Step |
Hyp |
Ref |
Expression |
1 |
|
addsdilem3.1 |
|
2 |
|
addsdilem3.2 |
|
3 |
|
addsdilem3.3 |
|
4 |
|
addsdilem3.4 |
Could not format ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) ) : No typesetting found for |- ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) ) with typecode |- |
5 |
|
addsdilem3.5 |
Could not format ( ph -> A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) ) : No typesetting found for |- ( ph -> A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) ) with typecode |- |
6 |
|
addsdilem3.6 |
Could not format ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) ) : No typesetting found for |- ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) ) with typecode |- |
7 |
|
addsdilem3.7 |
|
8 |
|
addsdilem3.8 |
|
9 |
|
oveq1 |
Could not format ( xO = X -> ( xO x.s ( B +s C ) ) = ( X x.s ( B +s C ) ) ) : No typesetting found for |- ( xO = X -> ( xO x.s ( B +s C ) ) = ( X x.s ( B +s C ) ) ) with typecode |- |
10 |
|
oveq1 |
Could not format ( xO = X -> ( xO x.s B ) = ( X x.s B ) ) : No typesetting found for |- ( xO = X -> ( xO x.s B ) = ( X x.s B ) ) with typecode |- |
11 |
|
oveq1 |
Could not format ( xO = X -> ( xO x.s C ) = ( X x.s C ) ) : No typesetting found for |- ( xO = X -> ( xO x.s C ) = ( X x.s C ) ) with typecode |- |
12 |
10 11
|
oveq12d |
Could not format ( xO = X -> ( ( xO x.s B ) +s ( xO x.s C ) ) = ( ( X x.s B ) +s ( X x.s C ) ) ) : No typesetting found for |- ( xO = X -> ( ( xO x.s B ) +s ( xO x.s C ) ) = ( ( X x.s B ) +s ( X x.s C ) ) ) with typecode |- |
13 |
9 12
|
eqeq12d |
Could not format ( xO = X -> ( ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) <-> ( X x.s ( B +s C ) ) = ( ( X x.s B ) +s ( X x.s C ) ) ) ) : No typesetting found for |- ( xO = X -> ( ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) <-> ( X x.s ( B +s C ) ) = ( ( X x.s B ) +s ( X x.s C ) ) ) ) with typecode |- |
14 |
4
|
adantr |
Could not format ( ( ph /\ ps ) -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) ) : No typesetting found for |- ( ( ph /\ ps ) -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) ) with typecode |- |
15 |
7
|
adantl |
|
16 |
13 14 15
|
rspcdva |
|
17 |
|
oveq1 |
Could not format ( yO = Y -> ( yO +s C ) = ( Y +s C ) ) : No typesetting found for |- ( yO = Y -> ( yO +s C ) = ( Y +s C ) ) with typecode |- |
18 |
17
|
oveq2d |
Could not format ( yO = Y -> ( A x.s ( yO +s C ) ) = ( A x.s ( Y +s C ) ) ) : No typesetting found for |- ( yO = Y -> ( A x.s ( yO +s C ) ) = ( A x.s ( Y +s C ) ) ) with typecode |- |
19 |
|
oveq2 |
Could not format ( yO = Y -> ( A x.s yO ) = ( A x.s Y ) ) : No typesetting found for |- ( yO = Y -> ( A x.s yO ) = ( A x.s Y ) ) with typecode |- |
20 |
19
|
oveq1d |
Could not format ( yO = Y -> ( ( A x.s yO ) +s ( A x.s C ) ) = ( ( A x.s Y ) +s ( A x.s C ) ) ) : No typesetting found for |- ( yO = Y -> ( ( A x.s yO ) +s ( A x.s C ) ) = ( ( A x.s Y ) +s ( A x.s C ) ) ) with typecode |- |
21 |
18 20
|
eqeq12d |
Could not format ( yO = Y -> ( ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) <-> ( A x.s ( Y +s C ) ) = ( ( A x.s Y ) +s ( A x.s C ) ) ) ) : No typesetting found for |- ( yO = Y -> ( ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) <-> ( A x.s ( Y +s C ) ) = ( ( A x.s Y ) +s ( A x.s C ) ) ) ) with typecode |- |
22 |
5
|
adantr |
Could not format ( ( ph /\ ps ) -> A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) ) : No typesetting found for |- ( ( ph /\ ps ) -> A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) ) with typecode |- |
23 |
8
|
adantl |
|
24 |
21 22 23
|
rspcdva |
|
25 |
16 24
|
oveq12d |
|
26 |
|
oveq1 |
Could not format ( xO = X -> ( xO x.s ( yO +s C ) ) = ( X x.s ( yO +s C ) ) ) : No typesetting found for |- ( xO = X -> ( xO x.s ( yO +s C ) ) = ( X x.s ( yO +s C ) ) ) with typecode |- |
27 |
|
oveq1 |
Could not format ( xO = X -> ( xO x.s yO ) = ( X x.s yO ) ) : No typesetting found for |- ( xO = X -> ( xO x.s yO ) = ( X x.s yO ) ) with typecode |- |
28 |
27 11
|
oveq12d |
Could not format ( xO = X -> ( ( xO x.s yO ) +s ( xO x.s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) ) : No typesetting found for |- ( xO = X -> ( ( xO x.s yO ) +s ( xO x.s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) ) with typecode |- |
29 |
26 28
|
eqeq12d |
Could not format ( xO = X -> ( ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) <-> ( X x.s ( yO +s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) ) ) : No typesetting found for |- ( xO = X -> ( ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) <-> ( X x.s ( yO +s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) ) ) with typecode |- |
30 |
17
|
oveq2d |
Could not format ( yO = Y -> ( X x.s ( yO +s C ) ) = ( X x.s ( Y +s C ) ) ) : No typesetting found for |- ( yO = Y -> ( X x.s ( yO +s C ) ) = ( X x.s ( Y +s C ) ) ) with typecode |- |
31 |
|
oveq2 |
Could not format ( yO = Y -> ( X x.s yO ) = ( X x.s Y ) ) : No typesetting found for |- ( yO = Y -> ( X x.s yO ) = ( X x.s Y ) ) with typecode |- |
32 |
31
|
oveq1d |
Could not format ( yO = Y -> ( ( X x.s yO ) +s ( X x.s C ) ) = ( ( X x.s Y ) +s ( X x.s C ) ) ) : No typesetting found for |- ( yO = Y -> ( ( X x.s yO ) +s ( X x.s C ) ) = ( ( X x.s Y ) +s ( X x.s C ) ) ) with typecode |- |
33 |
30 32
|
eqeq12d |
Could not format ( yO = Y -> ( ( X x.s ( yO +s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) <-> ( X x.s ( Y +s C ) ) = ( ( X x.s Y ) +s ( X x.s C ) ) ) ) : No typesetting found for |- ( yO = Y -> ( ( X x.s ( yO +s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) <-> ( X x.s ( Y +s C ) ) = ( ( X x.s Y ) +s ( X x.s C ) ) ) ) with typecode |- |
34 |
6
|
adantr |
Could not format ( ( ph /\ ps ) -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) ) : No typesetting found for |- ( ( ph /\ ps ) -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) ) with typecode |- |
35 |
29 33 34 15 23
|
rspc2dv |
|
36 |
25 35
|
oveq12d |
|
37 |
|
leftssno |
|
38 |
|
rightssno |
|
39 |
37 38
|
unssi |
|
40 |
39 7
|
sselid |
|
41 |
40
|
adantl |
|
42 |
2
|
adantr |
|
43 |
41 42
|
mulscld |
|
44 |
3
|
adantr |
|
45 |
41 44
|
mulscld |
|
46 |
|
pncans |
|
47 |
43 45 46
|
syl2anc |
|
48 |
47
|
oveq1d |
|
49 |
43 45
|
addscld |
|
50 |
1
|
adantr |
|
51 |
|
leftssno |
|
52 |
|
rightssno |
|
53 |
51 52
|
unssi |
|
54 |
53 8
|
sselid |
|
55 |
54
|
adantl |
|
56 |
50 55
|
mulscld |
|
57 |
1 3
|
mulscld |
|
58 |
57
|
adantr |
|
59 |
56 58
|
addscld |
|
60 |
49 59 45
|
addsubsd |
|
61 |
43 56 58
|
addsassd |
|
62 |
48 60 61
|
3eqtr4d |
|
63 |
62
|
oveq1d |
|
64 |
49 59
|
addscld |
|
65 |
40 54
|
mulscld |
|
66 |
65
|
adantl |
|
67 |
64 45 66
|
subsubs4d |
|
68 |
45 66
|
addscomd |
|
69 |
68
|
oveq2d |
|
70 |
67 69
|
eqtrd |
|
71 |
43 56
|
addscld |
|
72 |
71 58 66
|
addsubsd |
|
73 |
63 70 72
|
3eqtr3d |
|
74 |
36 73
|
eqtrd |
|