A >. ) ) -> A Btwn <. P , y >. )
3112, 20, 13, 19, 30btwncomand 32122 . . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) /\ P Btwn <. y , A >. ) ) -> A Btwn <. y , P >. )
32 simpr3 1069 . . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) /\ P Btwn <. y , A >. ) ) -> P Btwn <. y , A >. )
33 btwnswapid2 32125 . . . . . . . . . . . . . . . 16 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ y e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) -> ( ( A Btwn <. y , P >. /\ P Btwn <. y , A >. ) -> A = P ) )
3412, 20, 19, 13, 33syl13anc 1328 . . . . . . . . . . . . . . 15 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> ( ( A Btwn <. y , P >. /\ P Btwn <. y , A >. ) -> A = P ) )
3534adantr 481 . . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) /\ P Btwn <. y , A >. ) ) -> ( ( A Btwn <. y , P >. /\ P Btwn <. y , A >. ) -> A = P ) )
3631, 32, 35mp2and 715 . . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) /\ P Btwn <. y , A >. ) ) -> A = P )
3729, 36sylan2br 493 . . . . . . . . . . . 12 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) /\ P Btwn <. y , A >. ) ) -> A = P )
3837expr 643 . . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> ( P Btwn <. y , A >. -> A = P ) )
3928, 38mtod 189 . . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> -. P Btwn <. y , A >. )
40 broutsideof 32228 . . . . . . . . . 10 |- ( POutsideOf <. y , A >. <-> ( P Colinear <. y , A >. /\ -. P Btwn <. y , A >. ) )
4124, 39, 40sylanbrc 698 . . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> POutsideOf <. y , A >. )
42 simprrr 805 . . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> <. P , y >.Cgr <. P , B >. )
4341, 42jca 554 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> ( POutsideOf <. y , A >. /\ <. P , y >.Cgr <. P , B >. ) )
44 outsideofeq 32237 . . . . . . . . . 10 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ P e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ B e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> ( ( ( POutsideOf <. B , A >. /\ <. P , B >.Cgr <. P , B >. ) /\ ( POutsideOf <. y , A >. /\ <. P , y >.Cgr <. P , B >. ) ) -> B = y ) )
4512, 13, 20, 13, 14, 14, 19, 44syl133anc 1349 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> ( ( ( POutsideOf <. B , A >. /\ <. P , B >.Cgr <. P , B >. ) /\ ( POutsideOf <. y , A >. /\ <. P , y >.Cgr <. P , B >. ) ) -> B = y ) )
4645adantr 481 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> ( ( ( POutsideOf <. B , A >. /\ <. P , B >.Cgr <. P , B >. ) /\ ( POutsideOf <. y , A >. /\ <. P , y >.Cgr <. P , B >. ) ) -> B = y ) )
4717, 43, 46mp2and 715 . . . . . . 7 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> B = y )
48 opeq2 4403 . . . . . . . . 9 |- ( B = y -> <. P , B >. = <. P , y >. )
4948breq2d 4665 . . . . . . . 8 |- ( B = y -> ( A Btwn <. P , B >. <-> A Btwn <. P , y >. ) )
5018, 49syl5ibrcom 237 . . . . . . 7 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> ( B = y -> A Btwn <. P , B >. ) )
5147, 50mpd 15 . . . . . 6 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( POutsideOf <. A , B >. /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> A Btwn <. P , B >. )
5251an4s 869 . . . . 5 |- ( ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ POutsideOf <. A , B >. ) /\ ( y e. ( EE ` N ) /\ ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) ) ) -> A Btwn <. P , B >. )
5352rexlimdvaa 3032 . . . 4 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ POutsideOf <. A , B >. ) -> ( E. y e. ( EE ` N ) ( A Btwn <. P , y >. /\ <. P , y >.Cgr <. P , B >. ) -> A Btwn <. P , B >. ) )
547, 53sylbid 230 . . 3 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ POutsideOf <. A , B >. ) -> ( <. P , A >. Seg<_ <. P , B >. -> A Btwn <. P , B >. ) )
55 btwnsegle 32224 . . . 4 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A Btwn <. P , B >. -> <. P , A >. Seg<_ <. P , B >. ) )
5655adantr 481 . . 3 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ POutsideOf <. A , B >. ) -> ( A Btwn <. P , B >. -> <. P , A >. Seg<_ <. P , B >. ) )
5754, 56impbid 202 . 2 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ POutsideOf <. A , B >. ) -> ( <. P , A >. Seg<_ <. P , B >. <-> A Btwn <. P , B >. ) )
5857ex 450 1 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( POutsideOf <. A , B >. -> ( <. P , A >. Seg<_ <. P , B >. <-> A Btwn <. P , B >. ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770   Colinear ccolin 32144   Seg<_ csegle 32213  OutsideOfcoutsideof 32226
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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-ee 25771  df-btwn 25772  df-cgr 25773  df-ofs 32090  df-colinear 32146  df-ifs 32147  df-cgr3 32148  df-fs 32149  df-segle 32214  df-outsideof 32227
This theorem is referenced by: (None)
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