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Theorem ltexnq 8778
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 8729 . . . 4  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4859 . . 3  |-  ( A 
<Q  B  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
3 ordpinq 8746 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
4 elpqn 8728 . . . . . . . . 9  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
54adantr 452 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  e.  ( N. 
X.  N. ) )
6 xp1st 6308 . . . . . . . 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 8728 . . . . . . . . 9  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
98adantl 453 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  B  e.  ( N. 
X.  N. ) )
10 xp2nd 6309 . . . . . . . 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 8696 . . . . . . 7  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
137, 11, 12syl2anc 643 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
14 xp1st 6308 . . . . . . . 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 6309 . . . . . . . 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 8696 . . . . . . 7  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  e. 
N. )
1915, 17, 18syl2anc 643 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  e. 
N. )
20 ltexpi 8705 . . . . . 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 643 . . . . 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 4916 . . . . . . . . . . . 12  |-  Rel  ( N.  X.  N. )
234ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  A  e.  ( N.  X.  N. )
)
24 1st2nd 6325 . . . . . . . . . . . 12  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2522, 23, 24sylancr 645 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
2625oveq1d 6028 . . . . . . . . . 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 452 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 1st `  A
)  e.  N. )
2817adantr 452 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 2nd `  A
)  e.  N. )
29 simpr 448 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  y  e.  N. )
30 mulclpi 8696 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
3117, 11, 30syl2anc 643 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
3231adantr 452 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( 2nd `  A )  .N  ( 2nd `  B ) )  e.  N. )
33 addpipq 8740 . . . . . . . . . . 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 1185 . . . . . . . . . 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 2412 . . . . . . . . 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 6021 . . . . . . . . . . . 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 8701 . . . . . . . . . . . . 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 5675 . . . . . . . . . . . . . . 15  |-  ( 2nd `  A )  e.  _V
39 fvex 5675 . . . . . . . . . . . . . . 15  |-  ( 1st `  A )  e.  _V
40 fvex 5675 . . . . . . . . . . . . . . 15  |-  ( 2nd `  B )  e.  _V
41 mulcompi 8699 . . . . . . . . . . . . . . 15  |-  ( x  .N  y )  =  ( y  .N  x
)
42 mulasspi 8700 . . . . . . . . . . . . . . 15  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
4338, 39, 40, 41, 42caov12 6207 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  ( ( 1st `  A )  .N  ( 2nd `  B ) ) )  =  ( ( 1st `  A )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) )
44 mulcompi 8699 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  y )  =  ( y  .N  ( 2nd `  A ) )
4543, 44oveq12i 6025 . . . . . . . . . . . . 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 2401 . . . . . . . . . . . 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 8700 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 1st `  B
) ) )
48 mulcompi 8699 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )
4948oveq2i 6024 . . . . . . . . . . . . 13  |-  ( ( 2nd `  A )  .N  ( ( 2nd `  A )  .N  ( 1st `  B ) ) )  =  ( ( 2nd `  A )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) )
5047, 49eqtri 2400 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )
5136, 46, 503eqtr4g 2437 . . . . . . . . . . 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 8700 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) )
5352eqcomi 2384 . . . . . . . . . . . 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 3927 . . . . . . . . . 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 2391 . . . . . . . . 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 212 . . . . . . . 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 5661 . . . . . . . . 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 8759 . . . . . . . . . . 11  |-  ( ( /Q `  A )  +Q  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
)  =  ( /Q
`  ( A  +pQ  <.
y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
)
60 nqerid 8736 . . . . . . . . . . . . 13  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
6160ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( /Q `  A )  =  A )
6261oveq1d 6028 . . . . . . . . . . 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 2426 . . . . . . . . . 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 8696 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  A ) )  e. 
N. )
6517, 17, 64syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  e. 
N. )
6665adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( 2nd `  A )  .N  ( 2nd `  A ) )  e.  N. )
6715adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 1st `  B
)  e.  N. )
6811adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 2nd `  B
)  e.  N. )
69 mulcanenq 8763 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  B  e.  ( N.  X.  N. )
)
72 1st2nd 6325 . . . . . . . . . . . . . 14  |-  ( ( Rel  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
7322, 71, 72sylancr 645 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  B  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. )
7470, 73breqtrrd 4172 . . . . . . . . . . . 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 8696 . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 1st `  B ) )  e.  N. )
77 mulclpi 8696 . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 2nd `  B ) )  e.  N. )
79 opelxpi 4843 . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . 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 8738 . . . . . . . . . . . . 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 643 . . . . . . . . . . . 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 202 . . . . . . . . . . 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 8736 . . . . . . . . . . . 12  |-  ( B  e.  Q.  ->  ( /Q `  B )  =  B )
8584ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( /Q `  B )  =  B )
8683, 85eqtrd 2412 . . . . . . . . . 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 2394 . . . . . . . . 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 211 . . . . . . . 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 42 . . . . . . 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 5675 . . . . . . . 8  |-  ( /Q
`  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)  e.  _V
91 oveq2 6021 . . . . . . . . 9  |-  ( x  =  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )  ->  ( A  +Q  x
)  =  ( A  +Q  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
) )
9291eqeq1d 2388 . . . . . . . 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 2979 . . . . . . 7  |-  ( ( A  +Q  ( /Q
`  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
) )  =  B  ->  E. x ( A  +Q  x )  =  B )
9489, 93syl6 31 . . . . . 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 2766 . . . . 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 207 . . . 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 207 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  ->  E. x ( A  +Q  x )  =  B ) )
982, 97mpcom 34 . 2  |-  ( A 
<Q  B  ->  E. x
( A  +Q  x
)  =  B )
99 eleq1 2440 . . . . . . 7  |-  ( ( A  +Q  x )  =  B  ->  (
( A  +Q  x
)  e.  Q.  <->  B  e.  Q. ) )
10099biimparc 474 . . . . . 6  |-  ( ( B  e.  Q.  /\  ( A  +Q  x
)  =  B )  ->  ( A  +Q  x )  e.  Q. )
101 addnqf 8751 . . . . . . . 8  |-  +Q  :
( Q.  X.  Q. )
--> Q.
102101fdmi 5529 . . . . . . 7  |-  dom  +Q  =  ( Q.  X.  Q. )
103 0nnq 8727 . . . . . . 7  |-  -.  (/)  e.  Q.
104102, 103ndmovrcl 6165 . . . . . 6  |-  ( ( A  +Q  x )  e.  Q.  ->  ( A  e.  Q.  /\  x  e.  Q. ) )
105 ltaddnq 8777 . . . . . 6  |-  ( ( A  e.  Q.  /\  x  e.  Q. )  ->  A  <Q  ( A  +Q  x ) )
106100, 104, 1053syl 19 . . . . 5  |-  ( ( B  e.  Q.  /\  ( A  +Q  x
)  =  B )  ->  A  <Q  ( A  +Q  x ) )
107 simpr 448 . . . . 5  |-  ( ( B  e.  Q.  /\  ( A  +Q  x
)  =  B )  ->  ( A  +Q  x )  =  B )
108106, 107breqtrd 4170 . . . 4  |-  ( ( B  e.  Q.  /\  ( A  +Q  x
)  =  B )  ->  A  <Q  B )
109108ex 424 . . 3  |-  ( B  e.  Q.  ->  (
( A  +Q  x
)  =  B  ->  A  <Q  B ) )
110109exlimdv 1643 . 2  |-  ( B  e.  Q.  ->  ( E. x ( A  +Q  x )  =  B  ->  A  <Q  B ) )
11198, 110impbid2 196 1  |-  ( B  e.  Q.  ->  ( A  <Q  B  <->  E. x
( A  +Q  x
)  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   E.wrex 2643   <.cop 3753   class class class wbr 4146    X. cxp 4809   Rel wrel 4816   ` cfv 5387  (class class class)co 6013   1stc1st 6279   2ndc2nd 6280   N.cnpi 8645    +N cpli 8646    .N cmi 8647    <N clti 8648    +pQ cplpq 8649    ~Q ceq 8652   Q.cnq 8653   /Qcerq 8655    +Q cplq 8656    <Q cltq 8659
This theorem is referenced by:  ltbtwnnq  8781  prnmadd  8800  ltexprlem4  8842  ltexprlem7  8845  prlem936  8850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-omul 6658  df-er 6834  df-ni 8675  df-pli 8676  df-mi 8677  df-lti 8678  df-plpq 8711  df-mpq 8712  df-ltpq 8713  df-enq 8714  df-nq 8715  df-erq 8716  df-plq 8717  df-mq 8718  df-1nq 8719  df-ltnq 8721
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