ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltsrprg Unicode version
Theorem ltsrprg 7807
Description: Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)
Assertion
Ref Expression
ltsrprg |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) ) )
Proof of Theorem ltsrprg
Dummy variables x y z w v u f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enrex 7797 . 2 |- ~R e. _V
2 enrer 7795 . 2 |- ~R Er ( P. X. P. )
3 df-nr 7787 . 2 |- R. = ( ( P. X. P. ) /. ~R )
4 df-ltr 7790 . 2 |- <R = { <. x , y >. | ( ( x e. R. /\ y e. R. ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] ~R /\ y = [ <. v , u >. ] ~R ) /\ ( z +P. u ) <P ( w +P. v ) ) ) }
5 enreceq 7796 . . . . 5 |- ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) -> ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R <-> ( z +P. B ) = ( w +P. A ) ) )
6 enreceq 7796 . . . . . 6 |- ( ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. v , u >. ] ~R = [ <. C , D >. ] ~R <-> ( v +P. D ) = ( u +P. C ) ) )
7 eqcom 2195 . . . . . 6 |- ( ( v +P. D ) = ( u +P. C ) <-> ( u +P. C ) = ( v +P. D ) )
86, 7bitrdi 196 . . . . 5 |- ( ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. v , u >. ] ~R = [ <. C , D >. ] ~R <-> ( u +P. C ) = ( v +P. D ) ) )
95, 8bi2anan9 606 . . . 4 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R /\ [ <. v , u >. ] ~R = [ <. C , D >. ] ~R ) <-> ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) ) )
10 oveq12 5927 . . . . . . 7 |- ( ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) -> ( ( z +P. B ) +P. ( u +P. C ) ) = ( ( w +P. A ) +P. ( v +P. D ) ) )
1110adantl 277 . . . . . 6 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) ) -> ( ( z +P. B ) +P. ( u +P. C ) ) = ( ( w +P. A ) +P. ( v +P. D ) ) )
12 simprlr 538 . . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> u e. P. )
13 simplrr 536 . . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> B e. P. )
14 addcomprg 7638 . . . . . . . . . . . 12 |- ( ( u e. P. /\ B e. P. ) -> ( u +P. B ) = ( B +P. u ) )
1514oveq1d 5933 . . . . . . . . . . 11 |- ( ( u e. P. /\ B e. P. ) -> ( ( u +P. B ) +P. C ) = ( ( B +P. u ) +P. C ) )
1612, 13, 15syl2anc 411 . . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( u +P. B ) +P. C ) = ( ( B +P. u ) +P. C ) )
17 simprrl 539 . . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> C e. P. )
18 addassprg 7639 . . . . . . . . . . 11 |- ( ( u e. P. /\ B e. P. /\ C e. P. ) -> ( ( u +P. B ) +P. C ) = ( u +P. ( B +P. C ) ) )
1912, 13, 17, 18syl3anc 1249 . . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( u +P. B ) +P. C ) = ( u +P. ( B +P. C ) ) )
20 addassprg 7639 . . . . . . . . . . 11 |- ( ( B e. P. /\ u e. P. /\ C e. P. ) -> ( ( B +P. u ) +P. C ) = ( B +P. ( u +P. C ) ) )
2113, 12, 17, 20syl3anc 1249 . . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( B +P. u ) +P. C ) = ( B +P. ( u +P. C ) ) )
2216, 19, 213eqtr3d 2234 . . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( u +P. ( B +P. C ) ) = ( B +P. ( u +P. C ) ) )
2322oveq2d 5934 . . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( z +P. ( u +P. ( B +P. C ) ) ) = ( z +P. ( B +P. ( u +P. C ) ) ) )
24 simplll 533 . . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> z e. P. )
25 addclpr 7597 . . . . . . . . . . . . 13 |- ( ( w e. P. /\ v e. P. ) -> ( w +P. v ) e. P. )
2625ad2ant2lr 510 . . . . . . . . . . . 12 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w +P. v ) e. P. )
27 addclpr 7597 . . . . . . . . . . . . 13 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
2827ad2ant2lr 510 . . . . . . . . . . . 12 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( B +P. C ) e. P. )
2926, 28anim12ci 339 . . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( B +P. C ) e. P. /\ ( w +P. v ) e. P. ) )
3029an4s 588 . . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( B +P. C ) e. P. /\ ( w +P. v ) e. P. ) )
3130simpld 112 . . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( B +P. C ) e. P. )
32 addassprg 7639 . . . . . . . . 9 |- ( ( z e. P. /\ u e. P. /\ ( B +P. C ) e. P. ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( z +P. ( u +P. ( B +P. C ) ) ) )
3324, 12, 31, 32syl3anc 1249 . . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( z +P. ( u +P. ( B +P. C ) ) ) )
34 addclpr 7597 . . . . . . . . . 10 |- ( ( u e. P. /\ C e. P. ) -> ( u +P. C ) e. P. )
3512, 17, 34syl2anc 411 . . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( u +P. C ) e. P. )
36 addassprg 7639 . . . . . . . . 9 |- ( ( z e. P. /\ B e. P. /\ ( u +P. C ) e. P. ) -> ( ( z +P. B ) +P. ( u +P. C ) ) = ( z +P. ( B +P. ( u +P. C ) ) ) )
3724, 13, 35, 36syl3anc 1249 . . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( z +P. B ) +P. ( u +P. C ) ) = ( z +P. ( B +P. ( u +P. C ) ) ) )
3823, 33, 373eqtr4d 2236 . . . . . . 7 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( z +P. B ) +P. ( u +P. C ) ) )
3938adantr 276 . . . . . 6 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( z +P. B ) +P. ( u +P. C ) ) )
40 simprll 537 . . . . . . . . . . . 12 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> v e. P. )
41 simplrl 535 . . . . . . . . . . . 12 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> A e. P. )
42 addcomprg 7638 . . . . . . . . . . . 12 |- ( ( v e. P. /\ A e. P. ) -> ( v +P. A ) = ( A +P. v ) )
4340, 41, 42syl2anc 411 . . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( v +P. A ) = ( A +P. v ) )
4443oveq1d 5933 . . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( v +P. A ) +P. D ) = ( ( A +P. v ) +P. D ) )
45 simprrr 540 . . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> D e. P. )
46 addassprg 7639 . . . . . . . . . . 11 |- ( ( v e. P. /\ A e. P. /\ D e. P. ) -> ( ( v +P. A ) +P. D ) = ( v +P. ( A +P. D ) ) )
4740, 41, 45, 46syl3anc 1249 . . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( v +P. A ) +P. D ) = ( v +P. ( A +P. D ) ) )
48 addassprg 7639 . . . . . . . . . . 11 |- ( ( A e. P. /\ v e. P. /\ D e. P. ) -> ( ( A +P. v ) +P. D ) = ( A +P. ( v +P. D ) ) )
4941, 40, 45, 48syl3anc 1249 . . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( A +P. v ) +P. D ) = ( A +P. ( v +P. D ) ) )
5044, 47, 493eqtr3d 2234 . . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( v +P. ( A +P. D ) ) = ( A +P. ( v +P. D ) ) )
5150oveq2d 5934 . . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( w +P. ( v +P. ( A +P. D ) ) ) = ( w +P. ( A +P. ( v +P. D ) ) ) )
52 simpllr 534 . . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> w e. P. )
53 addclpr 7597 . . . . . . . . . 10 |- ( ( A e. P. /\ D e. P. ) -> ( A +P. D ) e. P. )
5441, 45, 53syl2anc 411 . . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( A +P. D ) e. P. )
55 addassprg 7639 . . . . . . . . 9 |- ( ( w e. P. /\ v e. P. /\ ( A +P. D ) e. P. ) -> ( ( w +P. v ) +P. ( A +P. D ) ) = ( w +P. ( v +P. ( A +P. D ) ) ) )
5652, 40, 54, 55syl3anc 1249 . . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( w +P. v ) +P. ( A +P. D ) ) = ( w +P. ( v +P. ( A +P. D ) ) ) )
57 addclpr 7597 . . . . . . . . . 10 |- ( ( v e. P. /\ D e. P. ) -> ( v +P. D ) e. P. )
5840, 45, 57syl2anc 411 . . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( v +P. D ) e. P. )
59 addassprg 7639 . . . . . . . . 9 |- ( ( w e. P. /\ A e. P. /\ ( v +P. D ) e. P. ) -> ( ( w +P. A ) +P. ( v +P. D ) ) = ( w +P. ( A +P. ( v +P. D ) ) ) )
6052, 41, 58, 59syl3anc 1249 . . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( w +P. A ) +P. ( v +P. D ) ) = ( w +P. ( A +P. ( v +P. D ) ) ) )
6151, 56, 603eqtr4d 2236 . . . . . . 7 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( w +P. v ) +P. ( A +P. D ) ) = ( ( w +P. A ) +P. ( v +P. D ) ) )
6261adantr 276 . . . . . 6 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) ) -> ( ( w +P. v ) +P. ( A +P. D ) ) = ( ( w +P. A ) +P. ( v +P. D ) ) )
6311, 39, 623eqtr4d 2236 . . . . 5 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( w +P. v ) +P. ( A +P. D ) ) )
6463ex 115 . . . 4 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( w +P. v ) +P. ( A +P. D ) ) ) )
659, 64sylbid 150 . . 3 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R /\ [ <. v , u >. ] ~R = [ <. C , D >. ] ~R ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( w +P. v ) +P. ( A +P. D ) ) ) )
66 ltaprg 7679 . . . . 5 |- ( ( x e. P. /\ y e. P. /\ f e. P. ) -> ( x <P y <-> ( f +P. x ) <P ( f +P. y ) ) )
6766adantl 277 . . . 4 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( x e. P. /\ y e. P. /\ f e. P. ) ) -> ( x <P y <-> ( f +P. x ) <P ( f +P. y ) ) )
68 addclpr 7597 . . . . 5 |- ( ( z e. P. /\ u e. P. ) -> ( z +P. u ) e. P. )
6924, 12, 68syl2anc 411 . . . 4 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( z +P. u ) e. P. )
7030simprd 114 . . . 4 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( w +P. v ) e. P. )
71 addcomprg 7638 . . . . 5 |- ( ( x e. P. /\ y e. P. ) -> ( x +P. y ) = ( y +P. x ) )
7271adantl 277 . . . 4 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( x e. P. /\ y e. P. ) ) -> ( x +P. y ) = ( y +P. x ) )
7367, 69, 31, 70, 72, 54caovord3d 6089 . . 3 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( w +P. v ) +P. ( A +P. D ) ) -> ( ( z +P. u ) <P ( w +P. v ) <-> ( A +P. D ) <P ( B +P. C ) ) ) )
7465, 73syld 45 . 2 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R /\ [ <. v , u >. ] ~R = [ <. C , D >. ] ~R ) -> ( ( z +P. u ) <P ( w +P. v ) <-> ( A +P. D ) <P ( B +P. C ) ) ) )
751, 2, 3, 4, 74brecop 6679 1 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   <.cop 3621   class class class wbr 4029  (class class class)co 5918   [cec 6585   P.cnp 7351   +P. cpp 7353   <P cltp 7355   ~R cer 7356   R.cnr 7357   <R cltr 7363
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530  df-enr 7786  df-nr 7787  df-ltr 7790
This theorem is referenced by:  gt0srpr  7808  lttrsr  7822  ltposr  7823  ltsosr  7824  0lt1sr  7825  ltasrg  7830  aptisr  7839  mulextsr1  7841  archsr  7842  prsrlt  7847  ltpsrprg  7863  mappsrprg  7864  map2psrprg  7865  pitoregt0  7909
  Copyright terms: Public domain W3C validator