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Theorem ltsrprg 8058
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 8048 . 2 |- ~R e. _V
2 enrer 8046 . 2 |- ~R Er ( P. X. P. )
3 df-nr 8038 . 2 |- R. = ( ( P. X. P. ) /. ~R )
4 df-ltr 8041 . 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 8047 . . . . 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 8047 . . . . . 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 2234 . . . . . 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 610 . . . 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 6058 . . . . . . 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 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. ) ) ) -> u e. P. )
13 simplrr 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. ) ) ) -> B e. P. )
14 addcomprg 7889 . . . . . . . . . . . 12 |- ( ( u e. P. /\ B e. P. ) -> ( u +P. B ) = ( B +P. u ) )
1514oveq1d 6064 . . . . . . . . . . 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 541 . . . . . . . . . . 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 7890 . . . . . . . . . . 11 |- ( ( u e. P. /\ B e. P. /\ C e. P. ) -> ( ( u +P. B ) +P. C ) = ( u +P. ( B +P. C ) ) )
1912, 13, 17, 18syl3anc 1274 . . . . . . . . . 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 7890 . . . . . . . . . . 11 |- ( ( B e. P. /\ u e. P. /\ C e. P. ) -> ( ( B +P. u ) +P. C ) = ( B +P. ( u +P. C ) ) )
2113, 12, 17, 20syl3anc 1274 . . . . . . . . . 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 2273 . . . . . . . . 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 6065 . . . . . . . 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 535 . . . . . . . . 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 7848 . . . . . . . . . . . . 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 7848 . . . . . . . . . . . . 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 592 . . . . . . . . . 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 7890 . . . . . . . . 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 1274 . . . . . . . 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 7848 . . . . . . . . . 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 7890 . . . . . . . . 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 1274 . . . . . . . 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 2275 . . . . . . 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 539 . . . . . . . . . . . 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 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. ) ) ) -> A e. P. )
42 addcomprg 7889 . . . . . . . . . . . 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 6064 . . . . . . . . . 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 542 . . . . . . . . . . 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 7890 . . . . . . . . . . 11 |- ( ( v e. P. /\ A e. P. /\ D e. P. ) -> ( ( v +P. A ) +P. D ) = ( v +P. ( A +P. D ) ) )
4740, 41, 45, 46syl3anc 1274 . . . . . . . . . 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 7890 . . . . . . . . . . 11 |- ( ( A e. P. /\ v e. P. /\ D e. P. ) -> ( ( A +P. v ) +P. D ) = ( A +P. ( v +P. D ) ) )
4941, 40, 45, 48syl3anc 1274 . . . . . . . . . 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 2273 . . . . . . . . 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 6065 . . . . . . . 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 536 . . . . . . . . 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 7848 . . . . . . . . . 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 7890 . . . . . . . . 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 1274 . . . . . . . 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 7848 . . . . . . . . . 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 7890 . . . . . . . . 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 1274 . . . . . . . 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 2275 . . . . . . 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 2275 . . . . 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 7930 . . . . 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 7848 . . . . 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 7889 . . . . 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 6224 . . 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 6858 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 1005   = wceq 1398   e. wcel 2203   <.cop 3691   class class class wbr 4108  (class class class)co 6049   [cec 6764   P.cnp 7602   +P. cpp 7604   <P cltp 7606   ~R cer 7607   R.cnr 7608   <R cltr 7614
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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709
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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7615  df-pli 7616  df-mi 7617  df-lti 7618  df-plpq 7655  df-mpq 7656  df-enq 7658  df-nqqs 7659  df-plqqs 7660  df-mqqs 7661  df-1nqqs 7662  df-rq 7663  df-ltnqqs 7664  df-enq0 7735  df-nq0 7736  df-0nq0 7737  df-plq0 7738  df-mq0 7739  df-inp 7777  df-iplp 7779  df-iltp 7781  df-enr 8037  df-nr 8038  df-ltr 8041
This theorem is referenced by:  gt0srpr  8059  lttrsr  8073  ltposr  8074  ltsosr  8075  0lt1sr  8076  ltasrg  8081  aptisr  8090  mulextsr1  8092  archsr  8093  prsrlt  8098  ltpsrprg  8114  mappsrprg  8115  map2psrprg  8116  pitoregt0  8160
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