Step |
Hyp |
Ref |
Expression |
1 |
|
pjpjpre.1 |
|- ( ph -> H e. CH ) |
2 |
|
pjpjpre.2 |
|- ( ph -> A e. ( H +H ( _|_ ` H ) ) ) |
3 |
|
chsh |
|- ( H e. CH -> H e. SH ) |
4 |
1 3
|
syl |
|- ( ph -> H e. SH ) |
5 |
|
shocsh |
|- ( H e. SH -> ( _|_ ` H ) e. SH ) |
6 |
4 5
|
syl |
|- ( ph -> ( _|_ ` H ) e. SH ) |
7 |
|
shsel |
|- ( ( H e. SH /\ ( _|_ ` H ) e. SH ) -> ( A e. ( H +H ( _|_ ` H ) ) <-> E. x e. H E. y e. ( _|_ ` H ) A = ( x +h y ) ) ) |
8 |
4 6 7
|
syl2anc |
|- ( ph -> ( A e. ( H +H ( _|_ ` H ) ) <-> E. x e. H E. y e. ( _|_ ` H ) A = ( x +h y ) ) ) |
9 |
2 8
|
mpbid |
|- ( ph -> E. x e. H E. y e. ( _|_ ` H ) A = ( x +h y ) ) |
10 |
|
simprr |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> A = ( x +h y ) ) |
11 |
|
simprll |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> x e. H ) |
12 |
|
simprlr |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> y e. ( _|_ ` H ) ) |
13 |
|
rspe |
|- ( ( y e. ( _|_ ` H ) /\ A = ( x +h y ) ) -> E. y e. ( _|_ ` H ) A = ( x +h y ) ) |
14 |
12 10 13
|
syl2anc |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> E. y e. ( _|_ ` H ) A = ( x +h y ) ) |
15 |
|
pjpreeq |
|- ( ( H e. CH /\ A e. ( H +H ( _|_ ` H ) ) ) -> ( ( ( projh ` H ) ` A ) = x <-> ( x e. H /\ E. y e. ( _|_ ` H ) A = ( x +h y ) ) ) ) |
16 |
1 2 15
|
syl2anc |
|- ( ph -> ( ( ( projh ` H ) ` A ) = x <-> ( x e. H /\ E. y e. ( _|_ ` H ) A = ( x +h y ) ) ) ) |
17 |
16
|
adantr |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> ( ( ( projh ` H ) ` A ) = x <-> ( x e. H /\ E. y e. ( _|_ ` H ) A = ( x +h y ) ) ) ) |
18 |
11 14 17
|
mpbir2and |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> ( ( projh ` H ) ` A ) = x ) |
19 |
|
shococss |
|- ( H e. SH -> H C_ ( _|_ ` ( _|_ ` H ) ) ) |
20 |
4 19
|
syl |
|- ( ph -> H C_ ( _|_ ` ( _|_ ` H ) ) ) |
21 |
20
|
adantr |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> H C_ ( _|_ ` ( _|_ ` H ) ) ) |
22 |
21 11
|
sseldd |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> x e. ( _|_ ` ( _|_ ` H ) ) ) |
23 |
1
|
adantr |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> H e. CH ) |
24 |
23 3
|
syl |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> H e. SH ) |
25 |
|
shel |
|- ( ( H e. SH /\ x e. H ) -> x e. ~H ) |
26 |
24 11 25
|
syl2anc |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> x e. ~H ) |
27 |
24 5
|
syl |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> ( _|_ ` H ) e. SH ) |
28 |
|
shel |
|- ( ( ( _|_ ` H ) e. SH /\ y e. ( _|_ ` H ) ) -> y e. ~H ) |
29 |
27 12 28
|
syl2anc |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> y e. ~H ) |
30 |
|
ax-hvcom |
|- ( ( x e. ~H /\ y e. ~H ) -> ( x +h y ) = ( y +h x ) ) |
31 |
26 29 30
|
syl2anc |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> ( x +h y ) = ( y +h x ) ) |
32 |
10 31
|
eqtrd |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> A = ( y +h x ) ) |
33 |
|
rspe |
|- ( ( x e. ( _|_ ` ( _|_ ` H ) ) /\ A = ( y +h x ) ) -> E. x e. ( _|_ ` ( _|_ ` H ) ) A = ( y +h x ) ) |
34 |
22 32 33
|
syl2anc |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> E. x e. ( _|_ ` ( _|_ ` H ) ) A = ( y +h x ) ) |
35 |
|
choccl |
|- ( H e. CH -> ( _|_ ` H ) e. CH ) |
36 |
1 35
|
syl |
|- ( ph -> ( _|_ ` H ) e. CH ) |
37 |
|
shocsh |
|- ( ( _|_ ` H ) e. SH -> ( _|_ ` ( _|_ ` H ) ) e. SH ) |
38 |
6 37
|
syl |
|- ( ph -> ( _|_ ` ( _|_ ` H ) ) e. SH ) |
39 |
|
shless |
|- ( ( ( H e. SH /\ ( _|_ ` ( _|_ ` H ) ) e. SH /\ ( _|_ ` H ) e. SH ) /\ H C_ ( _|_ ` ( _|_ ` H ) ) ) -> ( H +H ( _|_ ` H ) ) C_ ( ( _|_ ` ( _|_ ` H ) ) +H ( _|_ ` H ) ) ) |
40 |
4 38 6 20 39
|
syl31anc |
|- ( ph -> ( H +H ( _|_ ` H ) ) C_ ( ( _|_ ` ( _|_ ` H ) ) +H ( _|_ ` H ) ) ) |
41 |
|
shscom |
|- ( ( ( _|_ ` H ) e. SH /\ ( _|_ ` ( _|_ ` H ) ) e. SH ) -> ( ( _|_ ` H ) +H ( _|_ ` ( _|_ ` H ) ) ) = ( ( _|_ ` ( _|_ ` H ) ) +H ( _|_ ` H ) ) ) |
42 |
6 38 41
|
syl2anc |
|- ( ph -> ( ( _|_ ` H ) +H ( _|_ ` ( _|_ ` H ) ) ) = ( ( _|_ ` ( _|_ ` H ) ) +H ( _|_ ` H ) ) ) |
43 |
40 42
|
sseqtrrd |
|- ( ph -> ( H +H ( _|_ ` H ) ) C_ ( ( _|_ ` H ) +H ( _|_ ` ( _|_ ` H ) ) ) ) |
44 |
43 2
|
sseldd |
|- ( ph -> A e. ( ( _|_ ` H ) +H ( _|_ ` ( _|_ ` H ) ) ) ) |
45 |
|
pjpreeq |
|- ( ( ( _|_ ` H ) e. CH /\ A e. ( ( _|_ ` H ) +H ( _|_ ` ( _|_ ` H ) ) ) ) -> ( ( ( projh ` ( _|_ ` H ) ) ` A ) = y <-> ( y e. ( _|_ ` H ) /\ E. x e. ( _|_ ` ( _|_ ` H ) ) A = ( y +h x ) ) ) ) |
46 |
36 44 45
|
syl2anc |
|- ( ph -> ( ( ( projh ` ( _|_ ` H ) ) ` A ) = y <-> ( y e. ( _|_ ` H ) /\ E. x e. ( _|_ ` ( _|_ ` H ) ) A = ( y +h x ) ) ) ) |
47 |
46
|
adantr |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> ( ( ( projh ` ( _|_ ` H ) ) ` A ) = y <-> ( y e. ( _|_ ` H ) /\ E. x e. ( _|_ ` ( _|_ ` H ) ) A = ( y +h x ) ) ) ) |
48 |
12 34 47
|
mpbir2and |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> ( ( projh ` ( _|_ ` H ) ) ` A ) = y ) |
49 |
18 48
|
oveq12d |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> ( ( ( projh ` H ) ` A ) +h ( ( projh ` ( _|_ ` H ) ) ` A ) ) = ( x +h y ) ) |
50 |
10 49
|
eqtr4d |
|- ( ( ph /\ ( ( x e. H /\ y e. ( _|_ ` H ) ) /\ A = ( x +h y ) ) ) -> A = ( ( ( projh ` H ) ` A ) +h ( ( projh ` ( _|_ ` H ) ) ` A ) ) ) |
51 |
50
|
exp32 |
|- ( ph -> ( ( x e. H /\ y e. ( _|_ ` H ) ) -> ( A = ( x +h y ) -> A = ( ( ( projh ` H ) ` A ) +h ( ( projh ` ( _|_ ` H ) ) ` A ) ) ) ) ) |
52 |
51
|
rexlimdvv |
|- ( ph -> ( E. x e. H E. y e. ( _|_ ` H ) A = ( x +h y ) -> A = ( ( ( projh ` H ) ` A ) +h ( ( projh ` ( _|_ ` H ) ) ` A ) ) ) ) |
53 |
9 52
|
mpd |
|- ( ph -> A = ( ( ( projh ` H ) ` A ) +h ( ( projh ` ( _|_ ` H ) ) ` A ) ) ) |