Step |
Hyp |
Ref |
Expression |
1 |
|
an4 |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ∧ ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) |
2 |
|
reeanv |
⊢ ( ∃ 𝑦 ∈ 𝐵 ∃ 𝑤 ∈ 𝐵 ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ↔ ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) |
3 |
|
simpll1 |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → 𝐴 ∈ Sℋ ) |
4 |
|
simpll2 |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → 𝐵 ∈ Sℋ ) |
5 |
|
simpll3 |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → ( 𝐴 ∩ 𝐵 ) = 0ℋ ) |
6 |
|
simplrl |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → 𝑥 ∈ 𝐴 ) |
7 |
|
simprll |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → 𝑦 ∈ 𝐵 ) |
8 |
|
simplrr |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → 𝑧 ∈ 𝐴 ) |
9 |
|
simprlr |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → 𝑤 ∈ 𝐵 ) |
10 |
|
simprrl |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) |
11 |
|
simprrr |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) |
12 |
10 11
|
eqtr3d |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → ( 𝑥 +ℎ 𝑦 ) = ( 𝑧 +ℎ 𝑤 ) ) |
13 |
3 4 5 6 7 8 9 12
|
shuni |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) |
14 |
13
|
simpld |
⊢ ( ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) → 𝑥 = 𝑧 ) |
15 |
14
|
exp32 |
⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) → ( ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 = 𝑧 ) ) ) |
16 |
15
|
rexlimdvv |
⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ∃ 𝑦 ∈ 𝐵 ∃ 𝑤 ∈ 𝐵 ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 = 𝑧 ) ) |
17 |
2 16
|
syl5bir |
⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 = 𝑧 ) ) |
18 |
17
|
expimpd |
⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ∧ ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) → 𝑥 = 𝑧 ) ) |
19 |
1 18
|
syl5bir |
⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) → ( ( ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) → 𝑥 = 𝑧 ) ) |
20 |
19
|
alrimivv |
⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) → ∀ 𝑥 ∀ 𝑧 ( ( ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) → 𝑥 = 𝑧 ) ) |
21 |
|
eleq1w |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴 ) ) |
22 |
|
oveq1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 +ℎ 𝑦 ) = ( 𝑧 +ℎ 𝑦 ) ) |
23 |
22
|
eqeq2d |
⊢ ( 𝑥 = 𝑧 → ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ↔ 𝐶 = ( 𝑧 +ℎ 𝑦 ) ) ) |
24 |
23
|
rexbidv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ↔ ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑦 ) ) ) |
25 |
|
oveq2 |
⊢ ( 𝑦 = 𝑤 → ( 𝑧 +ℎ 𝑦 ) = ( 𝑧 +ℎ 𝑤 ) ) |
26 |
25
|
eqeq2d |
⊢ ( 𝑦 = 𝑤 → ( 𝐶 = ( 𝑧 +ℎ 𝑦 ) ↔ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) |
27 |
26
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑦 ) ↔ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) |
28 |
24 27
|
bitrdi |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ↔ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) |
29 |
21 28
|
anbi12d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) ) |
30 |
29
|
mo4 |
⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) ↔ ∀ 𝑥 ∀ 𝑧 ( ( ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) → 𝑥 = 𝑧 ) ) |
31 |
20 30
|
sylibr |
⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) → ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) ) |