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Theorem genprndu 6648
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 6639 . . . . . . . . 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 2359 . . . . . . . . 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 189 . . . . . . . 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 284 . . . . . . 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 455 . . . . . 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 6601 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
9 prnminu 6615 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  a  e.  ( 2nd `  A ) )  ->  E. c  e.  ( 2nd `  A ) c 
<Q  a )
108, 9sylan 271 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  a  e.  ( 2nd `  A ) )  ->  E. c  e.  ( 2nd `  A ) c 
<Q  a )
11 prop 6601 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnminu 6615 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  b  e.  ( 2nd `  B ) )  ->  E. d  e.  ( 2nd `  B ) d 
<Q  b )
1311, 12sylan 271 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  b  e.  ( 2nd `  B ) )  ->  E. d  e.  ( 2nd `  B ) d 
<Q  b )
1410, 13anim12i 325 . . . . . . . . . . . . . 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 530 . . . . . . . . . . . . 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 2494 . . . . . . . . . . . . 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 141 . . . . . . . . . . . 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 6636 . . . . . . . . . . . . . 14  |-  ( ( c  <Q  a  /\  d  <Q  b )  -> 
( c G d )  <Q  ( a G b ) )
2120reximi 2431 . . . . . . . . . . . . 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 2431 . . . . . . . . . . . 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 456 . . . . . . . . . 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 3793 . . . . . . . . . . . . . 14  |-  ( r  =  ( a G b )  ->  (
( c G d )  <Q  r  <->  ( c G d )  <Q 
( a G b ) ) )
2625biimprd 151 . . . . . . . . . . . . 13  |-  ( r  =  ( a G b )  ->  (
( c G d )  <Q  ( a G b )  -> 
( c G d )  <Q  r )
)
2726reximdv 2435 . . . . . . . . . . . 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 2435 . . . . . . . . . . 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 468 . . . . . . . . . 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 112 . . . . . . . 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 1791 . . . . . . 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 265 . . . . . 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 6641 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( c  e.  ( 2nd `  A
)  /\  d  e.  ( 2nd `  B ) )  ->  ( c G d )  e.  ( 2nd `  ( A F B ) ) ) )
3635imp 119 . . . . . . . 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 6608 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  c  e.  ( 2nd `  A ) )  -> 
c  e.  Q. )
388, 37sylan 271 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  c  e.  ( 2nd `  A ) )  -> 
c  e.  Q. )
39 elprnqu 6608 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  d  e.  ( 2nd `  B ) )  -> 
d  e.  Q. )
4011, 39sylan 271 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  d  e.  ( 2nd `  B ) )  -> 
d  e.  Q. )
4138, 40anim12i 325 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  c  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  d  e.  ( 2nd `  B ) ) )  ->  ( c  e. 
Q.  /\  d  e.  Q. ) )
4241an4s 530 . . . . . . . . . 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 3792 . . . . . . . . . . 11  |-  ( q  =  ( c G d )  ->  (
q  <Q  r  <->  ( c G d )  <Q 
r ) )
46 eleq1 2114 . . . . . . . . . . 11  |-  ( q  =  ( c G d )  ->  (
q  e.  ( 2nd `  ( A F B ) )  <->  ( c G d )  e.  ( 2nd `  ( A F B ) ) ) )
4745, 46anbi12d 450 . . . . . . . . . 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 266 . . . . . . . . 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 2675 . . . . . . . 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 412 . . . . . . 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 2455 . . . . . 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 265 . . . . 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 361 . . 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 6646 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 2nd `  ( A F B ) )  ->  ( q  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
5756alrimdv 1770 . . . . . . . . 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 3793 . . . . . . . . . . 11  |-  ( x  =  r  ->  (
q  <Q  x  <->  q  <Q  r ) )
59 eleq1 2114 . . . . . . . . . . 11  |-  ( x  =  r  ->  (
x  e.  ( 2nd `  ( A F B ) )  <->  r  e.  ( 2nd `  ( A F B ) ) ) )
6058, 59imbi12d 227 . . . . . . . . . 10  |-  ( x  =  r  ->  (
( q  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) )  <->  ( q  <Q  r  ->  r  e.  ( 2nd `  ( A F B ) ) ) ) )
6160cbvalv 1808 . . . . . . . . 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 154 . . . . . . . 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 1415 . . . . . . . 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 246 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  e.  ( 2nd `  ( A F B ) )  /\  q  <Q  r
)  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6665ancomsd 260 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  <Q 
r  /\  q  e.  ( 2nd `  ( A F B ) ) )  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6766ad2antrr 465 . . . 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 2448 . . 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 124 . 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 2407 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 101    <-> wb 102    /\ w3a 894   A.wal 1255    = wceq 1257   E.wex 1395    e. wcel 1407   A.wral 2321   E.wrex 2322   {crab 2325   <.cop 3403   class class class wbr 3789   ` cfv 4927  (class class class)co 5537    |-> cmpt2 5539   1stc1st 5790   2ndc2nd 5791   Q.cnq 6406    <Q cltq 6411   P.cnp 6417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-eprel 4051  df-id 4055  df-po 4058  df-iso 4059  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-mi 6432  df-lti 6433  df-enq 6473  df-nqqs 6474  df-ltnqqs 6479  df-inp 6592
This theorem is referenced by:  addclpr  6663  mulclpr  6698
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