Step |
Hyp |
Ref |
Expression |
1 |
|
cdjreu.1 |
⊢ 𝐴 ∈ Sℋ |
2 |
|
cdjreu.2 |
⊢ 𝐵 ∈ Sℋ |
3 |
1 2
|
shseli |
⊢ ( 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) |
4 |
3
|
biimpi |
⊢ ( 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) |
5 |
|
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 ) ) ) |
6 |
|
eqtr2 |
⊢ ( ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) → ( 𝑥 +ℎ 𝑦 ) = ( 𝑧 +ℎ 𝑤 ) ) |
7 |
1
|
sheli |
⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ ) |
8 |
2
|
sheli |
⊢ ( 𝑦 ∈ 𝐵 → 𝑦 ∈ ℋ ) |
9 |
7 8
|
anim12i |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) |
10 |
1
|
sheli |
⊢ ( 𝑧 ∈ 𝐴 → 𝑧 ∈ ℋ ) |
11 |
2
|
sheli |
⊢ ( 𝑤 ∈ 𝐵 → 𝑤 ∈ ℋ ) |
12 |
10 11
|
anim12i |
⊢ ( ( 𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) → ( 𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ ) ) |
13 |
|
hvaddsub4 |
⊢ ( ( ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( 𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ ) ) → ( ( 𝑥 +ℎ 𝑦 ) = ( 𝑧 +ℎ 𝑤 ) ↔ ( 𝑥 −ℎ 𝑧 ) = ( 𝑤 −ℎ 𝑦 ) ) ) |
14 |
9 12 13
|
syl2an |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑥 +ℎ 𝑦 ) = ( 𝑧 +ℎ 𝑤 ) ↔ ( 𝑥 −ℎ 𝑧 ) = ( 𝑤 −ℎ 𝑦 ) ) ) |
15 |
14
|
an4s |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑥 +ℎ 𝑦 ) = ( 𝑧 +ℎ 𝑤 ) ↔ ( 𝑥 −ℎ 𝑧 ) = ( 𝑤 −ℎ 𝑦 ) ) ) |
16 |
15
|
adantll |
⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑥 +ℎ 𝑦 ) = ( 𝑧 +ℎ 𝑤 ) ↔ ( 𝑥 −ℎ 𝑧 ) = ( 𝑤 −ℎ 𝑦 ) ) ) |
17 |
|
shsubcl |
⊢ ( ( 𝐵 ∈ Sℋ ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑤 −ℎ 𝑦 ) ∈ 𝐵 ) |
18 |
2 17
|
mp3an1 |
⊢ ( ( 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑤 −ℎ 𝑦 ) ∈ 𝐵 ) |
19 |
18
|
ancoms |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) → ( 𝑤 −ℎ 𝑦 ) ∈ 𝐵 ) |
20 |
|
eleq1 |
⊢ ( ( 𝑥 −ℎ 𝑧 ) = ( 𝑤 −ℎ 𝑦 ) → ( ( 𝑥 −ℎ 𝑧 ) ∈ 𝐵 ↔ ( 𝑤 −ℎ 𝑦 ) ∈ 𝐵 ) ) |
21 |
19 20
|
syl5ibrcom |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) → ( ( 𝑥 −ℎ 𝑧 ) = ( 𝑤 −ℎ 𝑦 ) → ( 𝑥 −ℎ 𝑧 ) ∈ 𝐵 ) ) |
22 |
21
|
adantl |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑥 −ℎ 𝑧 ) = ( 𝑤 −ℎ 𝑦 ) → ( 𝑥 −ℎ 𝑧 ) ∈ 𝐵 ) ) |
23 |
|
shsubcl |
⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( 𝑥 −ℎ 𝑧 ) ∈ 𝐴 ) |
24 |
1 23
|
mp3an1 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( 𝑥 −ℎ 𝑧 ) ∈ 𝐴 ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑥 −ℎ 𝑧 ) ∈ 𝐴 ) |
26 |
22 25
|
jctild |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑥 −ℎ 𝑧 ) = ( 𝑤 −ℎ 𝑦 ) → ( ( 𝑥 −ℎ 𝑧 ) ∈ 𝐴 ∧ ( 𝑥 −ℎ 𝑧 ) ∈ 𝐵 ) ) ) |
27 |
26
|
adantll |
⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑥 −ℎ 𝑧 ) = ( 𝑤 −ℎ 𝑦 ) → ( ( 𝑥 −ℎ 𝑧 ) ∈ 𝐴 ∧ ( 𝑥 −ℎ 𝑧 ) ∈ 𝐵 ) ) ) |
28 |
|
elin |
⊢ ( ( 𝑥 −ℎ 𝑧 ) ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( ( 𝑥 −ℎ 𝑧 ) ∈ 𝐴 ∧ ( 𝑥 −ℎ 𝑧 ) ∈ 𝐵 ) ) |
29 |
|
eleq2 |
⊢ ( ( 𝐴 ∩ 𝐵 ) = 0ℋ → ( ( 𝑥 −ℎ 𝑧 ) ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑥 −ℎ 𝑧 ) ∈ 0ℋ ) ) |
30 |
28 29
|
bitr3id |
⊢ ( ( 𝐴 ∩ 𝐵 ) = 0ℋ → ( ( ( 𝑥 −ℎ 𝑧 ) ∈ 𝐴 ∧ ( 𝑥 −ℎ 𝑧 ) ∈ 𝐵 ) ↔ ( 𝑥 −ℎ 𝑧 ) ∈ 0ℋ ) ) |
31 |
30
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( ( 𝑥 −ℎ 𝑧 ) ∈ 𝐴 ∧ ( 𝑥 −ℎ 𝑧 ) ∈ 𝐵 ) ↔ ( 𝑥 −ℎ 𝑧 ) ∈ 0ℋ ) ) |
32 |
27 31
|
sylibd |
⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑥 −ℎ 𝑧 ) = ( 𝑤 −ℎ 𝑦 ) → ( 𝑥 −ℎ 𝑧 ) ∈ 0ℋ ) ) |
33 |
|
elch0 |
⊢ ( ( 𝑥 −ℎ 𝑧 ) ∈ 0ℋ ↔ ( 𝑥 −ℎ 𝑧 ) = 0ℎ ) |
34 |
|
hvsubeq0 |
⊢ ( ( 𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ ) → ( ( 𝑥 −ℎ 𝑧 ) = 0ℎ ↔ 𝑥 = 𝑧 ) ) |
35 |
33 34
|
syl5bb |
⊢ ( ( 𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ ) → ( ( 𝑥 −ℎ 𝑧 ) ∈ 0ℋ ↔ 𝑥 = 𝑧 ) ) |
36 |
7 10 35
|
syl2an |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑥 −ℎ 𝑧 ) ∈ 0ℋ ↔ 𝑥 = 𝑧 ) ) |
37 |
36
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑥 −ℎ 𝑧 ) ∈ 0ℋ ↔ 𝑥 = 𝑧 ) ) |
38 |
32 37
|
sylibd |
⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑥 −ℎ 𝑧 ) = ( 𝑤 −ℎ 𝑦 ) → 𝑥 = 𝑧 ) ) |
39 |
16 38
|
sylbid |
⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑥 +ℎ 𝑦 ) = ( 𝑧 +ℎ 𝑤 ) → 𝑥 = 𝑧 ) ) |
40 |
6 39
|
syl5 |
⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 = 𝑧 ) ) |
41 |
40
|
rexlimdvva |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ∃ 𝑦 ∈ 𝐵 ∃ 𝑤 ∈ 𝐵 ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 = 𝑧 ) ) |
42 |
5 41
|
syl5bir |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 = 𝑧 ) ) |
43 |
42
|
ralrimivva |
⊢ ( ( 𝐴 ∩ 𝐵 ) = 0ℋ → ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 = 𝑧 ) ) |
44 |
4 43
|
anim12i |
⊢ ( ( 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 = 𝑧 ) ) ) |
45 |
|
oveq1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 +ℎ 𝑦 ) = ( 𝑧 +ℎ 𝑦 ) ) |
46 |
45
|
eqeq2d |
⊢ ( 𝑥 = 𝑧 → ( 𝐶 = ( 𝑥 +ℎ 𝑦 ) ↔ 𝐶 = ( 𝑧 +ℎ 𝑦 ) ) ) |
47 |
46
|
rexbidv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ↔ ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑦 ) ) ) |
48 |
|
oveq2 |
⊢ ( 𝑦 = 𝑤 → ( 𝑧 +ℎ 𝑦 ) = ( 𝑧 +ℎ 𝑤 ) ) |
49 |
48
|
eqeq2d |
⊢ ( 𝑦 = 𝑤 → ( 𝐶 = ( 𝑧 +ℎ 𝑦 ) ↔ 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) |
50 |
49
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑦 ) ↔ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) |
51 |
47 50
|
syl6bb |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ↔ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) ) |
52 |
51
|
reu4 |
⊢ ( ∃! 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ↔ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ∧ ∃ 𝑤 ∈ 𝐵 𝐶 = ( 𝑧 +ℎ 𝑤 ) ) → 𝑥 = 𝑧 ) ) ) |
53 |
44 52
|
sylibr |
⊢ ( ( 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) → ∃! 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) |