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Theorem ltsrprg 7709
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 7699 . 2 |- ~R e. _V
2 enrer 7697 . 2 |- ~R Er ( P. X. P. )
3 df-nr 7689 . 2 |- R. = ( ( P. X. P. ) /. ~R )
4 df-ltr 7692 . 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 7698 . . . . 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 7698 . . . . . 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 2172 . . . . . 6 |- ( ( v +P. D ) = ( u +P. C ) <-> ( u +P. C ) = ( v +P. D ) )
86, 7bitrdi 195 . . . . 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 601 . . . 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 5862 . . . . . . 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 275 . . . . . 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 533 . . . . . . . . . . 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 531 . . . . . . . . . . 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 7540 . . . . . . . . . . . 12 |- ( ( u e. P. /\ B e. P. ) -> ( u +P. B ) = ( B +P. u ) )
1514oveq1d 5868 . . . . . . . . . . 11 |- ( ( u e. P. /\ B e. P. ) -> ( ( u +P. B ) +P. C ) = ( ( B +P. u ) +P. C ) )
1612, 13, 15syl2anc 409 . . . . . . . . . 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 534 . . . . . . . . . . 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 7541 . . . . . . . . . . 11 |- ( ( u e. P. /\ B e. P. /\ C e. P. ) -> ( ( u +P. B ) +P. C ) = ( u +P. ( B +P. C ) ) )
1912, 13, 17, 18syl3anc 1233 . . . . . . . . . 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 7541 . . . . . . . . . . 11 |- ( ( B e. P. /\ u e. P. /\ C e. P. ) -> ( ( B +P. u ) +P. C ) = ( B +P. ( u +P. C ) ) )
2113, 12, 17, 20syl3anc 1233 . . . . . . . . . 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 2211 . . . . . . . . 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 5869 . . . . . . . 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 528 . . . . . . . . 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 7499 . . . . . . . . . . . . 13 |- ( ( w e. P. /\ v e. P. ) -> ( w +P. v ) e. P. )
2625ad2ant2lr 507 . . . . . . . . . . . 12 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w +P. v ) e. P. )
27 addclpr 7499 . . . . . . . . . . . . 13 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
2827ad2ant2lr 507 . . . . . . . . . . . 12 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( B +P. C ) e. P. )
2926, 28anim12ci 337 . . . . . . . . . . 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 583 . . . . . . . . . 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 111 . . . . . . . . 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 7541 . . . . . . . . 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 1233 . . . . . . . 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 7499 . . . . . . . . . 10 |- ( ( u e. P. /\ C e. P. ) -> ( u +P. C ) e. P. )
3512, 17, 34syl2anc 409 . . . . . . . . 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 7541 . . . . . . . . 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 1233 . . . . . . . 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 2213 . . . . . . 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 274 . . . . . 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 532 . . . . . . . . . . . 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 530 . . . . . . . . . . . 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 7540 . . . . . . . . . . . 12 |- ( ( v e. P. /\ A e. P. ) -> ( v +P. A ) = ( A +P. v ) )
4340, 41, 42syl2anc 409 . . . . . . . . . . 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 5868 . . . . . . . . . 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 535 . . . . . . . . . . 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 7541 . . . . . . . . . . 11 |- ( ( v e. P. /\ A e. P. /\ D e. P. ) -> ( ( v +P. A ) +P. D ) = ( v +P. ( A +P. D ) ) )
4740, 41, 45, 46syl3anc 1233 . . . . . . . . . 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 7541 . . . . . . . . . . 11 |- ( ( A e. P. /\ v e. P. /\ D e. P. ) -> ( ( A +P. v ) +P. D ) = ( A +P. ( v +P. D ) ) )
4941, 40, 45, 48syl3anc 1233 . . . . . . . . . 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 2211 . . . . . . . . 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 5869 . . . . . . . 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 529 . . . . . . . . 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 7499 . . . . . . . . . 10 |- ( ( A e. P. /\ D e. P. ) -> ( A +P. D ) e. P. )
5441, 45, 53syl2anc 409 . . . . . . . . 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 7541 . . . . . . . . 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 1233 . . . . . . . 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 7499 . . . . . . . . . 10 |- ( ( v e. P. /\ D e. P. ) -> ( v +P. D ) e. P. )
5840, 45, 57syl2anc 409 . . . . . . . . 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 7541 . . . . . . . . 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 1233 . . . . . . . 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 2213 . . . . . . 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 274 . . . . . 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 2213 . . . . 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 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. ) ) ) -> ( ( ( 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 149 . . 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 7581 . . . . 5 |- ( ( x e. P. /\ y e. P. /\ f e. P. ) -> ( x <P y <-> ( f +P. x ) <P ( f +P. y ) ) )
6766adantl 275 . . . 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 7499 . . . . 5 |- ( ( z e. P. /\ u e. P. ) -> ( z +P. u ) e. P. )
6924, 12, 68syl2anc 409 . . . 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 113 . . . 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 7540 . . . . 5 |- ( ( x e. P. /\ y e. P. ) -> ( x +P. y ) = ( y +P. x ) )
7271adantl 275 . . . 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 6023 . . 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 6603 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 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   <.cop 3586   class class class wbr 3989  (class class class)co 5853   [cec 6511   P.cnp 7253   +P. cpp 7255   <P cltp 7257   ~R cer 7258   R.cnr 7259   <R cltr 7265
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-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
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-ral 2453  df-rex 2454  df-reu 2455  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-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  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-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432  df-enr 7688  df-nr 7689  df-ltr 7692
This theorem is referenced by:  gt0srpr  7710  lttrsr  7724  ltposr  7725  ltsosr  7726  0lt1sr  7727  ltasrg  7732  aptisr  7741  mulextsr1  7743  archsr  7744  prsrlt  7749  ltpsrprg  7765  mappsrprg  7766  map2psrprg  7767  pitoregt0  7811
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