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Theorem genprndu 6620
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 6611 . . . . . . . . 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 2344 . . . . . . . . 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 185 . . . . . . . 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 280 . . . . . . 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 447 . . . . . 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 6573 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
9 prnminu 6587 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  a  e.  ( 2nd `  A ) )  ->  E. c  e.  ( 2nd `  A ) c 
<Q  a )
108, 9sylan 267 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  a  e.  ( 2nd `  A ) )  ->  E. c  e.  ( 2nd `  A ) c 
<Q  a )
11 prop 6573 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnminu 6587 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  b  e.  ( 2nd `  B ) )  ->  E. d  e.  ( 2nd `  B ) d 
<Q  b )
1311, 12sylan 267 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  b  e.  ( 2nd `  B ) )  ->  E. d  e.  ( 2nd `  B ) d 
<Q  b )
1410, 13anim12i 321 . . . . . . . . . . . . . 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 522 . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . 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 137 . . . . . . . . . . . 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 6608 . . . . . . . . . . . . . 14  |-  ( ( c  <Q  a  /\  d  <Q  b )  -> 
( c G d )  <Q  ( a G b ) )
2120reximi 2416 . . . . . . . . . . . . 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 2416 . . . . . . . . . . . 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 448 . . . . . . . . . 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 3768 . . . . . . . . . . . . . 14  |-  ( r  =  ( a G b )  ->  (
( c G d )  <Q  r  <->  ( c G d )  <Q 
( a G b ) ) )
2625biimprd 147 . . . . . . . . . . . . 13  |-  ( r  =  ( a G b )  ->  (
( c G d )  <Q  ( a G b )  -> 
( c G d )  <Q  r )
)
2726reximdv 2420 . . . . . . . . . . . 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 2420 . . . . . . . . . . 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 460 . . . . . . . . . 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 108 . . . . . . . 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 1777 . . . . . . 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 261 . . . . . 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 6613 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( c  e.  ( 2nd `  A
)  /\  d  e.  ( 2nd `  B ) )  ->  ( c G d )  e.  ( 2nd `  ( A F B ) ) ) )
3635imp 115 . . . . . . . 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 6580 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  c  e.  ( 2nd `  A ) )  -> 
c  e.  Q. )
388, 37sylan 267 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  c  e.  ( 2nd `  A ) )  -> 
c  e.  Q. )
39 elprnqu 6580 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  d  e.  ( 2nd `  B ) )  -> 
d  e.  Q. )
4011, 39sylan 267 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  d  e.  ( 2nd `  B ) )  -> 
d  e.  Q. )
4138, 40anim12i 321 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  c  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  d  e.  ( 2nd `  B ) ) )  ->  ( c  e. 
Q.  /\  d  e.  Q. ) )
4241an4s 522 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) ) )  ->  (
c  e.  Q.  /\  d  e.  Q. )
)
432caovcl 5655 . . . . . . . . . 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 3767 . . . . . . . . . . 11  |-  ( q  =  ( c G d )  ->  (
q  <Q  r  <->  ( c G d )  <Q 
r ) )
46 eleq1 2100 . . . . . . . . . . 11  |-  ( q  =  ( c G d )  ->  (
q  e.  ( 2nd `  ( A F B ) )  <->  ( c G d )  e.  ( 2nd `  ( A F B ) ) ) )
4745, 46anbi12d 442 . . . . . . . . . 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 262 . . . . . . . . 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 2660 . . . . . . . 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 404 . . . . . . 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 2440 . . . . . 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 261 . . . . 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 357 . . 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 6618 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 2nd `  ( A F B ) )  ->  ( q  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
5756alrimdv 1756 . . . . . . . . 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 3768 . . . . . . . . . . 11  |-  ( x  =  r  ->  (
q  <Q  x  <->  q  <Q  r ) )
59 eleq1 2100 . . . . . . . . . . 11  |-  ( x  =  r  ->  (
x  e.  ( 2nd `  ( A F B ) )  <->  r  e.  ( 2nd `  ( A F B ) ) ) )
6058, 59imbi12d 223 . . . . . . . . . 10  |-  ( x  =  r  ->  (
( q  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) )  <->  ( q  <Q  r  ->  r  e.  ( 2nd `  ( A F B ) ) ) ) )
6160cbvalv 1794 . . . . . . . . 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 150 . . . . . . . 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 1401 . . . . . . . 8  |-  ( A. r ( q  <Q 
r  ->  r  e.  ( 2nd `  ( A F B ) ) )  ->  ( q  <Q  r  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6462, 63syl6 29 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 2nd `  ( A F B ) )  ->  ( q  <Q 
r  ->  r  e.  ( 2nd `  ( A F B ) ) ) ) )
6564impd 242 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  e.  ( 2nd `  ( A F B ) )  /\  q  <Q  r
)  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6665ancomsd 256 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  <Q 
r  /\  q  e.  ( 2nd `  ( A F B ) ) )  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6766ad2antrr 457 . . . 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 2433 . . 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 120 . 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 2392 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 97    <-> wb 98    /\ w3a 885   A.wal 1241    = wceq 1243   E.wex 1381    e. wcel 1393   A.wral 2306   E.wrex 2307   {crab 2310   <.cop 3378   class class class wbr 3764   ` cfv 4902  (class class class)co 5512    |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378    <Q cltq 6383   P.cnp 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-lti 6405  df-enq 6445  df-nqqs 6446  df-ltnqqs 6451  df-inp 6564
This theorem is referenced by:  addclpr  6635  mulclpr  6670
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