Step |
Hyp |
Ref |
Expression |
1 |
|
pj1eu.a |
⊢ + = ( +g ‘ 𝐺 ) |
2 |
|
pj1eu.s |
⊢ ⊕ = ( LSSum ‘ 𝐺 ) |
3 |
|
pj1eu.o |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
|
pj1eu.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
5 |
|
pj1eu.2 |
⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) |
6 |
|
pj1eu.3 |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
7 |
|
pj1eu.4 |
⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) = { 0 } ) |
8 |
|
pj1eu.5 |
⊢ ( 𝜑 → 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) |
9 |
1 2
|
lsmelval |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ) ) |
10 |
5 6 9
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ) ) |
11 |
10
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ∃ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ) |
12 |
|
reeanv |
⊢ ( ∃ 𝑦 ∈ 𝑈 ∃ 𝑣 ∈ 𝑈 ( 𝑋 = ( 𝑥 + 𝑦 ) ∧ 𝑋 = ( 𝑢 + 𝑣 ) ) ↔ ( ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ∧ ∃ 𝑣 ∈ 𝑈 𝑋 = ( 𝑢 + 𝑣 ) ) ) |
13 |
|
eqtr2 |
⊢ ( ( 𝑋 = ( 𝑥 + 𝑦 ) ∧ 𝑋 = ( 𝑢 + 𝑣 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑢 + 𝑣 ) ) |
14 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) |
15 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
16 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → ( 𝑇 ∩ 𝑈 ) = { 0 } ) |
17 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) |
18 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → 𝑥 ∈ 𝑇 ) |
19 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → 𝑢 ∈ 𝑇 ) |
20 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → 𝑦 ∈ 𝑈 ) |
21 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → 𝑣 ∈ 𝑈 ) |
22 |
1 3 4 14 15 16 17 18 19 20 21
|
subgdisjb |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → ( ( 𝑥 + 𝑦 ) = ( 𝑢 + 𝑣 ) ↔ ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣 ) ) ) |
23 |
|
simpl |
⊢ ( ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣 ) → 𝑥 = 𝑢 ) |
24 |
22 23
|
syl6bi |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → ( ( 𝑥 + 𝑦 ) = ( 𝑢 + 𝑣 ) → 𝑥 = 𝑢 ) ) |
25 |
13 24
|
syl5 |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → ( ( 𝑋 = ( 𝑥 + 𝑦 ) ∧ 𝑋 = ( 𝑢 + 𝑣 ) ) → 𝑥 = 𝑢 ) ) |
26 |
25
|
rexlimdvva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) → ( ∃ 𝑦 ∈ 𝑈 ∃ 𝑣 ∈ 𝑈 ( 𝑋 = ( 𝑥 + 𝑦 ) ∧ 𝑋 = ( 𝑢 + 𝑣 ) ) → 𝑥 = 𝑢 ) ) |
27 |
12 26
|
syl5bir |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑢 ∈ 𝑇 ) ) → ( ( ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ∧ ∃ 𝑣 ∈ 𝑈 𝑋 = ( 𝑢 + 𝑣 ) ) → 𝑥 = 𝑢 ) ) |
28 |
27
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑇 ∀ 𝑢 ∈ 𝑇 ( ( ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ∧ ∃ 𝑣 ∈ 𝑈 𝑋 = ( 𝑢 + 𝑣 ) ) → 𝑥 = 𝑢 ) ) |
29 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ∀ 𝑥 ∈ 𝑇 ∀ 𝑢 ∈ 𝑇 ( ( ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ∧ ∃ 𝑣 ∈ 𝑈 𝑋 = ( 𝑢 + 𝑣 ) ) → 𝑥 = 𝑢 ) ) |
30 |
|
oveq1 |
⊢ ( 𝑥 = 𝑢 → ( 𝑥 + 𝑦 ) = ( 𝑢 + 𝑦 ) ) |
31 |
30
|
eqeq2d |
⊢ ( 𝑥 = 𝑢 → ( 𝑋 = ( 𝑥 + 𝑦 ) ↔ 𝑋 = ( 𝑢 + 𝑦 ) ) ) |
32 |
31
|
rexbidv |
⊢ ( 𝑥 = 𝑢 → ( ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ↔ ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑢 + 𝑦 ) ) ) |
33 |
|
oveq2 |
⊢ ( 𝑦 = 𝑣 → ( 𝑢 + 𝑦 ) = ( 𝑢 + 𝑣 ) ) |
34 |
33
|
eqeq2d |
⊢ ( 𝑦 = 𝑣 → ( 𝑋 = ( 𝑢 + 𝑦 ) ↔ 𝑋 = ( 𝑢 + 𝑣 ) ) ) |
35 |
34
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑢 + 𝑦 ) ↔ ∃ 𝑣 ∈ 𝑈 𝑋 = ( 𝑢 + 𝑣 ) ) |
36 |
32 35
|
bitrdi |
⊢ ( 𝑥 = 𝑢 → ( ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ↔ ∃ 𝑣 ∈ 𝑈 𝑋 = ( 𝑢 + 𝑣 ) ) ) |
37 |
36
|
reu4 |
⊢ ( ∃! 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ↔ ( ∃ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ∧ ∀ 𝑥 ∈ 𝑇 ∀ 𝑢 ∈ 𝑇 ( ( ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ∧ ∃ 𝑣 ∈ 𝑈 𝑋 = ( 𝑢 + 𝑣 ) ) → 𝑥 = 𝑢 ) ) ) |
38 |
11 29 37
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ∃! 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ) |