Step |
Hyp |
Ref |
Expression |
1 |
|
ltrelnq |
|- |
2 |
1
|
brel |
|- ( A ( A e. Q. /\ B e. Q. ) ) |
3 |
|
ordpinq |
|- ( ( A e. Q. /\ B e. Q. ) -> ( A ( ( 1st ` A ) .N ( 2nd ` B ) ) |
4 |
|
elpqn |
|- ( A e. Q. -> A e. ( N. X. N. ) ) |
5 |
4
|
adantr |
|- ( ( A e. Q. /\ B e. Q. ) -> A e. ( N. X. N. ) ) |
6 |
|
xp1st |
|- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. ) |
7 |
5 6
|
syl |
|- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` A ) e. N. ) |
8 |
|
elpqn |
|- ( B e. Q. -> B e. ( N. X. N. ) ) |
9 |
8
|
adantl |
|- ( ( A e. Q. /\ B e. Q. ) -> B e. ( N. X. N. ) ) |
10 |
|
xp2nd |
|- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. ) |
11 |
9 10
|
syl |
|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` B ) e. N. ) |
12 |
|
mulclpi |
|- ( ( ( 1st ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. ) |
13 |
7 11 12
|
syl2anc |
|- ( ( A e. Q. /\ B e. Q. ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. ) |
14 |
|
xp1st |
|- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. ) |
15 |
9 14
|
syl |
|- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` B ) e. N. ) |
16 |
|
xp2nd |
|- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. ) |
17 |
5 16
|
syl |
|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` A ) e. N. ) |
18 |
|
mulclpi |
|- ( ( ( 1st ` B ) e. N. /\ ( 2nd ` A ) e. N. ) -> ( ( 1st ` B ) .N ( 2nd ` A ) ) e. N. ) |
19 |
15 17 18
|
syl2anc |
|- ( ( A e. Q. /\ B e. Q. ) -> ( ( 1st ` B ) .N ( 2nd ` A ) ) e. N. ) |
20 |
|
ltexpi |
|- ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. /\ ( ( 1st ` B ) .N ( 2nd ` A ) ) e. N. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) E. y e. N. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) ) ) |
21 |
13 19 20
|
syl2anc |
|- ( ( A e. Q. /\ B e. Q. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) E. y e. N. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) ) ) |
22 |
|
relxp |
|- Rel ( N. X. N. ) |
23 |
4
|
ad2antrr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> A e. ( N. X. N. ) ) |
24 |
|
1st2nd |
|- ( ( Rel ( N. X. N. ) /\ A e. ( N. X. N. ) ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) |
25 |
22 23 24
|
sylancr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) |
26 |
25
|
oveq1d |
|- ( ( ( 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 ) ) >. ) ) |
27 |
7
|
adantr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( 1st ` A ) e. N. ) |
28 |
17
|
adantr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( 2nd ` A ) e. N. ) |
29 |
|
simpr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> y e. N. ) |
30 |
|
mulclpi |
|- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. ) |
31 |
17 11 30
|
syl2anc |
|- ( ( A e. Q. /\ B e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. ) |
32 |
31
|
adantr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. ) |
33 |
|
addpipq |
|- ( ( ( ( 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 ) ) ) >. ) |
34 |
27 28 29 32 33
|
syl22anc |
|- ( ( ( 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 ) ) ) >. ) |
35 |
26 34
|
eqtrd |
|- ( ( ( 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 |
|- ( ( ( ( 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 |
|- ( ( 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 |
|- ( 2nd ` A ) e. _V |
39 |
|
fvex |
|- ( 1st ` A ) e. _V |
40 |
|
fvex |
|- ( 2nd ` B ) e. _V |
41 |
|
mulcompi |
|- ( x .N y ) = ( y .N x ) |
42 |
|
mulasspi |
|- ( ( x .N y ) .N z ) = ( x .N ( y .N z ) ) |
43 |
38 39 40 41 42
|
caov12 |
|- ( ( 2nd ` A ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) = ( ( 1st ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) |
44 |
|
mulcompi |
|- ( ( 2nd ` A ) .N y ) = ( y .N ( 2nd ` A ) ) |
45 |
43 44
|
oveq12i |
|- ( ( ( 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 ) ) ) |
46 |
37 45
|
eqtr2i |
|- ( ( ( 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 |
|- ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 1st ` B ) ) ) |
48 |
|
mulcompi |
|- ( ( 2nd ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) |
49 |
48
|
oveq2i |
|- ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 1st ` B ) ) ) = ( ( 2nd ` A ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) |
50 |
47 49
|
eqtri |
|- ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) = ( ( 2nd ` A ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) |
51 |
36 46 50
|
3eqtr4g |
|- ( ( ( ( 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 |
|- ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) |
53 |
52
|
eqcomi |
|- ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) = ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) |
54 |
53
|
a1i |
|- ( ( ( ( 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 ) ) ) |
55 |
51 54
|
opeq12d |
|- ( ( ( ( 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 ) ) >. ) |
56 |
55
|
eqeq2d |
|- ( ( ( ( 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 ) ) >. ) ) |
57 |
35 56
|
syl5ibcom |
|- ( ( ( 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 |
|- ( ( 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 |
|- ( ( /Q ` A ) +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = ( /Q ` ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) |
60 |
|
nqerid |
|- ( A e. Q. -> ( /Q ` A ) = A ) |
61 |
60
|
ad2antrr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( /Q ` A ) = A ) |
62 |
61
|
oveq1d |
|- ( ( ( 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 ) ) >. ) ) ) |
63 |
59 62
|
eqtr3id |
|- ( ( ( 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 |
|- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` A ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` A ) ) e. N. ) |
65 |
17 17 64
|
syl2anc |
|- ( ( A e. Q. /\ B e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` A ) ) e. N. ) |
66 |
65
|
adantr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` A ) ) e. N. ) |
67 |
15
|
adantr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( 1st ` B ) e. N. ) |
68 |
11
|
adantr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( 2nd ` B ) e. N. ) |
69 |
|
mulcanenq |
|- ( ( ( ( 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 ) >. ) |
70 |
66 67 68 69
|
syl3anc |
|- ( ( ( 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 ) >. ) |
71 |
8
|
ad2antlr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> B e. ( N. X. N. ) ) |
72 |
|
1st2nd |
|- ( ( Rel ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. ) |
73 |
22 71 72
|
sylancr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. ) |
74 |
70 73
|
breqtrrd |
|- ( ( ( 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 |
|- ( ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) e. N. ) |
76 |
66 67 75
|
syl2anc |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) e. N. ) |
77 |
|
mulclpi |
|- ( ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) e. N. ) |
78 |
66 68 77
|
syl2anc |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) e. N. ) |
79 |
76 78
|
opelxpd |
|- ( ( ( 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. ) ) |
80 |
|
nqereq |
|- ( ( <. ( ( ( 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 ) ) ) |
81 |
79 71 80
|
syl2anc |
|- ( ( ( 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 ) ) ) |
82 |
74 81
|
mpbid |
|- ( ( ( 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 ) ) |
83 |
|
nqerid |
|- ( B e. Q. -> ( /Q ` B ) = B ) |
84 |
83
|
ad2antlr |
|- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( /Q ` B ) = B ) |
85 |
82 84
|
eqtrd |
|- ( ( ( 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 ) |
86 |
63 85
|
eqeq12d |
|- ( ( ( 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 ) ) |
87 |
58 86
|
syl5ib |
|- ( ( ( 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 ) ) |
88 |
57 87
|
syld |
|- ( ( ( 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 ) ) |
89 |
|
fvex |
|- ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) e. _V |
90 |
|
oveq2 |
|- ( x = ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) -> ( A +Q x ) = ( A +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) ) |
91 |
90
|
eqeq1d |
|- ( x = ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) -> ( ( A +Q x ) = B <-> ( A +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = B ) ) |
92 |
89 91
|
spcev |
|- ( ( A +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = B -> E. x ( A +Q x ) = B ) |
93 |
88 92
|
syl6 |
|- ( ( ( 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 ) ) |
94 |
93
|
rexlimdva |
|- ( ( 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 ) ) |
95 |
21 94
|
sylbid |
|- ( ( A e. Q. /\ B e. Q. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) E. x ( A +Q x ) = B ) ) |
96 |
3 95
|
sylbid |
|- ( ( A e. Q. /\ B e. Q. ) -> ( A E. x ( A +Q x ) = B ) ) |
97 |
2 96
|
mpcom |
|- ( A E. x ( A +Q x ) = B ) |
98 |
|
eleq1 |
|- ( ( A +Q x ) = B -> ( ( A +Q x ) e. Q. <-> B e. Q. ) ) |
99 |
98
|
biimparc |
|- ( ( B e. Q. /\ ( A +Q x ) = B ) -> ( A +Q x ) e. Q. ) |
100 |
|
addnqf |
|- +Q : ( Q. X. Q. ) --> Q. |
101 |
100
|
fdmi |
|- dom +Q = ( Q. X. Q. ) |
102 |
|
0nnq |
|- -. (/) e. Q. |
103 |
101 102
|
ndmovrcl |
|- ( ( A +Q x ) e. Q. -> ( A e. Q. /\ x e. Q. ) ) |
104 |
|
ltaddnq |
|- ( ( A e. Q. /\ x e. Q. ) -> A |
105 |
99 103 104
|
3syl |
|- ( ( B e. Q. /\ ( A +Q x ) = B ) -> A |
106 |
|
simpr |
|- ( ( B e. Q. /\ ( A +Q x ) = B ) -> ( A +Q x ) = B ) |
107 |
105 106
|
breqtrd |
|- ( ( B e. Q. /\ ( A +Q x ) = B ) -> A |
108 |
107
|
ex |
|- ( B e. Q. -> ( ( A +Q x ) = B -> A |
109 |
108
|
exlimdv |
|- ( B e. Q. -> ( E. x ( A +Q x ) = B -> A |
110 |
97 109
|
impbid2 |
|- ( B e. Q. -> ( A E. x ( A +Q x ) = B ) ) |