Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme26.b |
|- B = ( Base ` K ) |
2 |
|
cdleme26.l |
|- .<_ = ( le ` K ) |
3 |
|
cdleme26.j |
|- .\/ = ( join ` K ) |
4 |
|
cdleme26.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdleme26.a |
|- A = ( Atoms ` K ) |
6 |
|
cdleme26.h |
|- H = ( LHyp ` K ) |
7 |
|
cdleme27.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
cdleme27.f |
|- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) |
9 |
|
cdleme27.z |
|- Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) ) |
10 |
|
cdleme27.n |
|- N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) ) |
11 |
|
cdleme27.d |
|- D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) ) |
12 |
|
cdleme27.c |
|- C = if ( s .<_ ( P .\/ Q ) , D , F ) |
13 |
|
cdleme27.g |
|- G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) |
14 |
|
cdleme27.o |
|- O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) |
15 |
|
cdleme27.e |
|- E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) ) |
16 |
|
cdleme27.y |
|- Y = if ( t .<_ ( P .\/ Q ) , E , G ) |
17 |
|
simp211 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> ( K e. HL /\ W e. H ) ) |
18 |
|
simp221 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> ( P e. A /\ -. P .<_ W ) ) |
19 |
|
simp222 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
20 |
|
simp213 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> ( s e. A /\ -. s .<_ W ) ) |
21 |
|
simp223 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> ( t e. A /\ -. t .<_ W ) ) |
22 |
|
simp23r |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> ( V e. A /\ V .<_ W ) ) |
23 |
|
simp212 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> P =/= Q ) |
24 |
|
simp1l |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> s .<_ ( P .\/ Q ) ) |
25 |
|
simp1r |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> t .<_ ( P .\/ Q ) ) |
26 |
23 24 25
|
3jca |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> ( P =/= Q /\ s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) ) |
27 |
|
simp3 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> ( t .\/ V ) = ( P .\/ Q ) ) |
28 |
1 2 3 4 5 6 7 9 10 14 11 15
|
cdleme26ee |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ ( ( P =/= Q /\ s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) ) -> D .<_ ( E .\/ V ) ) |
29 |
17 18 19 20 21 22 26 27 28
|
syl332anc |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) = ( P .\/ Q ) ) -> D .<_ ( E .\/ V ) ) |
30 |
29
|
3expia |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( ( t .\/ V ) = ( P .\/ Q ) -> D .<_ ( E .\/ V ) ) ) |
31 |
|
simp1r |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> t .<_ ( P .\/ Q ) ) |
32 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> K e. HL ) |
33 |
32
|
3ad2ant2 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> K e. HL ) |
34 |
|
simp13l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> s e. A ) |
35 |
34
|
3ad2ant2 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> s e. A ) |
36 |
|
simp23l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> t e. A ) |
37 |
36
|
3ad2ant2 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> t e. A ) |
38 |
|
simp3ll |
|- ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> s =/= t ) |
39 |
38
|
3ad2ant2 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> s =/= t ) |
40 |
35 37 39
|
3jca |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> ( s e. A /\ t e. A /\ s =/= t ) ) |
41 |
|
simp21l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> P e. A ) |
42 |
41
|
3ad2ant2 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> P e. A ) |
43 |
|
simp22l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> Q e. A ) |
44 |
43
|
3ad2ant2 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> Q e. A ) |
45 |
|
simp212 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> P =/= Q ) |
46 |
|
simp3rl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> V e. A ) |
47 |
46
|
3ad2ant2 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> V e. A ) |
48 |
|
simp3 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> ( t .\/ V ) =/= ( P .\/ Q ) ) |
49 |
|
simp3lr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> s .<_ ( t .\/ V ) ) |
50 |
49
|
3ad2ant2 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> s .<_ ( t .\/ V ) ) |
51 |
|
simp1l |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> s .<_ ( P .\/ Q ) ) |
52 |
48 50 51
|
3jca |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> ( ( t .\/ V ) =/= ( P .\/ Q ) /\ s .<_ ( t .\/ V ) /\ s .<_ ( P .\/ Q ) ) ) |
53 |
2 3 4 5 6
|
cdleme22b |
|- ( ( ( K e. HL /\ ( s e. A /\ t e. A /\ s =/= t ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( V e. A /\ ( ( t .\/ V ) =/= ( P .\/ Q ) /\ s .<_ ( t .\/ V ) /\ s .<_ ( P .\/ Q ) ) ) ) -> -. t .<_ ( P .\/ Q ) ) |
54 |
33 40 42 44 45 47 52 53
|
syl232anc |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> -. t .<_ ( P .\/ Q ) ) |
55 |
31 54
|
pm2.21dd |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) /\ ( t .\/ V ) =/= ( P .\/ Q ) ) -> D .<_ ( E .\/ V ) ) |
56 |
55
|
3expia |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( ( t .\/ V ) =/= ( P .\/ Q ) -> D .<_ ( E .\/ V ) ) ) |
57 |
30 56
|
pm2.61dne |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> D .<_ ( E .\/ V ) ) |
58 |
|
iftrue |
|- ( s .<_ ( P .\/ Q ) -> if ( s .<_ ( P .\/ Q ) , D , F ) = D ) |
59 |
12 58
|
syl5eq |
|- ( s .<_ ( P .\/ Q ) -> C = D ) |
60 |
59
|
ad2antrr |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> C = D ) |
61 |
|
iftrue |
|- ( t .<_ ( P .\/ Q ) -> if ( t .<_ ( P .\/ Q ) , E , G ) = E ) |
62 |
16 61
|
syl5eq |
|- ( t .<_ ( P .\/ Q ) -> Y = E ) |
63 |
62
|
oveq1d |
|- ( t .<_ ( P .\/ Q ) -> ( Y .\/ V ) = ( E .\/ V ) ) |
64 |
63
|
ad2antlr |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( Y .\/ V ) = ( E .\/ V ) ) |
65 |
57 60 64
|
3brtr4d |
|- ( ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> C .<_ ( Y .\/ V ) ) |
66 |
65
|
ex |
|- ( ( s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) -> ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> C .<_ ( Y .\/ V ) ) ) |
67 |
|
simpr11 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
68 |
|
simpr12 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> P =/= Q ) |
69 |
|
simpll |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> s .<_ ( P .\/ Q ) ) |
70 |
68 69
|
jca |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( P =/= Q /\ s .<_ ( P .\/ Q ) ) ) |
71 |
|
simpr23 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( t e. A /\ -. t .<_ W ) ) |
72 |
|
simpr21 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
73 |
|
simpr22 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
74 |
|
simpr13 |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( s e. A /\ -. s .<_ W ) ) |
75 |
|
simplr |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> -. t .<_ ( P .\/ Q ) ) |
76 |
|
simpr3l |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( s =/= t /\ s .<_ ( t .\/ V ) ) ) |
77 |
|
simpr3r |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( V e. A /\ V .<_ W ) ) |
78 |
|
eqid |
|- ( ( P .\/ Q ) ./\ ( G .\/ ( ( s .\/ t ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( G .\/ ( ( s .\/ t ) ./\ W ) ) ) |
79 |
|
eqid |
|- ( iota_ u e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( G .\/ ( ( s .\/ t ) ./\ W ) ) ) ) ) = ( iota_ u e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( G .\/ ( ( s .\/ t ) ./\ W ) ) ) ) ) |
80 |
9 10 13 78 11 79
|
cdleme25cv |
|- D = ( iota_ u e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( G .\/ ( ( s .\/ t ) ./\ W ) ) ) ) ) |
81 |
1 2 3 4 5 6 7 13 78 80
|
cdleme26f |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ s .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( -. t .<_ ( P .\/ Q ) /\ ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> D .<_ ( G .\/ V ) ) |
82 |
67 70 71 72 73 74 75 76 77 81
|
syl333anc |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> D .<_ ( G .\/ V ) ) |
83 |
59
|
ad2antrr |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> C = D ) |
84 |
|
iffalse |
|- ( -. t .<_ ( P .\/ Q ) -> if ( t .<_ ( P .\/ Q ) , E , G ) = G ) |
85 |
16 84
|
syl5eq |
|- ( -. t .<_ ( P .\/ Q ) -> Y = G ) |
86 |
85
|
oveq1d |
|- ( -. t .<_ ( P .\/ Q ) -> ( Y .\/ V ) = ( G .\/ V ) ) |
87 |
86
|
ad2antlr |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( Y .\/ V ) = ( G .\/ V ) ) |
88 |
82 83 87
|
3brtr4d |
|- ( ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> C .<_ ( Y .\/ V ) ) |
89 |
88
|
ex |
|- ( ( s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) -> ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> C .<_ ( Y .\/ V ) ) ) |
90 |
|
simpr11 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
91 |
|
simpr12 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> P =/= Q ) |
92 |
|
simplr |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> t .<_ ( P .\/ Q ) ) |
93 |
91 92
|
jca |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( P =/= Q /\ t .<_ ( P .\/ Q ) ) ) |
94 |
|
simpr13 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( s e. A /\ -. s .<_ W ) ) |
95 |
|
simpr21 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
96 |
|
simpr22 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
97 |
|
simpr23 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( t e. A /\ -. t .<_ W ) ) |
98 |
|
simpll |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> -. s .<_ ( P .\/ Q ) ) |
99 |
|
simpr3l |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( s =/= t /\ s .<_ ( t .\/ V ) ) ) |
100 |
|
simpr3r |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( V e. A /\ V .<_ W ) ) |
101 |
|
eqid |
|- ( ( P .\/ Q ) ./\ ( F .\/ ( ( t .\/ s ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( F .\/ ( ( t .\/ s ) ./\ W ) ) ) |
102 |
|
eqid |
|- ( iota_ u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( F .\/ ( ( t .\/ s ) ./\ W ) ) ) ) ) = ( iota_ u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( F .\/ ( ( t .\/ s ) ./\ W ) ) ) ) ) |
103 |
9 14 8 101 15 102
|
cdleme25cv |
|- E = ( iota_ u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( F .\/ ( ( t .\/ s ) ./\ W ) ) ) ) ) |
104 |
1 2 3 4 5 6 7 8 101 103
|
cdleme26f2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ t .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> F .<_ ( E .\/ V ) ) |
105 |
90 93 94 95 96 97 98 99 100 104
|
syl333anc |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> F .<_ ( E .\/ V ) ) |
106 |
|
iffalse |
|- ( -. s .<_ ( P .\/ Q ) -> if ( s .<_ ( P .\/ Q ) , D , F ) = F ) |
107 |
12 106
|
syl5eq |
|- ( -. s .<_ ( P .\/ Q ) -> C = F ) |
108 |
107
|
ad2antrr |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> C = F ) |
109 |
63
|
ad2antlr |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( Y .\/ V ) = ( E .\/ V ) ) |
110 |
105 108 109
|
3brtr4d |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> C .<_ ( Y .\/ V ) ) |
111 |
110
|
ex |
|- ( ( -. s .<_ ( P .\/ Q ) /\ t .<_ ( P .\/ Q ) ) -> ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> C .<_ ( Y .\/ V ) ) ) |
112 |
|
simpr11 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
113 |
|
simpr23 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( t e. A /\ -. t .<_ W ) ) |
114 |
|
simplr |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> -. t .<_ ( P .\/ Q ) ) |
115 |
|
simpll |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> -. s .<_ ( P .\/ Q ) ) |
116 |
|
simpr12 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> P =/= Q ) |
117 |
114 115 116
|
3jca |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( -. t .<_ ( P .\/ Q ) /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) |
118 |
|
simpr21 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
119 |
|
simpr22 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
120 |
|
simpr13 |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( s e. A /\ -. s .<_ W ) ) |
121 |
|
simpr3l |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( s =/= t /\ s .<_ ( t .\/ V ) ) ) |
122 |
|
simpr3r |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( V e. A /\ V .<_ W ) ) |
123 |
2 3 4 5 6 7 8 13
|
cdleme22g |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( t e. A /\ -. t .<_ W ) /\ ( -. t .<_ ( P .\/ Q ) /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> F .<_ ( G .\/ V ) ) |
124 |
112 113 117 118 119 120 121 122 123
|
syl323anc |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> F .<_ ( G .\/ V ) ) |
125 |
107
|
ad2antrr |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> C = F ) |
126 |
86
|
ad2antlr |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> ( Y .\/ V ) = ( G .\/ V ) ) |
127 |
124 125 126
|
3brtr4d |
|- ( ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) ) -> C .<_ ( Y .\/ V ) ) |
128 |
127
|
ex |
|- ( ( -. s .<_ ( P .\/ Q ) /\ -. t .<_ ( P .\/ Q ) ) -> ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> C .<_ ( Y .\/ V ) ) ) |
129 |
66 89 111 128
|
4cases |
|- ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> C .<_ ( Y .\/ V ) ) |