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Theorem genprndu 7294
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 7285 . . . . . . . . 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 2430 . . . . . . . . 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 195 . . . . . . . 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 292 . . . . . . 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 467 . . . . . 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 7247 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
9 prnminu 7261 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  a  e.  ( 2nd `  A ) )  ->  E. c  e.  ( 2nd `  A ) c 
<Q  a )
108, 9sylan 279 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  a  e.  ( 2nd `  A ) )  ->  E. c  e.  ( 2nd `  A ) c 
<Q  a )
11 prop 7247 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnminu 7261 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  b  e.  ( 2nd `  B ) )  ->  E. d  e.  ( 2nd `  B ) d 
<Q  b )
1311, 12sylan 279 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  b  e.  ( 2nd `  B ) )  ->  E. d  e.  ( 2nd `  B ) d 
<Q  b )
1410, 13anim12i 334 . . . . . . . . . . . . . 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 560 . . . . . . . . . . . . 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 2575 . . . . . . . . . . . . 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 133 . . . . . . . . . . . 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 7282 . . . . . . . . . . . . . 14  |-  ( ( c  <Q  a  /\  d  <Q  b )  -> 
( c G d )  <Q  ( a G b ) )
2120reximi 2504 . . . . . . . . . . . . 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 2504 . . . . . . . . . . . 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 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 ) )
25 breq2 3901 . . . . . . . . . . . . . 14  |-  ( r  =  ( a G b )  ->  (
( c G d )  <Q  r  <->  ( c G d )  <Q 
( a G b ) ) )
2625biimprd 157 . . . . . . . . . . . . 13  |-  ( r  =  ( a G b )  ->  (
( c G d )  <Q  ( a G b )  -> 
( c G d )  <Q  r )
)
2726reximdv 2508 . . . . . . . . . . . 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 2508 . . . . . . . . . . 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 480 . . . . . . . . . 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 114 . . . . . . . 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 1851 . . . . . . 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 272 . . . . . 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 7287 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( c  e.  ( 2nd `  A
)  /\  d  e.  ( 2nd `  B ) )  ->  ( c G d )  e.  ( 2nd `  ( A F B ) ) ) )
3635imp 123 . . . . . . . 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 7254 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  c  e.  ( 2nd `  A ) )  -> 
c  e.  Q. )
388, 37sylan 279 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  c  e.  ( 2nd `  A ) )  -> 
c  e.  Q. )
39 elprnqu 7254 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  d  e.  ( 2nd `  B ) )  -> 
d  e.  Q. )
4011, 39sylan 279 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  d  e.  ( 2nd `  B ) )  -> 
d  e.  Q. )
4138, 40anim12i 334 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  c  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  d  e.  ( 2nd `  B ) ) )  ->  ( c  e. 
Q.  /\  d  e.  Q. ) )
4241an4s 560 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) ) )  ->  (
c  e.  Q.  /\  d  e.  Q. )
)
432caovcl 5891 . . . . . . . . . 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 3900 . . . . . . . . . . 11  |-  ( q  =  ( c G d )  ->  (
q  <Q  r  <->  ( c G d )  <Q 
r ) )
46 eleq1 2178 . . . . . . . . . . 11  |-  ( q  =  ( c G d )  ->  (
q  e.  ( 2nd `  ( A F B ) )  <->  ( c G d )  e.  ( 2nd `  ( A F B ) ) ) )
4745, 46anbi12d 462 . . . . . . . . . 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 273 . . . . . . . . 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 2765 . . . . . . . 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 422 . . . . . . 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 2532 . . . . . 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 272 . . . . 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 370 . . 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 7292 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 2nd `  ( A F B ) )  ->  ( q  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
5756alrimdv 1830 . . . . . . . . 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 3901 . . . . . . . . . . 11  |-  ( x  =  r  ->  (
q  <Q  x  <->  q  <Q  r ) )
59 eleq1 2178 . . . . . . . . . . 11  |-  ( x  =  r  ->  (
x  e.  ( 2nd `  ( A F B ) )  <->  r  e.  ( 2nd `  ( A F B ) ) ) )
6058, 59imbi12d 233 . . . . . . . . . 10  |-  ( x  =  r  ->  (
( q  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) )  <->  ( q  <Q  r  ->  r  e.  ( 2nd `  ( A F B ) ) ) ) )
6160cbvalv 1869 . . . . . . . . 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 160 . . . . . . . 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 1471 . . . . . . . 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 252 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  e.  ( 2nd `  ( A F B ) )  /\  q  <Q  r
)  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6665ancomsd 267 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  <Q 
r  /\  q  e.  ( 2nd `  ( A F B ) ) )  ->  r  e.  ( 2nd `  ( A F B ) ) ) )
6766ad2antrr 477 . . . 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 2524 . . 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 128 . 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 2480 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 103    <-> wb 104    /\ w3a 945   A.wal 1312    = wceq 1314   E.wex 1451    e. wcel 1463   A.wral 2391   E.wrex 2392   {crab 2395   <.cop 3498   class class class wbr 3897   ` cfv 5091  (class class class)co 5740    e. cmpo 5742   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052    <Q cltq 7057   P.cnp 7063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-mi 7078  df-lti 7079  df-enq 7119  df-nqqs 7120  df-ltnqqs 7125  df-inp 7238
This theorem is referenced by:  addclpr  7309  mulclpr  7344
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