Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem17.a |
|- ( ph -> A e. RR ) |
2 |
|
fourierdlem17.b |
|- ( ph -> B e. RR ) |
3 |
|
fourierdlem17.altb |
|- ( ph -> A < B ) |
4 |
|
fourierdlem17.l |
|- L = ( x e. ( A (,] B ) |-> if ( x = B , A , x ) ) |
5 |
1
|
leidd |
|- ( ph -> A <_ A ) |
6 |
1 2 3
|
ltled |
|- ( ph -> A <_ B ) |
7 |
1 2 1 5 6
|
eliccd |
|- ( ph -> A e. ( A [,] B ) ) |
8 |
7
|
ad2antrr |
|- ( ( ( ph /\ x e. ( A (,] B ) ) /\ x = B ) -> A e. ( A [,] B ) ) |
9 |
|
iocssicc |
|- ( A (,] B ) C_ ( A [,] B ) |
10 |
9
|
sseli |
|- ( x e. ( A (,] B ) -> x e. ( A [,] B ) ) |
11 |
10
|
ad2antlr |
|- ( ( ( ph /\ x e. ( A (,] B ) ) /\ -. x = B ) -> x e. ( A [,] B ) ) |
12 |
8 11
|
ifclda |
|- ( ( ph /\ x e. ( A (,] B ) ) -> if ( x = B , A , x ) e. ( A [,] B ) ) |
13 |
12 4
|
fmptd |
|- ( ph -> L : ( A (,] B ) --> ( A [,] B ) ) |