MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltexnq Structured version   Unicode version

Theorem ltexnq 9365
Description: Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltexnq  |-  ( B  e.  Q.  ->  ( A  <Q  B  <->  E. x
( A  +Q  x
)  =  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem ltexnq
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 9316 . . . 4  |-  <Q  C_  ( Q.  X.  Q. )
21brel 5054 . . 3  |-  ( A 
<Q  B  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
3 ordpinq 9333 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
4 elpqn 9315 . . . . . . . . 9  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
54adantr 465 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  e.  ( N. 
X.  N. ) )
6 xp1st 6825 . . . . . . . 8  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
75, 6syl 16 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 1st `  A
)  e.  N. )
8 elpqn 9315 . . . . . . . . 9  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
98adantl 466 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  B  e.  ( N. 
X.  N. ) )
10 xp2nd 6826 . . . . . . . 8  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
119, 10syl 16 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 2nd `  B
)  e.  N. )
12 mulclpi 9283 . . . . . . 7  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
137, 11, 12syl2anc 661 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
14 xp1st 6825 . . . . . . . 8  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
159, 14syl 16 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 1st `  B
)  e.  N. )
16 xp2nd 6826 . . . . . . . 8  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
175, 16syl 16 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 2nd `  A
)  e.  N. )
18 mulclpi 9283 . . . . . . 7  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  e. 
N. )
1915, 17, 18syl2anc 661 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  e. 
N. )
20 ltexpi 9292 . . . . . 6  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N.  /\  ( ( 1st `  B )  .N  ( 2nd `  A
) )  e.  N. )  ->  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) )  <->  E. y  e.  N.  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  +N  y
)  =  ( ( 1st `  B )  .N  ( 2nd `  A
) ) ) )
2113, 19, 20syl2anc 661 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( ( 1st `  A )  .N  ( 2nd `  B ) ) 
<N  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  <->  E. y  e.  N.  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  +N  y
)  =  ( ( 1st `  B )  .N  ( 2nd `  A
) ) ) )
22 relxp 5116 . . . . . . . . . . . 12  |-  Rel  ( N.  X.  N. )
234ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  A  e.  ( N.  X.  N. )
)
24 1st2nd 6841 . . . . . . . . . . . 12  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2522, 23, 24sylancr 663 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
2625oveq1d 6310 . . . . . . . . . 10  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( A  +pQ  <.
y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )  =  ( <. ( 1st `  A ) ,  ( 2nd `  A
) >.  +pQ  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
) )
277adantr 465 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 1st `  A
)  e.  N. )
2817adantr 465 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 2nd `  A
)  e.  N. )
29 simpr 461 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  y  e.  N. )
30 mulclpi 9283 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
3117, 11, 30syl2anc 661 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
3231adantr 465 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( 2nd `  A )  .N  ( 2nd `  B ) )  e.  N. )
33 addpipq 9327 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  /\  ( y  e.  N.  /\  ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. ) )  -> 
( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  +pQ  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>. )  =  <. ( ( ( 1st `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) )  +N  ( y  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) ) >. )
3427, 28, 29, 32, 33syl22anc 1229 . . . . . . . . . 10  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( <. ( 1st `  A ) ,  ( 2nd `  A
) >.  +pQ  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)  =  <. (
( ( 1st `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) )  +N  ( y  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) ) >. )
3526, 34eqtrd 2508 . . . . . . . . 9  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( A  +pQ  <.
y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )  =  <. ( ( ( 1st `  A )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) )  +N  ( y  .N  ( 2nd `  A
) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) >.
)
36 oveq2 6303 . . . . . . . . . . . 12  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  y )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  -> 
( ( 2nd `  A
)  .N  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  y ) )  =  ( ( 2nd `  A
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) ) )
37 distrpi 9288 . . . . . . . . . . . . 13  |-  ( ( 2nd `  A )  .N  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  +N  y
) )  =  ( ( ( 2nd `  A
)  .N  ( ( 1st `  A )  .N  ( 2nd `  B
) ) )  +N  ( ( 2nd `  A
)  .N  y ) )
38 fvex 5882 . . . . . . . . . . . . . . 15  |-  ( 2nd `  A )  e.  _V
39 fvex 5882 . . . . . . . . . . . . . . 15  |-  ( 1st `  A )  e.  _V
40 fvex 5882 . . . . . . . . . . . . . . 15  |-  ( 2nd `  B )  e.  _V
41 mulcompi 9286 . . . . . . . . . . . . . . 15  |-  ( x  .N  y )  =  ( y  .N  x
)
42 mulasspi 9287 . . . . . . . . . . . . . . 15  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
4338, 39, 40, 41, 42caov12 6498 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  ( ( 1st `  A )  .N  ( 2nd `  B ) ) )  =  ( ( 1st `  A )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) )
44 mulcompi 9286 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  y )  =  ( y  .N  ( 2nd `  A ) )
4543, 44oveq12i 6307 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  .N  ( ( 1st `  A )  .N  ( 2nd `  B
) ) )  +N  ( ( 2nd `  A
)  .N  y ) )  =  ( ( ( 1st `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) )  +N  ( y  .N  ( 2nd `  A ) ) )
4637, 45eqtr2i 2497 . . . . . . . . . . . 12  |-  ( ( ( 1st `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) )  +N  ( y  .N  ( 2nd `  A ) ) )  =  ( ( 2nd `  A )  .N  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  +N  y
) )
47 mulasspi 9287 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 1st `  B
) ) )
48 mulcompi 9286 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )
4948oveq2i 6306 . . . . . . . . . . . . 13  |-  ( ( 2nd `  A )  .N  ( ( 2nd `  A )  .N  ( 1st `  B ) ) )  =  ( ( 2nd `  A )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) )
5047, 49eqtri 2496 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )
5136, 46, 503eqtr4g 2533 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  y )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  -> 
( ( ( 1st `  A )  .N  (
( 2nd `  A
)  .N  ( 2nd `  B ) ) )  +N  ( y  .N  ( 2nd `  A
) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) )
52 mulasspi 9287 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) )
5352eqcomi 2480 . . . . . . . . . . . 12  |-  ( ( 2nd `  A )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) )
5453a1i 11 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  y )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  -> 
( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) )
5551, 54opeq12d 4227 . . . . . . . . . 10  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  y )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  ->  <. ( ( ( 1st `  A )  .N  (
( 2nd `  A
)  .N  ( 2nd `  B ) ) )  +N  ( y  .N  ( 2nd `  A
) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) >.  =  <. ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 1st `  B ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 2nd `  B ) )
>. )
5655eqeq2d 2481 . . . . . . . . 9  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  y )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  -> 
( ( A  +pQ  <.
y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )  =  <. ( ( ( 1st `  A )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) )  +N  ( y  .N  ( 2nd `  A
) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) >.  <->  ( A  +pQ  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>. )  =  <. ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >. )
)
5735, 56syl5ibcom 220 . . . . . . . 8  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  y )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  -> 
( A  +pQ  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>. )  =  <. ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >. )
)
58 fveq2 5872 . . . . . . . . 9  |-  ( ( A  +pQ  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>. )  =  <. ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >.  ->  ( /Q `  ( A  +pQ  <.
y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
)  =  ( /Q
`  <. ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 1st `  B ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 2nd `  B ) )
>. ) )
59 adderpq 9346 . . . . . . . . . . 11  |-  ( ( /Q `  A )  +Q  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
)  =  ( /Q
`  ( A  +pQ  <.
y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
)
60 nqerid 9323 . . . . . . . . . . . . 13  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
6160ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( /Q `  A )  =  A )
6261oveq1d 6310 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( /Q
`  A )  +Q  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>. ) )  =  ( A  +Q  ( /Q
`  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
) ) )
6359, 62syl5eqr 2522 . . . . . . . . . 10  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( /Q `  ( A  +pQ  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>. ) )  =  ( A  +Q  ( /Q
`  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
) ) )
64 mulclpi 9283 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  A ) )  e. 
N. )
6517, 17, 64syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  e. 
N. )
6665adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( 2nd `  A )  .N  ( 2nd `  A ) )  e.  N. )
6715adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 1st `  B
)  e.  N. )
6811adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 2nd `  B
)  e.  N. )
69 mulcanenq 9350 . . . . . . . . . . . . . 14  |-  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  e. 
N.  /\  ( 1st `  B )  e.  N.  /\  ( 2nd `  B
)  e.  N. )  -> 
<. ( ( ( 2nd `  A )  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >.  ~Q  <. ( 1st `  B ) ,  ( 2nd `  B
) >. )
7066, 67, 68, 69syl3anc 1228 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  <. ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 1st `  B ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 2nd `  B ) )
>.  ~Q  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
718ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  B  e.  ( N.  X.  N. )
)
72 1st2nd 6841 . . . . . . . . . . . . . 14  |-  ( ( Rel  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
7322, 71, 72sylancr 663 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  B  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. )
7470, 73breqtrrd 4479 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  <. ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 1st `  B ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 2nd `  B ) )
>.  ~Q  B )
75 mulclpi 9283 . . . . . . . . . . . . . . 15  |-  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  e. 
N.  /\  ( 1st `  B )  e.  N. )  ->  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 1st `  B ) )  e.  N. )
7666, 67, 75syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 1st `  B ) )  e.  N. )
77 mulclpi 9283 . . . . . . . . . . . . . . 15  |-  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  e. 
N.  /\  ( 2nd `  B )  e.  N. )  ->  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 2nd `  B ) )  e.  N. )
7866, 68, 77syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 2nd `  B ) )  e.  N. )
79 opelxpi 5037 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( 2nd `  A )  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) )  e.  N.  /\  ( ( ( 2nd `  A )  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) )  e.  N. )  ->  <. ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 1st `  B ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 2nd `  B ) )
>.  e.  ( N.  X.  N. ) )
8076, 78, 79syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  <. ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 1st `  B ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 2nd `  B ) )
>.  e.  ( N.  X.  N. ) )
81 nqereq 9325 . . . . . . . . . . . . 13  |-  ( (
<. ( ( ( 2nd `  A )  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >.  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( <. (
( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >.  ~Q  B  <->  ( /Q `  <. (
( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >. )  =  ( /Q `  B ) ) )
8280, 71, 81syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( <. (
( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >.  ~Q  B  <->  ( /Q `  <. (
( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >. )  =  ( /Q `  B ) ) )
8374, 82mpbid 210 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( /Q `  <. ( ( ( 2nd `  A )  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >. )  =  ( /Q `  B ) )
84 nqerid 9323 . . . . . . . . . . . 12  |-  ( B  e.  Q.  ->  ( /Q `  B )  =  B )
8584ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( /Q `  B )  =  B )
8683, 85eqtrd 2508 . . . . . . . . . 10  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( /Q `  <. ( ( ( 2nd `  A )  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >. )  =  B )
8763, 86eqeq12d 2489 . . . . . . . . 9  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( /Q
`  ( A  +pQ  <.
y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
)  =  ( /Q
`  <. ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 1st `  B ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 2nd `  B ) )
>. )  <->  ( A  +Q  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>. ) )  =  B ) )
8858, 87syl5ib 219 . . . . . . . 8  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( A 
+pQ  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)  =  <. (
( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) ) >.  ->  ( A  +Q  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
)  =  B ) )
8957, 88syld 44 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  y )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  -> 
( A  +Q  ( /Q `  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
) )  =  B ) )
90 fvex 5882 . . . . . . . 8  |-  ( /Q
`  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)  e.  _V
91 oveq2 6303 . . . . . . . . 9  |-  ( x  =  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )  ->  ( A  +Q  x
)  =  ( A  +Q  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
) )
9291eqeq1d 2469 . . . . . . . 8  |-  ( x  =  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )  ->  ( ( A  +Q  x )  =  B  <-> 
( A  +Q  ( /Q `  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
) )  =  B ) )
9390, 92spcev 3210 . . . . . . 7  |-  ( ( A  +Q  ( /Q
`  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
) )  =  B  ->  E. x ( A  +Q  x )  =  B )
9489, 93syl6 33 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  y )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  ->  E. x ( A  +Q  x )  =  B ) )
9594rexlimdva 2959 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( E. y  e. 
N.  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  +N  y
)  =  ( ( 1st `  B )  .N  ( 2nd `  A
) )  ->  E. x
( A  +Q  x
)  =  B ) )
9621, 95sylbid 215 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( ( 1st `  A )  .N  ( 2nd `  B ) ) 
<N  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  ->  E. x ( A  +Q  x )  =  B ) )
973, 96sylbid 215 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  ->  E. x ( A  +Q  x )  =  B ) )
982, 97mpcom 36 . 2  |-  ( A 
<Q  B  ->  E. x
( A  +Q  x
)  =  B )
99 eleq1 2539 . . . . . . 7  |-  ( ( A  +Q  x )  =  B  ->  (
( A  +Q  x
)  e.  Q.  <->  B  e.  Q. ) )
10099biimparc 487 . . . . . 6  |-  ( ( B  e.  Q.  /\  ( A  +Q  x
)  =  B )  ->  ( A  +Q  x )  e.  Q. )
101 addnqf 9338 . . . . . . . 8  |-  +Q  :
( Q.  X.  Q. )
--> Q.
102101fdmi 5742 . . . . . . 7  |-  dom  +Q  =  ( Q.  X.  Q. )
103 0nnq 9314 . . . . . . 7  |-  -.  (/)  e.  Q.
104102, 103ndmovrcl 6456 . . . . . 6  |-  ( ( A  +Q  x )  e.  Q.  ->  ( A  e.  Q.  /\  x  e.  Q. ) )
105 ltaddnq 9364 . . . . . 6  |-  ( ( A  e.  Q.  /\  x  e.  Q. )  ->  A  <Q  ( A  +Q  x ) )
106100, 104, 1053syl 20 . . . . 5  |-  ( ( B  e.  Q.  /\  ( A  +Q  x
)  =  B )  ->  A  <Q  ( A  +Q  x ) )
107 simpr 461 . . . . 5  |-  ( ( B  e.  Q.  /\  ( A  +Q  x
)  =  B )  ->  ( A  +Q  x )  =  B )
108106, 107breqtrd 4477 . . . 4  |-  ( ( B  e.  Q.  /\  ( A  +Q  x
)  =  B )  ->  A  <Q  B )
109108ex 434 . . 3  |-  ( B  e.  Q.  ->  (
( A  +Q  x
)  =  B  ->  A  <Q  B ) )
110109exlimdv 1700 . 2  |-  ( B  e.  Q.  ->  ( E. x ( A  +Q  x )  =  B  ->  A  <Q  B ) )
11198, 110impbid2 204 1  |-  ( B  e.  Q.  ->  ( A  <Q  B  <->  E. x
( A  +Q  x
)  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2818   <.cop 4039   class class class wbr 4453    X. cxp 5003   Rel wrel 5010   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   N.cnpi 9234    +N cpli 9235    .N cmi 9236    <N clti 9237    +pQ cplpq 9238    ~Q ceq 9241   Q.cnq 9242   /Qcerq 9244    +Q cplq 9245    <Q cltq 9248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-omul 7147  df-er 7323  df-ni 9262  df-pli 9263  df-mi 9264  df-lti 9265  df-plpq 9298  df-mpq 9299  df-ltpq 9300  df-enq 9301  df-nq 9302  df-erq 9303  df-plq 9304  df-mq 9305  df-1nq 9306  df-ltnq 9308
This theorem is referenced by:  ltbtwnnq  9368  prnmadd  9387  ltexprlem4  9429  ltexprlem7  9432  prlem936  9437
  Copyright terms: Public domain W3C validator