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Theorem genprndl 6677
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 6668 . . . . . . . . 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 2361 . . . . . . . . 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 189 . . . . . . . 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 284 . . . . . . 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 455 . . . . . 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 6631 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
9 prnmaxl 6644 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  a  e.  ( 1st `  A ) )  ->  E. c  e.  ( 1st `  A ) a 
<Q  c )
108, 9sylan 271 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  a  e.  ( 1st `  A ) )  ->  E. c  e.  ( 1st `  A ) a 
<Q  c )
11 prop 6631 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnmaxl 6644 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  b  e.  ( 1st `  B ) )  ->  E. d  e.  ( 1st `  B ) b 
<Q  d )
1311, 12sylan 271 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  b  e.  ( 1st `  B ) )  ->  E. d  e.  ( 1st `  B ) b 
<Q  d )
1410, 13anim12i 325 . . . . . . . . . . . . . 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 530 . . . . . . . . . . . . 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 2496 . . . . . . . . . . . . 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 141 . . . . . . . . . . . 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 6666 . . . . . . . . . . . . . 14  |-  ( ( a  <Q  c  /\  b  <Q  d )  -> 
( a G b )  <Q  ( c G d ) )
2120reximi 2433 . . . . . . . . . . . . 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 2433 . . . . . . . . . . . 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 456 . . . . . . . . . 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 3795 . . . . . . . . . . . . . 14  |-  ( q  =  ( a G b )  ->  (
q  <Q  ( c G d )  <->  ( a G b )  <Q 
( c G d ) ) )
2625biimprd 151 . . . . . . . . . . . . 13  |-  ( q  =  ( a G b )  ->  (
( a G b )  <Q  ( c G d )  -> 
q  <Q  ( c G d ) ) )
2726reximdv 2437 . . . . . . . . . . . 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 2437 . . . . . . . . . . 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 468 . . . . . . . . . 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 112 . . . . . . . 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 1793 . . . . . . 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 265 . . . . . 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 6670 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( c  e.  ( 1st `  A
)  /\  d  e.  ( 1st `  B ) )  ->  ( c G d )  e.  ( 1st `  ( A F B ) ) ) )
3635imp 119 . . . . . . . 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 6637 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  c  e.  ( 1st `  A ) )  -> 
c  e.  Q. )
388, 37sylan 271 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  c  e.  ( 1st `  A ) )  -> 
c  e.  Q. )
39 elprnql 6637 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  d  e.  ( 1st `  B ) )  -> 
d  e.  Q. )
4011, 39sylan 271 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  d  e.  ( 1st `  B ) )  -> 
d  e.  Q. )
4138, 40anim12i 325 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  c  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  d  e.  ( 1st `  B ) ) )  ->  ( c  e. 
Q.  /\  d  e.  Q. ) )
4241an4s 530 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( c  e.  ( 1st `  A )  /\  d  e.  ( 1st `  B ) ) )  ->  (
c  e.  Q.  /\  d  e.  Q. )
)
432caovcl 5683 . . . . . . . . . 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 3796 . . . . . . . . . . 11  |-  ( r  =  ( c G d )  ->  (
q  <Q  r  <->  q  <Q  ( c G d ) ) )
46 eleq1 2116 . . . . . . . . . . 11  |-  ( r  =  ( c G d )  ->  (
r  e.  ( 1st `  ( A F B ) )  <->  ( c G d )  e.  ( 1st `  ( A F B ) ) ) )
4745, 46anbi12d 450 . . . . . . . . . 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 266 . . . . . . . . 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 2677 . . . . . . . 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 412 . . . . . . 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 2457 . . . . . 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 265 . . . . 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 361 . . 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 6675 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( r  e.  ( 1st `  ( A F B ) )  ->  ( x  <Q  r  ->  x  e.  ( 1st `  ( A F B ) ) ) ) )
5756alrimdv 1772 . . . . . . . . 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 3795 . . . . . . . . . . 11  |-  ( x  =  q  ->  (
x  <Q  r  <->  q  <Q  r ) )
59 eleq1 2116 . . . . . . . . . . 11  |-  ( x  =  q  ->  (
x  e.  ( 1st `  ( A F B ) )  <->  q  e.  ( 1st `  ( A F B ) ) ) )
6058, 59imbi12d 227 . . . . . . . . . 10  |-  ( x  =  q  ->  (
( x  <Q  r  ->  x  e.  ( 1st `  ( A F B ) ) )  <->  ( q  <Q  r  ->  q  e.  ( 1st `  ( A F B ) ) ) ) )
6160cbvalv 1810 . . . . . . . . 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 154 . . . . . . . 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 1417 . . . . . . . 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 246 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( r  e.  ( 1st `  ( A F B ) )  /\  q  <Q  r
)  ->  q  e.  ( 1st `  ( A F B ) ) ) )
6665ancomsd 260 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  <Q 
r  /\  r  e.  ( 1st `  ( A F B ) ) )  ->  q  e.  ( 1st `  ( A F B ) ) ) )
6766ad2antrr 465 . . . 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 2450 . . 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 124 . 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 2409 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 101    <-> wb 102    /\ w3a 896   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   A.wral 2323   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540    |-> cmpt2 5542   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    <Q cltq 6441   P.cnp 6447
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 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-lti 6463  df-enq 6503  df-nqqs 6504  df-ltnqqs 6509  df-inp 6622
This theorem is referenced by:  addclpr  6693  mulclpr  6728
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