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Theorem genprndu 6763
Description: The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
Hypotheses
Ref Expression
genpelvl.1  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
genpelvl.2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
genprndu.ord  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z G x )  <Q 
( z G y ) ) )
genprndu.com  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x G y )  =  ( y G x ) )
genprndu.upper  |-  ( ( ( ( A  e. 
P.  /\  g  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 2nd `  B
) ) )  /\  x  e.  Q. )  ->  ( ( g G h )  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) )
Assertion
Ref Expression
genprndu  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. r  e.  Q.  ( r  e.  ( 2nd `  ( A F B ) )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) ) ) )
Distinct variable groups:    x, y, z, g, h, w, v, q, A    x, B, y, z, g, h, w, v, q    x, G, y, z, g, h, w, v, q    g, F, q    A, r, q, v, w, x, y, z    B, r, g, h   
h, F, r, v, w, x, y, z    G, r

Proof of Theorem genprndu
Dummy variables  a  b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 genpelvl.1 . . . . . . . . . 10  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
2 genpelvl.2 . . . . . . . . . 10  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
31, 2genpelvu 6754 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( r  e.  ( 2nd `  ( A F B ) )  <->  E. a  e.  ( 2nd `  A ) E. b  e.  ( 2nd `  B ) r  =  ( a G b ) ) )
4 r2ex 2387 . . . . . . . . 9  |-  ( E. a  e.  ( 2nd `  A ) E. b  e.  ( 2nd `  B
) r  =  ( a G b )  <->  E. a E. b ( ( a  e.  ( 2nd `  A )  /\  b  e.  ( 2nd `  B ) )  /\  r  =  ( a G b ) ) )
53, 4syl6bb 194 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( r  e.  ( 2nd `  ( A F B ) )  <->  E. a E. b ( ( a  e.  ( 2nd `  A )  /\  b  e.  ( 2nd `  B ) )  /\  r  =  ( a G b ) ) ) )
65biimpa 290 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  r  e.  ( 2nd `  ( A F B ) ) )  ->  E. a E. b
( ( a  e.  ( 2nd `  A
)  /\  b  e.  ( 2nd `  B ) )  /\  r  =  ( a G b ) ) )
76adantrl 462 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( r  e.  Q.  /\  r  e.  ( 2nd `  ( A F B ) ) ) )  ->  E. a E. b
( ( a  e.  ( 2nd `  A
)  /\  b  e.  ( 2nd `  B ) )  /\  r  =  ( a G b ) ) )
8 prop 6716 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
9 prnminu 6730 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  a  e.  ( 2nd `  A ) )  ->  E. c  e.  ( 2nd `  A ) c 
<Q  a )
108, 9sylan 277 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  a  e.  ( 2nd `  A ) )  ->  E. c  e.  ( 2nd `  A ) c 
<Q  a )
11 prop 6716 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnminu 6730 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  b  e.  ( 2nd `  B ) )  ->  E. d  e.  ( 2nd `  B ) d 
<Q  b )
1311, 12sylan 277 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  b  e.  ( 2nd `  B ) )  ->  E. d  e.  ( 2nd `  B ) d 
<Q  b )
1410, 13anim12i 331 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  a  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  b  e.  ( 2nd `  B ) ) )  ->  ( E. c  e.  ( 2nd `  A
) c  <Q  a  /\  E. d  e.  ( 2nd `  B ) d  <Q  b )
)
1514an4s 553 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( a  e.  ( 2nd `  A )  /\  b  e.  ( 2nd `  B ) ) )  ->  ( E. c  e.  ( 2nd `  A ) c 
<Q  a  /\  E. d  e.  ( 2nd `  B
) d  <Q  b
) )
16 reeanv 2524 . . . . . . . . . . . . 13  |-  ( E. c  e.  ( 2nd `  A ) E. d  e.  ( 2nd `  B
) ( c  <Q 
a  /\  d  <Q  b )  <->  ( E. c  e.  ( 2nd `  A
) c  <Q  a  /\  E. d  e.  ( 2nd `  B ) d  <Q  b )
)
1715, 16sylibr 132 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( a  e.  ( 2nd `  A )  /\  b  e.  ( 2nd `  B ) ) )  ->  E. c  e.  ( 2nd `  A
) E. d  e.  ( 2nd `  B
) ( c  <Q 
a  /\  d  <Q  b ) )
18 genprndu.ord . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z G x )  <Q 
( z G y ) ) )
19 genprndu.com . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x G y )  =  ( y G x ) )
2018, 19genplt2i 6751 . . . . . . . . . . . . . 14  |-  ( ( c  <Q  a  /\  d  <Q  b )  -> 
( c G d )  <Q  ( a G b ) )
2120reximi 2459 . . . . . . . . . . . . 13  |-  ( E. d  e.  ( 2nd `  B ) ( c 
<Q  a  /\  d  <Q  b )  ->  E. d  e.  ( 2nd `  B
) ( c G d )  <Q  (
a G b ) )
2221reximi 2459 . . . . . . . . . . . 12  |-  ( E. c  e.  ( 2nd `  A ) E. d  e.  ( 2nd `  B
) ( c  <Q 
a  /\  d  <Q  b )  ->  E. c  e.  ( 2nd `  A
) E. d  e.  ( 2nd `  B
) ( c G d )  <Q  (
a G b ) )
2317, 22syl 14 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( a  e.  ( 2nd `  A )  /\  b  e.  ( 2nd `  B ) ) )  ->  E. c  e.  ( 2nd `  A
) E. d  e.  ( 2nd `  B
) ( c G d )  <Q  (
a G b ) )
2423adantrr 463 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( ( a  e.  ( 2nd `  A
)  /\  b  e.  ( 2nd `  B ) )  /\  r  =  ( a G b ) ) )  ->  E. c  e.  ( 2nd `  A ) E. d  e.  ( 2nd `  B ) ( c G d )  <Q 
( a G b ) )
25 breq2 3791 . . . . . . . . . . . . . 14  |-  ( r  =  ( a G b )  ->  (
( c G d )  <Q  r  <->  ( c G d )  <Q 
( a G b ) ) )
2625biimprd 156 . . . . . . . . . . . . 13  |-  ( r  =  ( a G b )  ->  (
( c G d )  <Q  ( a G b )  -> 
( c G d )  <Q  r )
)
2726reximdv 2463 . . . . . . . . . . . 12  |-  ( r  =  ( a G b )  ->  ( E. d  e.  ( 2nd `  B ) ( c G d ) 
<Q  ( a G b )  ->  E. d  e.  ( 2nd `  B
) ( c G d )  <Q  r
) )
2827reximdv 2463 . . . . . . . . . . 11  |-  ( r  =  ( a G b )  ->  ( E. c  e.  ( 2nd `  A ) E. d  e.  ( 2nd `  B ) ( c G d )  <Q 
( a G b )  ->  E. c  e.  ( 2nd `  A
) E. d  e.  ( 2nd `  B
) ( c G d )  <Q  r
) )
2928ad2antll 475 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( ( a  e.  ( 2nd `  A
)  /\  b  e.  ( 2nd `  B ) )  /\  r  =  ( a G b ) ) )  -> 
( E. c  e.  ( 2nd `  A
) E. d  e.  ( 2nd `  B
) ( c G d )  <Q  (
a G b )  ->  E. c  e.  ( 2nd `  A ) E. d  e.  ( 2nd `  B ) ( c G d )  <Q  r )
)
3024, 29mpd 13 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( ( a  e.  ( 2nd `  A
)  /\  b  e.  ( 2nd `  B ) )  /\  r  =  ( a G b ) ) )  ->  E. c  e.  ( 2nd `  A ) E. d  e.  ( 2nd `  B ) ( c G d )  <Q 
r )
3130ex 113 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( a  e.  ( 2nd `  A
)  /\  b  e.  ( 2nd `  B ) )  /\  r  =  ( a G b ) )  ->  E. c  e.  ( 2nd `  A
) E. d  e.  ( 2nd `  B
) ( c G d )  <Q  r
) )
3231exlimdvv 1819 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. a E. b ( ( a  e.  ( 2nd `  A
)  /\  b  e.  ( 2nd `  B ) )  /\  r  =  ( a G b ) )  ->  E. c  e.  ( 2nd `  A
) E. d  e.  ( 2nd `  B
) ( c G d )  <Q  r
) )
3332adantr 270 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( r  e.  Q.  /\  r  e.  ( 2nd `  ( A F B ) ) ) )  ->  ( E. a E. b ( ( a  e.  ( 2nd `  A
)  /\  b  e.  ( 2nd `  B ) )  /\  r  =  ( a G b ) )  ->  E. c  e.  ( 2nd `  A
) E. d  e.  ( 2nd `  B
) ( c G d )  <Q  r
) )
347, 33mpd 13 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( r  e.  Q.  /\  r  e.  ( 2nd `  ( A F B ) ) ) )  ->  E. c  e.  ( 2nd `  A ) E. d  e.  ( 2nd `  B ) ( c G d )  <Q  r )
351, 2genppreclu 6756 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( c  e.  ( 2nd `  A
)  /\  d  e.  ( 2nd `  B ) )  ->  ( c G d )  e.  ( 2nd `  ( A F B ) ) ) )
3635imp 122 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) ) )  ->  (
c G d )  e.  ( 2nd `  ( A F B ) ) )
37 elprnqu 6723 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  c  e.  ( 2nd `  A ) )  -> 
c  e.  Q. )
388, 37sylan 277 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  c  e.  ( 2nd `  A ) )  -> 
c  e.  Q. )
39 elprnqu 6723 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  d  e.  ( 2nd `  B ) )  -> 
d  e.  Q. )
4011, 39sylan 277 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  d  e.  ( 2nd `  B ) )  -> 
d  e.  Q. )
4138, 40anim12i 331 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  c  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  d  e.  ( 2nd `  B ) ) )  ->  ( c  e. 
Q.  /\  d  e.  Q. ) )
4241an4s 553 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) ) )  ->  (
c  e.  Q.  /\  d  e.  Q. )
)
432caovcl 5680 . . . . . . . . . 10  |-  ( ( c  e.  Q.  /\  d  e.  Q. )  ->  ( c G d )  e.  Q. )
4442, 43syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) ) )  ->  (
c G d )  e.  Q. )
45 breq1 3790 . . . . . . . . . . 11  |-  ( q  =  ( c G d )  ->  (
q  <Q  r  <->  ( c G d )  <Q 
r ) )
46 eleq1 2142 . . . . . . . . . . 11  |-  ( q  =  ( c G d )  ->  (
q  e.  ( 2nd `  ( A F B ) )  <->  ( c G d )  e.  ( 2nd `  ( A F B ) ) ) )
4745, 46anbi12d 457 . . . . . . . . . 10  |-  ( q  =  ( c G d )  ->  (
( q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) )  <-> 
( ( c G d )  <Q  r  /\  ( c G d )  e.  ( 2nd `  ( A F B ) ) ) ) )
4847adantl 271 . . . . . . . . 9  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B
) ) )  /\  q  =  ( c G d ) )  ->  ( ( q 
<Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) )  <->  ( ( c G d )  <Q 
r  /\  ( c G d )  e.  ( 2nd `  ( A F B ) ) ) ) )
4944, 48rspcedv 2706 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) ) )  ->  (
( ( c G d )  <Q  r  /\  ( c G d )  e.  ( 2nd `  ( A F B ) ) )  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) ) ) )
5036, 49mpan2d 419 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) ) )  ->  (
( c G d )  <Q  r  ->  E. q  e.  Q.  (
q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) ) ) )
5150rexlimdvva 2485 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. c  e.  ( 2nd `  A
) E. d  e.  ( 2nd `  B
) ( c G d )  <Q  r  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) ) ) )
5251adantr 270 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( r  e.  Q.  /\  r  e.  ( 2nd `  ( A F B ) ) ) )  ->  ( E. c  e.  ( 2nd `  A
) E. d  e.  ( 2nd `  B
) ( c G d )  <Q  r  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) ) ) )
5334, 52mpd 13 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( r  e.  Q.  /\  r  e.  ( 2nd `  ( A F B ) ) ) )  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) ) )
5453expr 367 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  r  e.  Q. )  ->  ( r  e.  ( 2nd `  ( A F B ) )  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) ) ) )
55 genprndu.upper . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
P.  /\  g  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 2nd `  B
) ) )  /\  x  e.  Q. )  ->  ( ( g G h )  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) )
561, 2, 55genpcuu 6761 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 2nd `  ( A F B ) )  ->  ( q  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
5756alrimdv 1798 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 2nd `  ( A F B ) )  ->  A. x ( q 
<Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
58 breq2 3791 . . . . . . . . . . 11  |-  ( x  =  r  ->  (
q  <Q  x  <->  q  <Q  r ) )
59 eleq1 2142 . . . . . . . . . . 11  |-  ( x  =  r  ->  (
x  e.  ( 2nd `  ( A F B ) )  <->  r  e.  ( 2nd `  ( A F B ) ) ) )
6058, 59imbi12d 232 . . . . . . . . . 10  |-  ( x  =  r  ->  (
( q  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) )  <->  ( q  <Q  r  ->  r  e.  ( 2nd `  ( A F B ) ) ) ) )
6160cbvalv 1836 . . . . . . . . 9  |-  ( A. x ( q  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) )  <->  A. r ( q 
<Q  r  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6257, 61syl6ib 159 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 2nd `  ( A F B ) )  ->  A. r ( q 
<Q  r  ->  r  e.  ( 2nd `  ( A F B ) ) ) ) )
63 sp 1442 . . . . . . . 8  |-  ( A. r ( q  <Q 
r  ->  r  e.  ( 2nd `  ( A F B ) ) )  ->  ( q  <Q  r  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6462, 63syl6 33 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 2nd `  ( A F B ) )  ->  ( q  <Q 
r  ->  r  e.  ( 2nd `  ( A F B ) ) ) ) )
6564impd 251 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  e.  ( 2nd `  ( A F B ) )  /\  q  <Q  r
)  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6665ancomsd 265 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  <Q 
r  /\  q  e.  ( 2nd `  ( A F B ) ) )  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6766ad2antrr 472 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  r  e.  Q. )  /\  q  e.  Q. )  ->  (
( q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) )  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6867rexlimdva 2478 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  r  e.  Q. )  ->  ( E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  ( A F B ) ) )  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6954, 68impbid 127 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  r  e.  Q. )  ->  ( r  e.  ( 2nd `  ( A F B ) )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) ) ) )
7069ralrimiva 2435 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. r  e.  Q.  ( r  e.  ( 2nd `  ( A F B ) )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   A.wral 2349   E.wrex 2350   {crab 2353   <.cop 3403   class class class wbr 3787   ` cfv 4926  (class class class)co 5537    |-> cmpt2 5539   1stc1st 5790   2ndc2nd 5791   Q.cnq 6521    <Q cltq 6526   P.cnp 6532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-mi 6547  df-lti 6548  df-enq 6588  df-nqqs 6589  df-ltnqqs 6594  df-inp 6707
This theorem is referenced by:  addclpr  6778  mulclpr  6813
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