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Theorem genprndl 7297
Description: The lower 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. )
genprndl.ord  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z G x )  <Q 
( z G y ) ) )
genprndl.com  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x G y )  =  ( y G x ) )
genprndl.lower  |-  ( ( ( ( A  e. 
P.  /\  g  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B
) ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g G h )  ->  x  e.  ( 1st `  ( A F B ) ) ) )
Assertion
Ref Expression
genprndl  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. q  e.  Q.  ( q  e.  ( 1st `  ( A F B ) )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( 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 genprndl
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, 2genpelvl 7288 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 1st `  ( A F B ) )  <->  E. a  e.  ( 1st `  A ) E. b  e.  ( 1st `  B ) q  =  ( a G b ) ) )
4 r2ex 2432 . . . . . . . . 9  |-  ( E. a  e.  ( 1st `  A ) E. b  e.  ( 1st `  B
) q  =  ( a G b )  <->  E. a E. b ( ( a  e.  ( 1st `  A )  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) ) )
53, 4syl6bb 195 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 1st `  ( A F B ) )  <->  E. a E. b ( ( a  e.  ( 1st `  A )  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) ) ) )
65biimpa 294 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  q  e.  ( 1st `  ( A F B ) ) )  ->  E. a E. b
( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) ) )
76adantrl 469 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  q  e.  ( 1st `  ( A F B ) ) ) )  ->  E. a E. b
( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) ) )
8 prop 7251 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
9 prnmaxl 7264 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  a  e.  ( 1st `  A ) )  ->  E. c  e.  ( 1st `  A ) a 
<Q  c )
108, 9sylan 281 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  a  e.  ( 1st `  A ) )  ->  E. c  e.  ( 1st `  A ) a 
<Q  c )
11 prop 7251 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnmaxl 7264 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  b  e.  ( 1st `  B ) )  ->  E. d  e.  ( 1st `  B ) b 
<Q  d )
1311, 12sylan 281 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  b  e.  ( 1st `  B ) )  ->  E. d  e.  ( 1st `  B ) b 
<Q  d )
1410, 13anim12i 336 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  a  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  b  e.  ( 1st `  B ) ) )  ->  ( E. c  e.  ( 1st `  A
) a  <Q  c  /\  E. d  e.  ( 1st `  B ) b  <Q  d )
)
1514an4s 562 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( a  e.  ( 1st `  A )  /\  b  e.  ( 1st `  B ) ) )  ->  ( E. c  e.  ( 1st `  A ) a 
<Q  c  /\  E. d  e.  ( 1st `  B
) b  <Q  d
) )
16 reeanv 2577 . . . . . . . . . . . . 13  |-  ( E. c  e.  ( 1st `  A ) E. d  e.  ( 1st `  B
) ( a  <Q 
c  /\  b  <Q  d )  <->  ( E. c  e.  ( 1st `  A
) a  <Q  c  /\  E. d  e.  ( 1st `  B ) b  <Q  d )
)
1715, 16sylibr 133 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( a  e.  ( 1st `  A )  /\  b  e.  ( 1st `  B ) ) )  ->  E. c  e.  ( 1st `  A
) E. d  e.  ( 1st `  B
) ( a  <Q 
c  /\  b  <Q  d ) )
18 genprndl.ord . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z G x )  <Q 
( z G y ) ) )
19 genprndl.com . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x G y )  =  ( y G x ) )
2018, 19genplt2i 7286 . . . . . . . . . . . . . 14  |-  ( ( a  <Q  c  /\  b  <Q  d )  -> 
( a G b )  <Q  ( c G d ) )
2120reximi 2506 . . . . . . . . . . . . 13  |-  ( E. d  e.  ( 1st `  B ) ( a 
<Q  c  /\  b  <Q  d )  ->  E. d  e.  ( 1st `  B
) ( a G b )  <Q  (
c G d ) )
2221reximi 2506 . . . . . . . . . . . 12  |-  ( E. c  e.  ( 1st `  A ) E. d  e.  ( 1st `  B
) ( a  <Q 
c  /\  b  <Q  d )  ->  E. c  e.  ( 1st `  A
) E. d  e.  ( 1st `  B
) ( a G b )  <Q  (
c G d ) )
2317, 22syl 14 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( a  e.  ( 1st `  A )  /\  b  e.  ( 1st `  B ) ) )  ->  E. c  e.  ( 1st `  A
) E. d  e.  ( 1st `  B
) ( a G b )  <Q  (
c G d ) )
2423adantrr 470 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) ) )  ->  E. c  e.  ( 1st `  A ) E. d  e.  ( 1st `  B ) ( a G b )  <Q 
( c G d ) )
25 breq1 3902 . . . . . . . . . . . . . 14  |-  ( q  =  ( a G b )  ->  (
q  <Q  ( c G d )  <->  ( a G b )  <Q 
( c G d ) ) )
2625biimprd 157 . . . . . . . . . . . . 13  |-  ( q  =  ( a G b )  ->  (
( a G b )  <Q  ( c G d )  -> 
q  <Q  ( c G d ) ) )
2726reximdv 2510 . . . . . . . . . . . 12  |-  ( q  =  ( a G b )  ->  ( E. d  e.  ( 1st `  B ) ( a G b ) 
<Q  ( c G d )  ->  E. d  e.  ( 1st `  B
) q  <Q  (
c G d ) ) )
2827reximdv 2510 . . . . . . . . . . 11  |-  ( q  =  ( a G b )  ->  ( E. c  e.  ( 1st `  A ) E. d  e.  ( 1st `  B ) ( a G b )  <Q 
( c G d )  ->  E. c  e.  ( 1st `  A
) E. d  e.  ( 1st `  B
) q  <Q  (
c G d ) ) )
2928ad2antll 482 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) ) )  -> 
( E. c  e.  ( 1st `  A
) E. d  e.  ( 1st `  B
) ( a G b )  <Q  (
c G d )  ->  E. c  e.  ( 1st `  A ) E. d  e.  ( 1st `  B ) q  <Q  ( c G d ) ) )
3024, 29mpd 13 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) ) )  ->  E. c  e.  ( 1st `  A ) E. d  e.  ( 1st `  B ) q  <Q 
( c G d ) )
3130ex 114 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  ->  E. c  e.  ( 1st `  A
) E. d  e.  ( 1st `  B
) q  <Q  (
c G d ) ) )
3231exlimdvv 1853 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. a E. b ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  ->  E. c  e.  ( 1st `  A
) E. d  e.  ( 1st `  B
) q  <Q  (
c G d ) ) )
3332adantr 274 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  q  e.  ( 1st `  ( A F B ) ) ) )  ->  ( E. a E. b ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  ->  E. c  e.  ( 1st `  A
) E. d  e.  ( 1st `  B
) q  <Q  (
c G d ) ) )
347, 33mpd 13 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  q  e.  ( 1st `  ( A F B ) ) ) )  ->  E. c  e.  ( 1st `  A ) E. d  e.  ( 1st `  B ) q  <Q  ( c G d ) )
351, 2genpprecll 7290 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( c  e.  ( 1st `  A
)  /\  d  e.  ( 1st `  B ) )  ->  ( c G d )  e.  ( 1st `  ( A F B ) ) ) )
3635imp 123 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 1st `  A )  /\  d  e.  ( 1st `  B ) ) )  ->  (
c G d )  e.  ( 1st `  ( A F B ) ) )
37 elprnql 7257 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  c  e.  ( 1st `  A ) )  -> 
c  e.  Q. )
388, 37sylan 281 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  c  e.  ( 1st `  A ) )  -> 
c  e.  Q. )
39 elprnql 7257 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  d  e.  ( 1st `  B ) )  -> 
d  e.  Q. )
4011, 39sylan 281 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  d  e.  ( 1st `  B ) )  -> 
d  e.  Q. )
4138, 40anim12i 336 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  c  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  d  e.  ( 1st `  B ) ) )  ->  ( c  e. 
Q.  /\  d  e.  Q. ) )
4241an4s 562 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 1st `  A )  /\  d  e.  ( 1st `  B ) ) )  ->  (
c  e.  Q.  /\  d  e.  Q. )
)
432caovcl 5893 . . . . . . . . . 10  |-  ( ( c  e.  Q.  /\  d  e.  Q. )  ->  ( c G d )  e.  Q. )
4442, 43syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 1st `  A )  /\  d  e.  ( 1st `  B ) ) )  ->  (
c G d )  e.  Q. )
45 breq2 3903 . . . . . . . . . . 11  |-  ( r  =  ( c G d )  ->  (
q  <Q  r  <->  q  <Q  ( c G d ) ) )
46 eleq1 2180 . . . . . . . . . . 11  |-  ( r  =  ( c G d )  ->  (
r  e.  ( 1st `  ( A F B ) )  <->  ( c G d )  e.  ( 1st `  ( A F B ) ) ) )
4745, 46anbi12d 464 . . . . . . . . . 10  |-  ( r  =  ( c G d )  ->  (
( q  <Q  r  /\  r  e.  ( 1st `  ( A F B ) ) )  <-> 
( q  <Q  (
c G d )  /\  ( c G d )  e.  ( 1st `  ( A F B ) ) ) ) )
4847adantl 275 . . . . . . . . 9  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
c  e.  ( 1st `  A )  /\  d  e.  ( 1st `  B
) ) )  /\  r  =  ( c G d ) )  ->  ( ( q 
<Q  r  /\  r  e.  ( 1st `  ( A F B ) ) )  <->  ( q  <Q 
( c G d )  /\  ( c G d )  e.  ( 1st `  ( A F B ) ) ) ) )
4944, 48rspcedv 2767 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 1st `  A )  /\  d  e.  ( 1st `  B ) ) )  ->  (
( q  <Q  (
c G d )  /\  ( c G d )  e.  ( 1st `  ( A F B ) ) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  ( A F B ) ) ) ) )
5036, 49mpan2d 424 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 1st `  A )  /\  d  e.  ( 1st `  B ) ) )  ->  (
q  <Q  ( c G d )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  ( A F B ) ) ) ) )
5150rexlimdvva 2534 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. c  e.  ( 1st `  A
) E. d  e.  ( 1st `  B
) q  <Q  (
c G d )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( A F B ) ) ) ) )
5251adantr 274 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  q  e.  ( 1st `  ( A F B ) ) ) )  ->  ( E. c  e.  ( 1st `  A
) E. d  e.  ( 1st `  B
) q  <Q  (
c G d )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( A F B ) ) ) ) )
5334, 52mpd 13 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  q  e.  ( 1st `  ( A F B ) ) ) )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( A F B ) ) ) )
5453expr 372 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  q  e.  Q. )  ->  ( q  e.  ( 1st `  ( A F B ) )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( A F B ) ) ) ) )
55 genprndl.lower . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
P.  /\  g  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B
) ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g G h )  ->  x  e.  ( 1st `  ( A F B ) ) ) )
561, 2, 55genpcdl 7295 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( r  e.  ( 1st `  ( A F B ) )  ->  ( x  <Q  r  ->  x  e.  ( 1st `  ( A F B ) ) ) ) )
5756alrimdv 1832 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( r  e.  ( 1st `  ( A F B ) )  ->  A. x ( x 
<Q  r  ->  x  e.  ( 1st `  ( A F B ) ) ) ) )
58 breq1 3902 . . . . . . . . . . 11  |-  ( x  =  q  ->  (
x  <Q  r  <->  q  <Q  r ) )
59 eleq1 2180 . . . . . . . . . . 11  |-  ( x  =  q  ->  (
x  e.  ( 1st `  ( A F B ) )  <->  q  e.  ( 1st `  ( A F B ) ) ) )
6058, 59imbi12d 233 . . . . . . . . . 10  |-  ( x  =  q  ->  (
( x  <Q  r  ->  x  e.  ( 1st `  ( A F B ) ) )  <->  ( q  <Q  r  ->  q  e.  ( 1st `  ( A F B ) ) ) ) )
6160cbvalv 1871 . . . . . . . . 9  |-  ( A. x ( x  <Q  r  ->  x  e.  ( 1st `  ( A F B ) ) )  <->  A. q ( q 
<Q  r  ->  q  e.  ( 1st `  ( A F B ) ) ) )
6257, 61syl6ib 160 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( r  e.  ( 1st `  ( A F B ) )  ->  A. q ( q 
<Q  r  ->  q  e.  ( 1st `  ( A F B ) ) ) ) )
63 sp 1473 . . . . . . . 8  |-  ( A. q ( q  <Q 
r  ->  q  e.  ( 1st `  ( A F B ) ) )  ->  ( q  <Q  r  ->  q  e.  ( 1st `  ( A F B ) ) ) )
6462, 63syl6 33 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( r  e.  ( 1st `  ( A F B ) )  ->  ( q  <Q 
r  ->  q  e.  ( 1st `  ( A F B ) ) ) ) )
6564impd 252 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( r  e.  ( 1st `  ( A F B ) )  /\  q  <Q  r
)  ->  q  e.  ( 1st `  ( A F B ) ) ) )
6665ancomsd 267 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  <Q 
r  /\  r  e.  ( 1st `  ( A F B ) ) )  ->  q  e.  ( 1st `  ( A F B ) ) ) )
6766ad2antrr 479 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  q  e.  Q. )  /\  r  e.  Q. )  ->  (
( q  <Q  r  /\  r  e.  ( 1st `  ( A F B ) ) )  ->  q  e.  ( 1st `  ( A F B ) ) ) )
6867rexlimdva 2526 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  q  e.  Q. )  ->  ( E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  ( A F B ) ) )  ->  q  e.  ( 1st `  ( A F B ) ) ) )
6954, 68impbid 128 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  q  e.  Q. )  ->  ( q  e.  ( 1st `  ( A F B ) )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( A F B ) ) ) ) )
7069ralrimiva 2482 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. q  e.  Q.  ( q  e.  ( 1st `  ( A F B ) )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( A F B ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947   A.wal 1314    = wceq 1316   E.wex 1453    e. wcel 1465   A.wral 2393   E.wrex 2394   {crab 2397   <.cop 3500   class class class wbr 3899   ` cfv 5093  (class class class)co 5742    e. cmpo 5744   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056    <Q cltq 7061   P.cnp 7067
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-mi 7082  df-lti 7083  df-enq 7123  df-nqqs 7124  df-ltnqqs 7129  df-inp 7242
This theorem is referenced by:  addclpr  7313  mulclpr  7348
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