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Theorem ltexnq 9156
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 9107 . . . 4  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4899 . . 3  |-  ( A 
<Q  B  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
3 ordpinq 9124 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
4 elpqn 9106 . . . . . . . . 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 6618 . . . . . . . 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 9106 . . . . . . . . 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 6619 . . . . . . . 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 9074 . . . . . . 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 6618 . . . . . . . 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 6619 . . . . . . . 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 9074 . . . . . . 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 9083 . . . . . 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 4959 . . . . . . . . . . . 12  |-  Rel  ( N.  X.  N. )
234ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  A  e.  ( N.  X.  N. )
)
24 1st2nd 6632 . . . . . . . . . . . 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 6118 . . . . . . . . . 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 9074 . . . . . . . . . . . . 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 9118 . . . . . . . . . . 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 1219 . . . . . . . . . 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 2475 . . . . . . . . 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 6111 . . . . . . . . . . . 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 9079 . . . . . . . . . . . . 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 5713 . . . . . . . . . . . . . . 15  |-  ( 2nd `  A )  e.  _V
39 fvex 5713 . . . . . . . . . . . . . . 15  |-  ( 1st `  A )  e.  _V
40 fvex 5713 . . . . . . . . . . . . . . 15  |-  ( 2nd `  B )  e.  _V
41 mulcompi 9077 . . . . . . . . . . . . . . 15  |-  ( x  .N  y )  =  ( y  .N  x
)
42 mulasspi 9078 . . . . . . . . . . . . . . 15  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
4338, 39, 40, 41, 42caov12 6303 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  ( ( 1st `  A )  .N  ( 2nd `  B ) ) )  =  ( ( 1st `  A )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) )
44 mulcompi 9077 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  y )  =  ( y  .N  ( 2nd `  A ) )
4543, 44oveq12i 6115 . . . . . . . . . . . . 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 2464 . . . . . . . . . . . 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 9078 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 1st `  B
) ) )
48 mulcompi 9077 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )
4948oveq2i 6114 . . . . . . . . . . . . 13  |-  ( ( 2nd `  A )  .N  ( ( 2nd `  A )  .N  ( 1st `  B ) ) )  =  ( ( 2nd `  A )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) )
5047, 49eqtri 2463 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )
5136, 46, 503eqtr4g 2500 . . . . . . . . . . 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 9078 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) )
5352eqcomi 2447 . . . . . . . . . . . 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 4079 . . . . . . . . . 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 2454 . . . . . . . . 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 5703 . . . . . . . . 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 9137 . . . . . . . . . . 11  |-  ( ( /Q `  A )  +Q  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
)  =  ( /Q
`  ( A  +pQ  <.
y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
)
60 nqerid 9114 . . . . . . . . . . . . 13  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
6160ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( /Q `  A )  =  A )
6261oveq1d 6118 . . . . . . . . . . 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 2489 . . . . . . . . . 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 9074 . . . . . . . . . . . . . . . 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 9141 . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . 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 6632 . . . . . . . . . . . . . 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 4330 . . . . . . . . . . . 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 9074 . . . . . . . . . . . . . . 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 9074 . . . . . . . . . . . . . . 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 4883 . . . . . . . . . . . . . 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 9116 . . . . . . . . . . . . 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 9114 . . . . . . . . . . . 12  |-  ( B  e.  Q.  ->  ( /Q `  B )  =  B )
8584ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( /Q `  B )  =  B )
8683, 85eqtrd 2475 . . . . . . . . . 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 2457 . . . . . . . . 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 5713 . . . . . . . 8  |-  ( /Q
`  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)  e.  _V
91 oveq2 6111 . . . . . . . . 9  |-  ( x  =  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )  ->  ( A  +Q  x
)  =  ( A  +Q  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
) )
9291eqeq1d 2451 . . . . . . . 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 3076 . . . . . . 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 2853 . . . . 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 2503 . . . . . . 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 9129 . . . . . . . 8  |-  +Q  :
( Q.  X.  Q. )
--> Q.
102101fdmi 5576 . . . . . . 7  |-  dom  +Q  =  ( Q.  X.  Q. )
103 0nnq 9105 . . . . . . 7  |-  -.  (/)  e.  Q.
104102, 103ndmovrcl 6261 . . . . . 6  |-  ( ( A  +Q  x )  e.  Q.  ->  ( A  e.  Q.  /\  x  e.  Q. ) )
105 ltaddnq 9155 . . . . . 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 4328 . . . 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 1690 . 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 1369   E.wex 1586    e. wcel 1756   E.wrex 2728   <.cop 3895   class class class wbr 4304    X. cxp 4850   Rel wrel 4857   ` cfv 5430  (class class class)co 6103   1stc1st 6587   2ndc2nd 6588   N.cnpi 9023    +N cpli 9024    .N cmi 9025    <N clti 9026    +pQ cplpq 9027    ~Q ceq 9030   Q.cnq 9031   /Qcerq 9033    +Q cplq 9034    <Q cltq 9037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-omul 6937  df-er 7113  df-ni 9053  df-pli 9054  df-mi 9055  df-lti 9056  df-plpq 9089  df-mpq 9090  df-ltpq 9091  df-enq 9092  df-nq 9093  df-erq 9094  df-plq 9095  df-mq 9096  df-1nq 9097  df-ltnq 9099
This theorem is referenced by:  ltbtwnnq  9159  prnmadd  9178  ltexprlem4  9220  ltexprlem7  9223  prlem936  9228
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