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Theorem genprndu 7071
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 7062 . . . . . . . . 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 2398 . . . . . . . . 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 7024 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
9 prnminu 7038 . . . . . . . . . . . . . . . 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 7024 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnminu 7038 . . . . . . . . . . . . . . . 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 555 . . . . . . . . . . . . 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 2536 . . . . . . . . . . . . 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 7059 . . . . . . . . . . . . . 14  |-  ( ( c  <Q  a  /\  d  <Q  b )  -> 
( c G d )  <Q  ( a G b ) )
2120reximi 2470 . . . . . . . . . . . . 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 2470 . . . . . . . . . . . 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 3847 . . . . . . . . . . . . . 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 2474 . . . . . . . . . . . 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 2474 . . . . . . . . . . 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 1825 . . . . . . 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 7064 . . . . . . . . 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 7031 . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . 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 555 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) ) )  ->  (
c  e.  Q.  /\  d  e.  Q. )
)
432caovcl 5791 . . . . . . . . . 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 3846 . . . . . . . . . . 11  |-  ( q  =  ( c G d )  ->  (
q  <Q  r  <->  ( c G d )  <Q 
r ) )
46 eleq1 2150 . . . . . . . . . . 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 2726 . . . . . . . 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 2496 . . . . . 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 7069 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 2nd `  ( A F B ) )  ->  ( q  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
5756alrimdv 1804 . . . . . . . . 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 3847 . . . . . . . . . . 11  |-  ( x  =  r  ->  (
q  <Q  x  <->  q  <Q  r ) )
59 eleq1 2150 . . . . . . . . . . 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 1842 . . . . . . . . 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 1446 . . . . . . . 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 2489 . . 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 2446 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 924   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3447   class class class wbr 3843   ` cfv 5010  (class class class)co 5644    |-> cmpt2 5646   1stc1st 5901   2ndc2nd 5902   Q.cnq 6829    <Q cltq 6834   P.cnp 6840
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-eprel 4114  df-id 4118  df-po 4121  df-iso 4122  df-iord 4191  df-on 4193  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-irdg 6127  df-oadd 6177  df-omul 6178  df-er 6282  df-ec 6284  df-qs 6288  df-ni 6853  df-mi 6855  df-lti 6856  df-enq 6896  df-nqqs 6897  df-ltnqqs 6902  df-inp 7015
This theorem is referenced by:  addclpr  7086  mulclpr  7121
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