Step |
Hyp |
Ref |
Expression |
1 |
|
icccmp.1 |
|- J = ( topGen ` ran (,) ) |
2 |
|
icccmp.2 |
|- T = ( J |`t ( A [,] B ) ) |
3 |
|
icccmp.3 |
|- D = ( ( abs o. - ) |` ( RR X. RR ) ) |
4 |
|
icccmp.4 |
|- S = { x e. ( A [,] B ) | E. z e. ( ~P U i^i Fin ) ( A [,] x ) C_ U. z } |
5 |
|
icccmp.5 |
|- ( ph -> A e. RR ) |
6 |
|
icccmp.6 |
|- ( ph -> B e. RR ) |
7 |
|
icccmp.7 |
|- ( ph -> A <_ B ) |
8 |
|
icccmp.8 |
|- ( ph -> U C_ J ) |
9 |
|
icccmp.9 |
|- ( ph -> ( A [,] B ) C_ U. U ) |
10 |
5
|
rexrd |
|- ( ph -> A e. RR* ) |
11 |
6
|
rexrd |
|- ( ph -> B e. RR* ) |
12 |
|
lbicc2 |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) ) |
13 |
10 11 7 12
|
syl3anc |
|- ( ph -> A e. ( A [,] B ) ) |
14 |
9 13
|
sseldd |
|- ( ph -> A e. U. U ) |
15 |
|
eluni2 |
|- ( A e. U. U <-> E. u e. U A e. u ) |
16 |
14 15
|
sylib |
|- ( ph -> E. u e. U A e. u ) |
17 |
|
snssi |
|- ( u e. U -> { u } C_ U ) |
18 |
17
|
ad2antrl |
|- ( ( ph /\ ( u e. U /\ A e. u ) ) -> { u } C_ U ) |
19 |
|
snex |
|- { u } e. _V |
20 |
19
|
elpw |
|- ( { u } e. ~P U <-> { u } C_ U ) |
21 |
18 20
|
sylibr |
|- ( ( ph /\ ( u e. U /\ A e. u ) ) -> { u } e. ~P U ) |
22 |
|
snfi |
|- { u } e. Fin |
23 |
22
|
a1i |
|- ( ( ph /\ ( u e. U /\ A e. u ) ) -> { u } e. Fin ) |
24 |
21 23
|
elind |
|- ( ( ph /\ ( u e. U /\ A e. u ) ) -> { u } e. ( ~P U i^i Fin ) ) |
25 |
10
|
adantr |
|- ( ( ph /\ ( u e. U /\ A e. u ) ) -> A e. RR* ) |
26 |
|
iccid |
|- ( A e. RR* -> ( A [,] A ) = { A } ) |
27 |
25 26
|
syl |
|- ( ( ph /\ ( u e. U /\ A e. u ) ) -> ( A [,] A ) = { A } ) |
28 |
|
snssi |
|- ( A e. u -> { A } C_ u ) |
29 |
28
|
ad2antll |
|- ( ( ph /\ ( u e. U /\ A e. u ) ) -> { A } C_ u ) |
30 |
27 29
|
eqsstrd |
|- ( ( ph /\ ( u e. U /\ A e. u ) ) -> ( A [,] A ) C_ u ) |
31 |
|
unieq |
|- ( z = { u } -> U. z = U. { u } ) |
32 |
|
vex |
|- u e. _V |
33 |
32
|
unisn |
|- U. { u } = u |
34 |
31 33
|
eqtrdi |
|- ( z = { u } -> U. z = u ) |
35 |
34
|
sseq2d |
|- ( z = { u } -> ( ( A [,] A ) C_ U. z <-> ( A [,] A ) C_ u ) ) |
36 |
35
|
rspcev |
|- ( ( { u } e. ( ~P U i^i Fin ) /\ ( A [,] A ) C_ u ) -> E. z e. ( ~P U i^i Fin ) ( A [,] A ) C_ U. z ) |
37 |
24 30 36
|
syl2anc |
|- ( ( ph /\ ( u e. U /\ A e. u ) ) -> E. z e. ( ~P U i^i Fin ) ( A [,] A ) C_ U. z ) |
38 |
16 37
|
rexlimddv |
|- ( ph -> E. z e. ( ~P U i^i Fin ) ( A [,] A ) C_ U. z ) |
39 |
|
oveq2 |
|- ( x = A -> ( A [,] x ) = ( A [,] A ) ) |
40 |
39
|
sseq1d |
|- ( x = A -> ( ( A [,] x ) C_ U. z <-> ( A [,] A ) C_ U. z ) ) |
41 |
40
|
rexbidv |
|- ( x = A -> ( E. z e. ( ~P U i^i Fin ) ( A [,] x ) C_ U. z <-> E. z e. ( ~P U i^i Fin ) ( A [,] A ) C_ U. z ) ) |
42 |
41 4
|
elrab2 |
|- ( A e. S <-> ( A e. ( A [,] B ) /\ E. z e. ( ~P U i^i Fin ) ( A [,] A ) C_ U. z ) ) |
43 |
13 38 42
|
sylanbrc |
|- ( ph -> A e. S ) |
44 |
4
|
ssrab3 |
|- S C_ ( A [,] B ) |
45 |
44
|
sseli |
|- ( y e. S -> y e. ( A [,] B ) ) |
46 |
|
elicc2 |
|- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) ) |
47 |
5 6 46
|
syl2anc |
|- ( ph -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) ) |
48 |
47
|
biimpa |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( y e. RR /\ A <_ y /\ y <_ B ) ) |
49 |
48
|
simp3d |
|- ( ( ph /\ y e. ( A [,] B ) ) -> y <_ B ) |
50 |
45 49
|
sylan2 |
|- ( ( ph /\ y e. S ) -> y <_ B ) |
51 |
50
|
ralrimiva |
|- ( ph -> A. y e. S y <_ B ) |
52 |
43 51
|
jca |
|- ( ph -> ( A e. S /\ A. y e. S y <_ B ) ) |