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 |
|
pj1f.p |
⊢ 𝑃 = ( proj1 ‘ 𝐺 ) |
10 |
|
subgrcl |
⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
11 |
5 10
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
13 |
12
|
subgss |
⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 𝑇 ⊆ ( Base ‘ 𝐺 ) ) |
14 |
5 13
|
syl |
⊢ ( 𝜑 → 𝑇 ⊆ ( Base ‘ 𝐺 ) ) |
15 |
12
|
subgss |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 𝑈 ⊆ ( Base ‘ 𝐺 ) ) |
16 |
6 15
|
syl |
⊢ ( 𝜑 → 𝑈 ⊆ ( Base ‘ 𝐺 ) ) |
17 |
11 14 16
|
3jca |
⊢ ( 𝜑 → ( 𝐺 ∈ Grp ∧ 𝑇 ⊆ ( Base ‘ 𝐺 ) ∧ 𝑈 ⊆ ( Base ‘ 𝐺 ) ) ) |
18 |
12 1 2 9
|
pj1val |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑇 ⊆ ( Base ‘ 𝐺 ) ∧ 𝑈 ⊆ ( Base ‘ 𝐺 ) ) ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) = ( ℩ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ) ) |
19 |
17 18
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) = ( ℩ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ) ) |
20 |
1 2 3 4 5 6 7 8
|
pj1eu |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ∃! 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ) |
21 |
|
riotacl2 |
⊢ ( ∃! 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) → ( ℩ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ) ∈ { 𝑥 ∈ 𝑇 ∣ ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) } ) |
22 |
20 21
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ( ℩ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ) ∈ { 𝑥 ∈ 𝑇 ∣ ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) } ) |
23 |
19 22
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ∈ { 𝑥 ∈ 𝑇 ∣ ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) } ) |
24 |
|
oveq1 |
⊢ ( 𝑥 = ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) → ( 𝑥 + 𝑦 ) = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) |
25 |
24
|
eqeq2d |
⊢ ( 𝑥 = ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) → ( 𝑋 = ( 𝑥 + 𝑦 ) ↔ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) |
26 |
25
|
rexbidv |
⊢ ( 𝑥 = ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) → ( ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) ↔ ∃ 𝑦 ∈ 𝑈 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) |
27 |
26
|
elrab |
⊢ ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ∈ { 𝑥 ∈ 𝑇 ∣ ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) } ↔ ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ∈ 𝑇 ∧ ∃ 𝑦 ∈ 𝑈 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) |
28 |
27
|
simprbi |
⊢ ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ∈ { 𝑥 ∈ 𝑇 ∣ ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑥 + 𝑦 ) } → ∃ 𝑦 ∈ 𝑈 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) |
29 |
23 28
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ∃ 𝑦 ∈ 𝑈 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) |
30 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) |
31 |
11
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → 𝐺 ∈ Grp ) |
32 |
16
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → 𝑈 ⊆ ( Base ‘ 𝐺 ) ) |
33 |
14
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → 𝑇 ⊆ ( Base ‘ 𝐺 ) ) |
34 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) |
35 |
2 4
|
lsmcom2 |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) ) |
36 |
5 6 8 35
|
syl3anc |
⊢ ( 𝜑 → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) ) |
37 |
36
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) ) |
38 |
34 37
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → 𝑋 ∈ ( 𝑈 ⊕ 𝑇 ) ) |
39 |
12 1 2 9
|
pj1val |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑈 ⊆ ( Base ‘ 𝐺 ) ∧ 𝑇 ⊆ ( Base ‘ 𝐺 ) ) ∧ 𝑋 ∈ ( 𝑈 ⊕ 𝑇 ) ) → ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑋 ) = ( ℩ 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑢 + 𝑣 ) ) ) |
40 |
31 32 33 38 39
|
syl31anc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑋 ) = ( ℩ 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑢 + 𝑣 ) ) ) |
41 |
1 2 3 4 5 6 7 8 9
|
pj1f |
⊢ ( 𝜑 → ( 𝑇 𝑃 𝑈 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑇 ) |
42 |
41
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ( 𝑇 𝑃 𝑈 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑇 ) |
43 |
42 34
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ∈ 𝑇 ) |
44 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) |
45 |
44 43
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ∈ ( 𝑍 ‘ 𝑈 ) ) |
46 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → 𝑦 ∈ 𝑈 ) |
47 |
1 4
|
cntzi |
⊢ ( ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ∈ ( 𝑍 ‘ 𝑈 ) ∧ 𝑦 ∈ 𝑈 ) → ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) = ( 𝑦 + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ) ) |
48 |
45 46 47
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) = ( 𝑦 + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ) ) |
49 |
30 48
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → 𝑋 = ( 𝑦 + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ) ) |
50 |
|
oveq2 |
⊢ ( 𝑣 = ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) → ( 𝑦 + 𝑣 ) = ( 𝑦 + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ) ) |
51 |
50
|
rspceeqv |
⊢ ( ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ∈ 𝑇 ∧ 𝑋 = ( 𝑦 + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) ) ) → ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑦 + 𝑣 ) ) |
52 |
43 49 51
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑦 + 𝑣 ) ) |
53 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → 𝜑 ) |
54 |
|
incom |
⊢ ( 𝑈 ∩ 𝑇 ) = ( 𝑇 ∩ 𝑈 ) |
55 |
54 7
|
syl5eq |
⊢ ( 𝜑 → ( 𝑈 ∩ 𝑇 ) = { 0 } ) |
56 |
4 5 6 8
|
cntzrecd |
⊢ ( 𝜑 → 𝑈 ⊆ ( 𝑍 ‘ 𝑇 ) ) |
57 |
1 2 3 4 6 5 55 56
|
pj1eu |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑈 ⊕ 𝑇 ) ) → ∃! 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑢 + 𝑣 ) ) |
58 |
53 38 57
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ∃! 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑢 + 𝑣 ) ) |
59 |
|
oveq1 |
⊢ ( 𝑢 = 𝑦 → ( 𝑢 + 𝑣 ) = ( 𝑦 + 𝑣 ) ) |
60 |
59
|
eqeq2d |
⊢ ( 𝑢 = 𝑦 → ( 𝑋 = ( 𝑢 + 𝑣 ) ↔ 𝑋 = ( 𝑦 + 𝑣 ) ) ) |
61 |
60
|
rexbidv |
⊢ ( 𝑢 = 𝑦 → ( ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑢 + 𝑣 ) ↔ ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑦 + 𝑣 ) ) ) |
62 |
61
|
riota2 |
⊢ ( ( 𝑦 ∈ 𝑈 ∧ ∃! 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑢 + 𝑣 ) ) → ( ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑦 + 𝑣 ) ↔ ( ℩ 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑢 + 𝑣 ) ) = 𝑦 ) ) |
63 |
46 58 62
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ( ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑦 + 𝑣 ) ↔ ( ℩ 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑢 + 𝑣 ) ) = 𝑦 ) ) |
64 |
52 63
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ( ℩ 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑇 𝑋 = ( 𝑢 + 𝑣 ) ) = 𝑦 ) |
65 |
40 64
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑋 ) = 𝑦 ) |
66 |
65
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑋 ) ) = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) |
67 |
30 66
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + 𝑦 ) ) ) → 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑋 ) ) ) |
68 |
29 67
|
rexlimddv |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ) → 𝑋 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑋 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑋 ) ) ) |