Step |
Hyp |
Ref |
Expression |
1 |
|
diblsmopel.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
diblsmopel.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
diblsmopel.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
diblsmopel.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
diblsmopel.o |
⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
6 |
|
diblsmopel.v |
⊢ 𝑉 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
diblsmopel.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
diblsmopel.q |
⊢ ⊕ = ( LSSum ‘ 𝑉 ) |
9 |
|
diblsmopel.p |
⊢ ✚ = ( LSSum ‘ 𝑈 ) |
10 |
|
diblsmopel.j |
⊢ 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
diblsmopel.i |
⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
|
diblsmopel.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
13 |
|
diblsmopel.x |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) |
14 |
|
diblsmopel.y |
⊢ ( 𝜑 → ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) |
15 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
16 |
1 2 3 7 11 15
|
diblss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
17 |
12 13 16
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
18 |
1 2 3 7 11 15
|
diblss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
19 |
12 14 18
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
20 |
|
eqid |
⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 ) |
21 |
3 7 20 15 9
|
dvhopellsm |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐼 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( ( 𝐼 ‘ 𝑋 ) ✚ ( 𝐼 ‘ 𝑌 ) ) ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ) |
22 |
12 17 19 21
|
syl3anc |
⊢ ( 𝜑 → ( 〈 𝐹 , 𝑆 〉 ∈ ( ( 𝐼 ‘ 𝑋 ) ✚ ( 𝐼 ‘ 𝑌 ) ) ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ) |
23 |
|
excom |
⊢ ( ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) |
24 |
1 2 3 4 5 10 11
|
dibopelval2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑦 = 𝑂 ) ) ) |
25 |
12 13 24
|
syl2anc |
⊢ ( 𝜑 → ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑦 = 𝑂 ) ) ) |
26 |
1 2 3 4 5 10 11
|
dibopelval2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ↔ ( 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ∧ 𝑤 = 𝑂 ) ) ) |
27 |
12 14 26
|
syl2anc |
⊢ ( 𝜑 → ( 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ↔ ( 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ∧ 𝑤 = 𝑂 ) ) ) |
28 |
25 27
|
anbi12d |
⊢ ( 𝜑 → ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ↔ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑦 = 𝑂 ) ∧ ( 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ∧ 𝑤 = 𝑂 ) ) ) ) |
29 |
|
an4 |
⊢ ( ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑦 = 𝑂 ) ∧ ( 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ∧ 𝑤 = 𝑂 ) ) ↔ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ) ) ) |
30 |
|
ancom |
⊢ ( ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ) ) ↔ ( ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ) ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) ) |
31 |
29 30
|
bitri |
⊢ ( ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑦 = 𝑂 ) ∧ ( 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ∧ 𝑤 = 𝑂 ) ) ↔ ( ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ) ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) ) |
32 |
28 31
|
bitrdi |
⊢ ( 𝜑 → ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ↔ ( ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ) ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) ) ) |
33 |
32
|
anbi1d |
⊢ ( 𝜑 → ( ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ( ( ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ) ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ) |
34 |
|
anass |
⊢ ( ( ( ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ) ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ( ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ) ∧ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ) |
35 |
|
df-3an |
⊢ ( ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ↔ ( ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ) ∧ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ) |
36 |
34 35
|
bitr4i |
⊢ ( ( ( ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ) ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ) |
37 |
33 36
|
bitrdi |
⊢ ( 𝜑 → ( ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ) ) |
38 |
37
|
2exbidv |
⊢ ( 𝜑 → ( ∃ 𝑦 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ∃ 𝑦 ∃ 𝑤 ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ) ) |
39 |
4
|
fvexi |
⊢ 𝑇 ∈ V |
40 |
39
|
mptex |
⊢ ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) ∈ V |
41 |
5 40
|
eqeltri |
⊢ 𝑂 ∈ V |
42 |
|
opeq2 |
⊢ ( 𝑦 = 𝑂 → 〈 𝑥 , 𝑦 〉 = 〈 𝑥 , 𝑂 〉 ) |
43 |
42
|
oveq1d |
⊢ ( 𝑦 = 𝑂 → ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) |
44 |
43
|
eqeq2d |
⊢ ( 𝑦 = 𝑂 → ( 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ↔ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) |
45 |
44
|
anbi2d |
⊢ ( 𝑦 = 𝑂 → ( ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ) |
46 |
|
opeq2 |
⊢ ( 𝑤 = 𝑂 → 〈 𝑧 , 𝑤 〉 = 〈 𝑧 , 𝑂 〉 ) |
47 |
46
|
oveq2d |
⊢ ( 𝑤 = 𝑂 → ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑂 〉 ) ) |
48 |
47
|
eqeq2d |
⊢ ( 𝑤 = 𝑂 → ( 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ↔ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑂 〉 ) ) ) |
49 |
48
|
anbi2d |
⊢ ( 𝑤 = 𝑂 → ( ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑂 〉 ) ) ) ) |
50 |
41 41 45 49
|
ceqsex2v |
⊢ ( ∃ 𝑦 ∃ 𝑤 ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ↔ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑂 〉 ) ) ) |
51 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
52 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) |
53 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ) |
54 |
1 2 3 4 10
|
diael |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ) → 𝑥 ∈ 𝑇 ) |
55 |
51 52 53 54
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → 𝑥 ∈ 𝑇 ) |
56 |
|
eqid |
⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
57 |
1 3 4 56 5
|
tendo0cl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
58 |
51 57
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → 𝑂 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
59 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) |
60 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) |
61 |
1 2 3 4 10
|
diael |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) → 𝑧 ∈ 𝑇 ) |
62 |
51 59 60 61
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → 𝑧 ∈ 𝑇 ) |
63 |
|
eqid |
⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 ) |
64 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( +g ‘ ( Scalar ‘ 𝑈 ) ) |
65 |
3 4 56 7 63 20 64
|
dvhopvadd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑂 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ ( 𝑧 ∈ 𝑇 ∧ 𝑂 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑂 〉 ) = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑂 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) 〉 ) |
66 |
51 55 58 62 58 65
|
syl122anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑂 〉 ) = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑂 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) 〉 ) |
67 |
66
|
eqeq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑂 〉 ) ↔ 〈 𝐹 , 𝑆 〉 = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑂 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) 〉 ) ) |
68 |
|
vex |
⊢ 𝑥 ∈ V |
69 |
|
vex |
⊢ 𝑧 ∈ V |
70 |
68 69
|
coex |
⊢ ( 𝑥 ∘ 𝑧 ) ∈ V |
71 |
|
ovex |
⊢ ( 𝑂 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) ∈ V |
72 |
70 71
|
opth2 |
⊢ ( 〈 𝐹 , 𝑆 〉 = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑂 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) 〉 ↔ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = ( 𝑂 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) ) ) |
73 |
|
eqid |
⊢ ( +g ‘ 𝑉 ) = ( +g ‘ 𝑉 ) |
74 |
3 4 6 73
|
dvavadd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇 ) ) → ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) = ( 𝑥 ∘ 𝑧 ) ) |
75 |
51 55 62 74
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) = ( 𝑥 ∘ 𝑧 ) ) |
76 |
75
|
eqeq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ↔ 𝐹 = ( 𝑥 ∘ 𝑧 ) ) ) |
77 |
76
|
bicomd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ↔ 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ) |
78 |
|
eqid |
⊢ ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) = ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) |
79 |
3 4 56 7 63 78 64
|
dvhfplusr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ) |
80 |
51 79
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ) |
81 |
80
|
oveqd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 𝑂 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) = ( 𝑂 ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 𝑂 ) ) |
82 |
1 3 4 56 5 78
|
tendo0pl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑂 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) → ( 𝑂 ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 𝑂 ) = 𝑂 ) |
83 |
51 58 82
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 𝑂 ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 𝑂 ) = 𝑂 ) |
84 |
81 83
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 𝑂 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) = 𝑂 ) |
85 |
84
|
eqeq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 𝑆 = ( 𝑂 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) ↔ 𝑆 = 𝑂 ) ) |
86 |
77 85
|
anbi12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = ( 𝑂 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) ) ↔ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ) |
87 |
72 86
|
syl5bb |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 〈 𝐹 , 𝑆 〉 = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑂 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) 〉 ↔ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ) |
88 |
67 87
|
bitrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ) → ( 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑂 〉 ) ↔ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ) |
89 |
88
|
pm5.32da |
⊢ ( 𝜑 → ( ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑂 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑂 〉 ) ) ↔ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ) ) |
90 |
50 89
|
syl5bb |
⊢ ( 𝜑 → ( ∃ 𝑦 ∃ 𝑤 ( 𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ) ↔ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ) ) |
91 |
38 90
|
bitrd |
⊢ ( 𝜑 → ( ∃ 𝑦 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ) ) |
92 |
91
|
exbidv |
⊢ ( 𝜑 → ( ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ) ) |
93 |
23 92
|
syl5bb |
⊢ ( 𝜑 → ( ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ) ) |
94 |
93
|
exbidv |
⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐼 ‘ 𝑋 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝐼 ‘ 𝑌 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 ( +g ‘ 𝑈 ) 〈 𝑧 , 𝑤 〉 ) ) ↔ ∃ 𝑥 ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ) ) |
95 |
|
anass |
⊢ ( ( ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ∧ 𝑆 = 𝑂 ) ↔ ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ) |
96 |
95
|
bicomi |
⊢ ( ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ↔ ( ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ∧ 𝑆 = 𝑂 ) ) |
97 |
96
|
2exbii |
⊢ ( ∃ 𝑥 ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ↔ ∃ 𝑥 ∃ 𝑧 ( ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ∧ 𝑆 = 𝑂 ) ) |
98 |
|
19.41vv |
⊢ ( ∃ 𝑥 ∃ 𝑧 ( ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ∧ 𝑆 = 𝑂 ) ↔ ( ∃ 𝑥 ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ∧ 𝑆 = 𝑂 ) ) |
99 |
97 98
|
bitri |
⊢ ( ∃ 𝑥 ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ↔ ( ∃ 𝑥 ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ∧ 𝑆 = 𝑂 ) ) |
100 |
3 6
|
dvalvec |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑉 ∈ LVec ) |
101 |
|
lveclmod |
⊢ ( 𝑉 ∈ LVec → 𝑉 ∈ LMod ) |
102 |
|
eqid |
⊢ ( LSubSp ‘ 𝑉 ) = ( LSubSp ‘ 𝑉 ) |
103 |
102
|
lsssssubg |
⊢ ( 𝑉 ∈ LMod → ( LSubSp ‘ 𝑉 ) ⊆ ( SubGrp ‘ 𝑉 ) ) |
104 |
12 100 101 103
|
4syl |
⊢ ( 𝜑 → ( LSubSp ‘ 𝑉 ) ⊆ ( SubGrp ‘ 𝑉 ) ) |
105 |
1 2 3 6 10 102
|
dialss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑉 ) ) |
106 |
12 13 105
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑉 ) ) |
107 |
104 106
|
sseldd |
⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝑉 ) ) |
108 |
1 2 3 6 10 102
|
dialss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑉 ) ) |
109 |
12 14 108
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑉 ) ) |
110 |
104 109
|
sseldd |
⊢ ( 𝜑 → ( 𝐽 ‘ 𝑌 ) ∈ ( SubGrp ‘ 𝑉 ) ) |
111 |
73 8
|
lsmelval |
⊢ ( ( ( 𝐽 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝑉 ) ∧ ( 𝐽 ‘ 𝑌 ) ∈ ( SubGrp ‘ 𝑉 ) ) → ( 𝐹 ∈ ( ( 𝐽 ‘ 𝑋 ) ⊕ ( 𝐽 ‘ 𝑌 ) ) ↔ ∃ 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∃ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ) |
112 |
107 110 111
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ∈ ( ( 𝐽 ‘ 𝑋 ) ⊕ ( 𝐽 ‘ 𝑌 ) ) ↔ ∃ 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∃ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ) |
113 |
|
r2ex |
⊢ ( ∃ 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∃ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ↔ ∃ 𝑥 ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ) |
114 |
112 113
|
bitrdi |
⊢ ( 𝜑 → ( 𝐹 ∈ ( ( 𝐽 ‘ 𝑋 ) ⊕ ( 𝐽 ‘ 𝑌 ) ) ↔ ∃ 𝑥 ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ) ) |
115 |
114
|
anbi1d |
⊢ ( 𝜑 → ( ( 𝐹 ∈ ( ( 𝐽 ‘ 𝑋 ) ⊕ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝑆 = 𝑂 ) ↔ ( ∃ 𝑥 ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ∧ 𝑆 = 𝑂 ) ) ) |
116 |
115
|
bicomd |
⊢ ( 𝜑 → ( ( ∃ 𝑥 ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ) ∧ 𝑆 = 𝑂 ) ↔ ( 𝐹 ∈ ( ( 𝐽 ‘ 𝑋 ) ⊕ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝑆 = 𝑂 ) ) ) |
117 |
99 116
|
syl5bb |
⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑧 ( ( 𝑥 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝐽 ‘ 𝑌 ) ) ∧ ( 𝐹 = ( 𝑥 ( +g ‘ 𝑉 ) 𝑧 ) ∧ 𝑆 = 𝑂 ) ) ↔ ( 𝐹 ∈ ( ( 𝐽 ‘ 𝑋 ) ⊕ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝑆 = 𝑂 ) ) ) |
118 |
22 94 117
|
3bitrd |
⊢ ( 𝜑 → ( 〈 𝐹 , 𝑆 〉 ∈ ( ( 𝐼 ‘ 𝑋 ) ✚ ( 𝐼 ‘ 𝑌 ) ) ↔ ( 𝐹 ∈ ( ( 𝐽 ‘ 𝑋 ) ⊕ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝑆 = 𝑂 ) ) ) |