Step |
Hyp |
Ref |
Expression |
1 |
|
ltrelnq |
|- |
2 |
1
|
brel |
|- ( A ( A e. Q. /\ B e. Q. ) ) |
3 |
2
|
simprd |
|- ( A B e. Q. ) |
4 |
|
ltexnq |
|- ( B e. Q. -> ( A E. y ( A +Q y ) = B ) ) |
5 |
|
eleq1 |
|- ( ( A +Q y ) = B -> ( ( A +Q y ) e. Q. <-> B e. Q. ) ) |
6 |
5
|
biimparc |
|- ( ( B e. Q. /\ ( A +Q y ) = B ) -> ( A +Q y ) e. Q. ) |
7 |
|
addnqf |
|- +Q : ( Q. X. Q. ) --> Q. |
8 |
7
|
fdmi |
|- dom +Q = ( Q. X. Q. ) |
9 |
|
0nnq |
|- -. (/) e. Q. |
10 |
8 9
|
ndmovrcl |
|- ( ( A +Q y ) e. Q. -> ( A e. Q. /\ y e. Q. ) ) |
11 |
6 10
|
syl |
|- ( ( B e. Q. /\ ( A +Q y ) = B ) -> ( A e. Q. /\ y e. Q. ) ) |
12 |
11
|
simprd |
|- ( ( B e. Q. /\ ( A +Q y ) = B ) -> y e. Q. ) |
13 |
|
nsmallnq |
|- ( y e. Q. -> E. z z |
14 |
11
|
simpld |
|- ( ( B e. Q. /\ ( A +Q y ) = B ) -> A e. Q. ) |
15 |
1
|
brel |
|- ( z ( z e. Q. /\ y e. Q. ) ) |
16 |
15
|
simpld |
|- ( z z e. Q. ) |
17 |
|
ltaddnq |
|- ( ( A e. Q. /\ z e. Q. ) -> A |
18 |
14 16 17
|
syl2an |
|- ( ( ( B e. Q. /\ ( A +Q y ) = B ) /\ z A |
19 |
|
ltanq |
|- ( A e. Q. -> ( z ( A +Q z ) |
20 |
19
|
biimpa |
|- ( ( A e. Q. /\ z ( A +Q z ) |
21 |
14 20
|
sylan |
|- ( ( ( B e. Q. /\ ( A +Q y ) = B ) /\ z ( A +Q z ) |
22 |
|
simplr |
|- ( ( ( B e. Q. /\ ( A +Q y ) = B ) /\ z ( A +Q y ) = B ) |
23 |
21 22
|
breqtrd |
|- ( ( ( B e. Q. /\ ( A +Q y ) = B ) /\ z ( A +Q z ) |
24 |
|
ovex |
|- ( A +Q z ) e. _V |
25 |
|
breq2 |
|- ( x = ( A +Q z ) -> ( A A |
26 |
|
breq1 |
|- ( x = ( A +Q z ) -> ( x ( A +Q z ) |
27 |
25 26
|
anbi12d |
|- ( x = ( A +Q z ) -> ( ( A ( A |
28 |
24 27
|
spcev |
|- ( ( A E. x ( A |
29 |
18 23 28
|
syl2anc |
|- ( ( ( B e. Q. /\ ( A +Q y ) = B ) /\ z E. x ( A |
30 |
29
|
ex |
|- ( ( B e. Q. /\ ( A +Q y ) = B ) -> ( z E. x ( A |
31 |
30
|
exlimdv |
|- ( ( B e. Q. /\ ( A +Q y ) = B ) -> ( E. z z E. x ( A |
32 |
13 31
|
syl5 |
|- ( ( B e. Q. /\ ( A +Q y ) = B ) -> ( y e. Q. -> E. x ( A |
33 |
12 32
|
mpd |
|- ( ( B e. Q. /\ ( A +Q y ) = B ) -> E. x ( A |
34 |
33
|
ex |
|- ( B e. Q. -> ( ( A +Q y ) = B -> E. x ( A |
35 |
34
|
exlimdv |
|- ( B e. Q. -> ( E. y ( A +Q y ) = B -> E. x ( A |
36 |
4 35
|
sylbid |
|- ( B e. Q. -> ( A E. x ( A |
37 |
3 36
|
mpcom |
|- ( A E. x ( A |
38 |
|
ltsonq |
|- |
39 |
38 1
|
sotri |
|- ( ( A A |
40 |
39
|
exlimiv |
|- ( E. x ( A A |
41 |
37 40
|
impbii |
|- ( A E. x ( A |