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Theorem genprndl 7551
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 7542 . . . . . . . . 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 2510 . . . . . . . . 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, 4bitrdi 196 . . . . . . . 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 296 . . . . . . 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 478 . . . . . 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 7505 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
9 prnmaxl 7518 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  a  e.  ( 1st `  A ) )  ->  E. c  e.  ( 1st `  A ) a 
<Q  c )
108, 9sylan 283 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  a  e.  ( 1st `  A ) )  ->  E. c  e.  ( 1st `  A ) a 
<Q  c )
11 prop 7505 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnmaxl 7518 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  b  e.  ( 1st `  B ) )  ->  E. d  e.  ( 1st `  B ) b 
<Q  d )
1311, 12sylan 283 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  b  e.  ( 1st `  B ) )  ->  E. d  e.  ( 1st `  B ) b 
<Q  d )
1410, 13anim12i 338 . . . . . . . . . . . . . 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 588 . . . . . . . . . . . . 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 2660 . . . . . . . . . . . . 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 134 . . . . . . . . . . . 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 7540 . . . . . . . . . . . . . 14  |-  ( ( a  <Q  c  /\  b  <Q  d )  -> 
( a G b )  <Q  ( c G d ) )
2120reximi 2587 . . . . . . . . . . . . 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 2587 . . . . . . . . . . . 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 479 . . . . . . . . . 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 4021 . . . . . . . . . . . . . 14  |-  ( q  =  ( a G b )  ->  (
q  <Q  ( c G d )  <->  ( a G b )  <Q 
( c G d ) ) )
2625biimprd 158 . . . . . . . . . . . . 13  |-  ( q  =  ( a G b )  ->  (
( a G b )  <Q  ( c G d )  -> 
q  <Q  ( c G d ) ) )
2726reximdv 2591 . . . . . . . . . . . 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 2591 . . . . . . . . . . 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 491 . . . . . . . . . 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 115 . . . . . . . 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 1909 . . . . . . 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 276 . . . . . 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 7544 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( c  e.  ( 1st `  A
)  /\  d  e.  ( 1st `  B ) )  ->  ( c G d )  e.  ( 1st `  ( A F B ) ) ) )
3635imp 124 . . . . . . . 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 7511 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  c  e.  ( 1st `  A ) )  -> 
c  e.  Q. )
388, 37sylan 283 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  c  e.  ( 1st `  A ) )  -> 
c  e.  Q. )
39 elprnql 7511 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  d  e.  ( 1st `  B ) )  -> 
d  e.  Q. )
4011, 39sylan 283 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  d  e.  ( 1st `  B ) )  -> 
d  e.  Q. )
4138, 40anim12i 338 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  c  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  d  e.  ( 1st `  B ) ) )  ->  ( c  e. 
Q.  /\  d  e.  Q. ) )
4241an4s 588 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 1st `  A )  /\  d  e.  ( 1st `  B ) ) )  ->  (
c  e.  Q.  /\  d  e.  Q. )
)
432caovcl 6052 . . . . . . . . . 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 4022 . . . . . . . . . . 11  |-  ( r  =  ( c G d )  ->  (
q  <Q  r  <->  q  <Q  ( c G d ) ) )
46 eleq1 2252 . . . . . . . . . . 11  |-  ( r  =  ( c G d )  ->  (
r  e.  ( 1st `  ( A F B ) )  <->  ( c G d )  e.  ( 1st `  ( A F B ) ) ) )
4745, 46anbi12d 473 . . . . . . . . . 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 277 . . . . . . . . 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 2860 . . . . . . . 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 428 . . . . . . 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 2615 . . . . . 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 276 . . . . 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 375 . . 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 7549 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( r  e.  ( 1st `  ( A F B ) )  ->  ( x  <Q  r  ->  x  e.  ( 1st `  ( A F B ) ) ) ) )
5756alrimdv 1887 . . . . . . . . 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 4021 . . . . . . . . . . 11  |-  ( x  =  q  ->  (
x  <Q  r  <->  q  <Q  r ) )
59 eleq1 2252 . . . . . . . . . . 11  |-  ( x  =  q  ->  (
x  e.  ( 1st `  ( A F B ) )  <->  q  e.  ( 1st `  ( A F B ) ) ) )
6058, 59imbi12d 234 . . . . . . . . . 10  |-  ( x  =  q  ->  (
( x  <Q  r  ->  x  e.  ( 1st `  ( A F B ) ) )  <->  ( q  <Q  r  ->  q  e.  ( 1st `  ( A F B ) ) ) ) )
6160cbvalv 1929 . . . . . . . . 9  |-  ( A. x ( x  <Q  r  ->  x  e.  ( 1st `  ( A F B ) ) )  <->  A. q ( q 
<Q  r  ->  q  e.  ( 1st `  ( A F B ) ) ) )
6257, 61imbitrdi 161 . . . . . . . 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 1522 . . . . . . . 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 254 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( r  e.  ( 1st `  ( A F B ) )  /\  q  <Q  r
)  ->  q  e.  ( 1st `  ( A F B ) ) ) )
6665ancomsd 269 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  <Q 
r  /\  r  e.  ( 1st `  ( A F B ) ) )  ->  q  e.  ( 1st `  ( A F B ) ) ) )
6766ad2antrr 488 . . . 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 2607 . . 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 129 . 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 2563 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 104    <-> wb 105    /\ w3a 980   A.wal 1362    = wceq 1364   E.wex 1503    e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472   <.cop 3610   class class class wbr 4018   ` cfv 5235  (class class class)co 5897    e. cmpo 5899   1stc1st 6164   2ndc2nd 6165   Q.cnq 7310    <Q cltq 7315   P.cnp 7321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-mi 7336  df-lti 7337  df-enq 7377  df-nqqs 7378  df-ltnqqs 7383  df-inp 7496
This theorem is referenced by:  addclpr  7567  mulclpr  7602
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