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Theorem ltexnq 8479
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
StepHypRef Expression
1 ltrelnq 8430 . . . 4  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4644 . . 3  |-  ( A 
<Q  B  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
3 ordpinq 8447 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
4 elpqn 8429 . . . . . . . . 9  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
54adantr 453 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  e.  ( N. 
X.  N. ) )
6 xp1st 6001 . . . . . . . 8  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
75, 6syl 17 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 1st `  A
)  e.  N. )
8 elpqn 8429 . . . . . . . . 9  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
98adantl 454 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  B  e.  ( N. 
X.  N. ) )
10 xp2nd 6002 . . . . . . . 8  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
119, 10syl 17 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 2nd `  B
)  e.  N. )
12 mulclpi 8397 . . . . . . 7  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
137, 11, 12syl2anc 645 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
14 xp1st 6001 . . . . . . . 8  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
159, 14syl 17 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 1st `  B
)  e.  N. )
16 xp2nd 6002 . . . . . . . 8  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
175, 16syl 17 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 2nd `  A
)  e.  N. )
18 mulclpi 8397 . . . . . . 7  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  e. 
N. )
1915, 17, 18syl2anc 645 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 1st `  B
)  .N  ( 2nd `  A ) )  e. 
N. )
20 ltexpi 8406 . . . . . 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 645 . . . . 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 4701 . . . . . . . . . . . 12  |-  Rel  ( N.  X.  N. )
234ad2antrr 709 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  A  e.  ( N.  X.  N. )
)
24 1st2nd 6018 . . . . . . . . . . . 12  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2522, 23, 24sylancr 647 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
2625oveq1d 5725 . . . . . . . . . 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 453 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 1st `  A
)  e.  N. )
2817adantr 453 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 2nd `  A
)  e.  N. )
29 simpr 449 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  y  e.  N. )
30 mulclpi 8397 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
3117, 11, 30syl2anc 645 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
3231adantr 453 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( 2nd `  A )  .N  ( 2nd `  B ) )  e.  N. )
33 addpipq 8441 . . . . . . . . . . 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 1188 . . . . . . . . . 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 2285 . . . . . . . . 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 5718 . . . . . . . . . . . 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 8402 . . . . . . . . . . . . 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 5391 . . . . . . . . . . . . . . 15  |-  ( 2nd `  A )  e.  _V
39 fvex 5391 . . . . . . . . . . . . . . 15  |-  ( 1st `  A )  e.  _V
40 fvex 5391 . . . . . . . . . . . . . . 15  |-  ( 2nd `  B )  e.  _V
41 mulcompi 8400 . . . . . . . . . . . . . . 15  |-  ( x  .N  y )  =  ( y  .N  x
)
42 mulasspi 8401 . . . . . . . . . . . . . . 15  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
4338, 39, 40, 41, 42caov12 5900 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  ( ( 1st `  A )  .N  ( 2nd `  B ) ) )  =  ( ( 1st `  A )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) )
44 mulcompi 8400 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  y )  =  ( y  .N  ( 2nd `  A ) )
4543, 44oveq12i 5722 . . . . . . . . . . . . 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 2274 . . . . . . . . . . . 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 8401 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 1st `  B
) ) )
48 mulcompi 8400 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )
4948oveq2i 5721 . . . . . . . . . . . . 13  |-  ( ( 2nd `  A )  .N  ( ( 2nd `  A )  .N  ( 1st `  B ) ) )  =  ( ( 2nd `  A )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) )
5047, 49eqtri 2273 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )
5136, 46, 503eqtr4g 2310 . . . . . . . . . . 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 8401 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) )
5352eqcomi 2257 . . . . . . . . . . . 12  |-  ( ( 2nd `  A )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  B
) )
5453a1i 12 . . . . . . . . . . 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 3704 . . . . . . . . . 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 2264 . . . . . . . . 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 213 . . . . . . . 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 5377 . . . . . . . . 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 8460 . . . . . . . . . . 11  |-  ( ( /Q `  A )  +Q  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
)  =  ( /Q
`  ( A  +pQ  <.
y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
)
60 nqerid 8437 . . . . . . . . . . . . 13  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
6160ad2antrr 709 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( /Q `  A )  =  A )
6261oveq1d 5725 . . . . . . . . . . 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 2299 . . . . . . . . . 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 8397 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  A ) )  e. 
N. )
6517, 17, 64syl2anc 645 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( 2nd `  A
)  .N  ( 2nd `  A ) )  e. 
N. )
6665adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( 2nd `  A )  .N  ( 2nd `  A ) )  e.  N. )
6715adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 1st `  B
)  e.  N. )
6811adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( 2nd `  B
)  e.  N. )
69 mulcanenq 8464 . . . . . . . . . . . . . 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 1187 . . . . . . . . . . . . 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 710 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  B  e.  ( N.  X.  N. )
)
72 1st2nd 6018 . . . . . . . . . . . . . 14  |-  ( ( Rel  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
7322, 71, 72sylancr 647 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  B  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. )
7470, 73breqtrrd 3946 . . . . . . . . . . . 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 8397 . . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 1st `  B ) )  e.  N. )
77 mulclpi 8397 . . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( ( ( 2nd `  A )  .N  ( 2nd `  A
) )  .N  ( 2nd `  B ) )  e.  N. )
79 opelxpi 4628 . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . 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 8439 . . . . . . . . . . . . 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 645 . . . . . . . . . . . 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 203 . . . . . . . . . . 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 8437 . . . . . . . . . . . 12  |-  ( B  e.  Q.  ->  ( /Q `  B )  =  B )
8584ad2antlr 710 . . . . . . . . . . 11  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  y  e.  N. )  ->  ( /Q `  B )  =  B )
8683, 85eqtrd 2285 . . . . . . . . . 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 2267 . . . . . . . . 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 212 . . . . . . . 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 5391 . . . . . . . 8  |-  ( /Q
`  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)  e.  _V
91 oveq2 5718 . . . . . . . . 9  |-  ( x  =  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )  ->  ( A  +Q  x
)  =  ( A  +Q  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
) )
9291eqeq1d 2261 . . . . . . . 8  |-  ( x  =  ( /Q `  <. y ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )  ->  ( ( A  +Q  x )  =  B  <-> 
( A  +Q  ( /Q `  <. y ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
) )  =  B ) )
9390, 92cla4ev 2812 . . . . . . 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 2629 . . . . 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 208 . . . 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 208 . . 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 2313 . . . . . . 7  |-  ( ( A  +Q  x )  =  B  ->  (
( A  +Q  x
)  e.  Q.  <->  B  e.  Q. ) )
10099biimparc 475 . . . . . 6  |-  ( ( B  e.  Q.  /\  ( A  +Q  x
)  =  B )  ->  ( A  +Q  x )  e.  Q. )
101 addnqf 8452 . . . . . . . 8  |-  +Q  :
( Q.  X.  Q. )
--> Q.
102101fdmi 5251 . . . . . . 7  |-  dom  +Q  =  ( Q.  X.  Q. )
103 0nnq 8428 . . . . . . 7  |-  -.  (/)  e.  Q.
104102, 103ndmovrcl 5858 . . . . . 6  |-  ( ( A  +Q  x )  e.  Q.  ->  ( A  e.  Q.  /\  x  e.  Q. ) )
105 ltaddnq 8478 . . . . . 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 449 . . . . 5  |-  ( ( B  e.  Q.  /\  ( A  +Q  x
)  =  B )  ->  ( A  +Q  x )  =  B )
108106, 107breqtrd 3944 . . . 4  |-  ( ( B  e.  Q.  /\  ( A  +Q  x
)  =  B )  ->  A  <Q  B )
109108ex 425 . . 3  |-  ( B  e.  Q.  ->  (
( A  +Q  x
)  =  B  ->  A  <Q  B ) )
110109exlimdv 1932 . 2  |-  ( B  e.  Q.  ->  ( E. x ( A  +Q  x )  =  B  ->  A  <Q  B ) )
11198, 110impbid2 197 1  |-  ( B  e.  Q.  ->  ( A  <Q  B  <->  E. x
( A  +Q  x
)  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   E.wrex 2510   <.cop 3547   class class class wbr 3920    X. cxp 4578   Rel wrel 4585   ` cfv 4592  (class class class)co 5710   1stc1st 5972   2ndc2nd 5973   N.cnpi 8346    +N cpli 8347    .N cmi 8348    <N clti 8349    +pQ cplpq 8350    ~Q ceq 8353   Q.cnq 8354   /Qcerq 8356    +Q cplq 8357    <Q cltq 8360
This theorem is referenced by:  ltbtwnnq  8482  prnmadd  8501  ltexprlem4  8543  ltexprlem7  8546  prlem936  8551
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-omul 6370  df-er 6546  df-ni 8376  df-pli 8377  df-mi 8378  df-lti 8379  df-plpq 8412  df-mpq 8413  df-ltpq 8414  df-enq 8415  df-nq 8416  df-erq 8417  df-plq 8418  df-mq 8419  df-1nq 8420  df-ltnq 8422
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