Step |
Hyp |
Ref |
Expression |
1 |
|
lcfl6.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
lcfl6.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
lcfl6.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
lcfl6.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
lcfl6.a |
⊢ + = ( +g ‘ 𝑈 ) |
6 |
|
lcfl6.t |
⊢ · = ( ·𝑠 ‘ 𝑈 ) |
7 |
|
lcfl6.s |
⊢ 𝑆 = ( Scalar ‘ 𝑈 ) |
8 |
|
lcfl6.r |
⊢ 𝑅 = ( Base ‘ 𝑆 ) |
9 |
|
lcfl6.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
10 |
|
lcfl6.f |
⊢ 𝐹 = ( LFnl ‘ 𝑈 ) |
11 |
|
lcfl6.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
12 |
|
lcfl6.c |
⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } |
13 |
|
lcfl6.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
14 |
|
lcfl6.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
15 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
lcfl6 |
⊢ ( 𝜑 → ( 𝐺 ∈ 𝐶 ↔ ( ( 𝐿 ‘ 𝐺 ) = 𝑉 ∨ ∃ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ) ) ) |
16 |
13
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∧ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ) ) ∧ ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
17 |
|
eqid |
⊢ ( 𝑢 ∈ 𝑉 ↦ ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑥 ) ) ) ) = ( 𝑢 ∈ 𝑉 ↦ ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑥 ) ) ) ) |
18 |
|
eqid |
⊢ ( 𝑢 ∈ 𝑉 ↦ ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑦 ) ) ) ) = ( 𝑢 ∈ 𝑉 ↦ ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑦 ) ) ) ) |
19 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∧ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ) ) ∧ ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) ) → 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) |
20 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∧ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ) ) ∧ ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) ) → 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ) |
21 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∧ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ) ) ∧ ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) ) → 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ) |
22 |
|
eqeq1 |
⊢ ( 𝑣 = 𝑢 → ( 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ↔ 𝑢 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) |
23 |
22
|
rexbidv |
⊢ ( 𝑣 = 𝑢 → ( ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ↔ ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) |
24 |
23
|
riotabidv |
⊢ ( 𝑣 = 𝑢 → ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) = ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) |
25 |
|
oveq1 |
⊢ ( 𝑘 = 𝑙 → ( 𝑘 · 𝑥 ) = ( 𝑙 · 𝑥 ) ) |
26 |
25
|
oveq2d |
⊢ ( 𝑘 = 𝑙 → ( 𝑤 + ( 𝑘 · 𝑥 ) ) = ( 𝑤 + ( 𝑙 · 𝑥 ) ) ) |
27 |
26
|
eqeq2d |
⊢ ( 𝑘 = 𝑙 → ( 𝑢 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ↔ 𝑢 = ( 𝑤 + ( 𝑙 · 𝑥 ) ) ) ) |
28 |
27
|
rexbidv |
⊢ ( 𝑘 = 𝑙 → ( ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ↔ ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑤 + ( 𝑙 · 𝑥 ) ) ) ) |
29 |
|
oveq1 |
⊢ ( 𝑤 = 𝑧 → ( 𝑤 + ( 𝑙 · 𝑥 ) ) = ( 𝑧 + ( 𝑙 · 𝑥 ) ) ) |
30 |
29
|
eqeq2d |
⊢ ( 𝑤 = 𝑧 → ( 𝑢 = ( 𝑤 + ( 𝑙 · 𝑥 ) ) ↔ 𝑢 = ( 𝑧 + ( 𝑙 · 𝑥 ) ) ) ) |
31 |
30
|
cbvrexvw |
⊢ ( ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑤 + ( 𝑙 · 𝑥 ) ) ↔ ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑥 ) ) ) |
32 |
28 31
|
bitrdi |
⊢ ( 𝑘 = 𝑙 → ( ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ↔ ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑥 ) ) ) ) |
33 |
32
|
cbvriotavw |
⊢ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) = ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑥 ) ) ) |
34 |
24 33
|
eqtrdi |
⊢ ( 𝑣 = 𝑢 → ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) = ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑥 ) ) ) ) |
35 |
34
|
cbvmptv |
⊢ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) = ( 𝑢 ∈ 𝑉 ↦ ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑥 ) ) ) ) |
36 |
21 35
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∧ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ) ) ∧ ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) ) → 𝐺 = ( 𝑢 ∈ 𝑉 ↦ ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑥 ) ) ) ) ) |
37 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∧ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ) ) ∧ ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) ) → 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) |
38 |
|
eqeq1 |
⊢ ( 𝑣 = 𝑢 → ( 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ↔ 𝑢 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) |
39 |
38
|
rexbidv |
⊢ ( 𝑣 = 𝑢 → ( ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ↔ ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) |
40 |
39
|
riotabidv |
⊢ ( 𝑣 = 𝑢 → ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) = ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) |
41 |
|
oveq1 |
⊢ ( 𝑘 = 𝑙 → ( 𝑘 · 𝑦 ) = ( 𝑙 · 𝑦 ) ) |
42 |
41
|
oveq2d |
⊢ ( 𝑘 = 𝑙 → ( 𝑤 + ( 𝑘 · 𝑦 ) ) = ( 𝑤 + ( 𝑙 · 𝑦 ) ) ) |
43 |
42
|
eqeq2d |
⊢ ( 𝑘 = 𝑙 → ( 𝑢 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ↔ 𝑢 = ( 𝑤 + ( 𝑙 · 𝑦 ) ) ) ) |
44 |
43
|
rexbidv |
⊢ ( 𝑘 = 𝑙 → ( ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ↔ ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑤 + ( 𝑙 · 𝑦 ) ) ) ) |
45 |
|
oveq1 |
⊢ ( 𝑤 = 𝑧 → ( 𝑤 + ( 𝑙 · 𝑦 ) ) = ( 𝑧 + ( 𝑙 · 𝑦 ) ) ) |
46 |
45
|
eqeq2d |
⊢ ( 𝑤 = 𝑧 → ( 𝑢 = ( 𝑤 + ( 𝑙 · 𝑦 ) ) ↔ 𝑢 = ( 𝑧 + ( 𝑙 · 𝑦 ) ) ) ) |
47 |
46
|
cbvrexvw |
⊢ ( ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑤 + ( 𝑙 · 𝑦 ) ) ↔ ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑦 ) ) ) |
48 |
44 47
|
bitrdi |
⊢ ( 𝑘 = 𝑙 → ( ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ↔ ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑦 ) ) ) ) |
49 |
48
|
cbvriotavw |
⊢ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) = ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑦 ) ) ) |
50 |
40 49
|
eqtrdi |
⊢ ( 𝑣 = 𝑢 → ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) = ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑦 ) ) ) ) |
51 |
50
|
cbvmptv |
⊢ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) = ( 𝑢 ∈ 𝑉 ↦ ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑦 ) ) ) ) |
52 |
37 51
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∧ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ) ) ∧ ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) ) → 𝐺 = ( 𝑢 ∈ 𝑉 ↦ ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑦 ) ) ) ) ) |
53 |
36 52
|
eqtr3d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∧ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ) ) ∧ ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) ) → ( 𝑢 ∈ 𝑉 ↦ ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑥 ) ) ) ) = ( 𝑢 ∈ 𝑉 ↦ ( ℩ 𝑙 ∈ 𝑅 ∃ 𝑧 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑢 = ( 𝑧 + ( 𝑙 · 𝑦 ) ) ) ) ) |
54 |
1 2 3 4 5 6 7 8 9 10 11 16 17 18 19 20 53
|
lcfl7lem |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∧ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ) ) ∧ ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) ) → 𝑥 = 𝑦 ) |
55 |
54
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∧ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ) ) → ( ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) → 𝑥 = 𝑦 ) ) |
56 |
55
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∀ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ( ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) → 𝑥 = 𝑦 ) ) |
57 |
56
|
a1d |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) → ∀ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∀ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ( ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) → 𝑥 = 𝑦 ) ) ) |
58 |
57
|
ancld |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) → ( ∃ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ ∀ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∀ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ( ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) → 𝑥 = 𝑦 ) ) ) ) |
59 |
|
sneq |
⊢ ( 𝑥 = 𝑦 → { 𝑥 } = { 𝑦 } ) |
60 |
59
|
fveq2d |
⊢ ( 𝑥 = 𝑦 → ( ⊥ ‘ { 𝑥 } ) = ( ⊥ ‘ { 𝑦 } ) ) |
61 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑘 · 𝑥 ) = ( 𝑘 · 𝑦 ) ) |
62 |
61
|
oveq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝑤 + ( 𝑘 · 𝑥 ) ) = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) |
63 |
62
|
eqeq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ↔ 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) |
64 |
60 63
|
rexeqbidv |
⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ↔ ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) |
65 |
64
|
riotabidv |
⊢ ( 𝑥 = 𝑦 → ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) = ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) |
66 |
65
|
mpteq2dv |
⊢ ( 𝑥 = 𝑦 → ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) |
67 |
66
|
eqeq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ↔ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) ) |
68 |
67
|
reu4 |
⊢ ( ∃! 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ↔ ( ∃ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ ∀ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∀ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ( ( 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ∧ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑦 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑦 ) ) ) ) ) → 𝑥 = 𝑦 ) ) ) |
69 |
58 68
|
syl6ibr |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) → ∃! 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ) ) |
70 |
|
reurex |
⊢ ( ∃! 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) → ∃ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ) |
71 |
69 70
|
impbid1 |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ↔ ∃! 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ) ) |
72 |
71
|
orbi2d |
⊢ ( 𝜑 → ( ( ( 𝐿 ‘ 𝐺 ) = 𝑉 ∨ ∃ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ) ↔ ( ( 𝐿 ‘ 𝐺 ) = 𝑉 ∨ ∃! 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ) ) ) |
73 |
15 72
|
bitrd |
⊢ ( 𝜑 → ( 𝐺 ∈ 𝐶 ↔ ( ( 𝐿 ‘ 𝐺 ) = 𝑉 ∨ ∃! 𝑥 ∈ ( 𝑉 ∖ { 0 } ) 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) ) ) ) |