Step |
Hyp |
Ref |
Expression |
1 |
|
hdmapval0.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hdmapval0.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hdmapval0.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
4 |
|
hdmapval0.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
hdmapval0.q |
⊢ 𝑄 = ( 0g ‘ 𝐶 ) |
6 |
|
hdmapval0.s |
⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
hdmapval0.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
9 |
|
eqid |
⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
11 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
|
eqid |
⊢ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 = 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 |
13 |
1 10 11 2 8 3 12 7
|
dvheveccl |
⊢ ( 𝜑 → 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ∈ ( ( Base ‘ 𝑈 ) ∖ { 0 } ) ) |
14 |
13
|
eldifad |
⊢ ( 𝜑 → 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ∈ ( Base ‘ 𝑈 ) ) |
15 |
1 2 7
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
16 |
8 3
|
lmod0vcl |
⊢ ( 𝑈 ∈ LMod → 0 ∈ ( Base ‘ 𝑈 ) ) |
17 |
15 16
|
syl |
⊢ ( 𝜑 → 0 ∈ ( Base ‘ 𝑈 ) ) |
18 |
1 2 8 9 7 14 17
|
dvh3dim |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ( Base ‘ 𝑈 ) ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) |
19 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
20 |
|
eqid |
⊢ ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) |
21 |
|
eqid |
⊢ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) |
22 |
7
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
23 |
17
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → 0 ∈ ( Base ‘ 𝑈 ) ) |
24 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → 𝑥 ∈ ( Base ‘ 𝑈 ) ) |
25 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
26 |
8 25 9 15 14 17
|
lspprcl |
⊢ ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
27 |
8 9 15 14 17
|
lspprid1 |
⊢ ( 𝜑 → 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) |
28 |
25 9 15 26 27
|
lspsnel5a |
⊢ ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ⊆ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) |
29 |
8 9 15 14 17
|
lspprid2 |
⊢ ( 𝜑 → 0 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) |
30 |
25 9 15 26 29
|
lspsnel5a |
⊢ ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ { 0 } ) ⊆ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) |
31 |
28 30
|
unssd |
⊢ ( 𝜑 → ( ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 0 } ) ) ⊆ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) |
32 |
31
|
ssneld |
⊢ ( 𝜑 → ( ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) → ¬ 𝑥 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 0 } ) ) ) ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ) → ( ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) → ¬ 𝑥 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 0 } ) ) ) ) |
34 |
33
|
3impia |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ¬ 𝑥 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 0 } ) ) ) |
35 |
1 12 2 8 9 4 19 20 21 6 22 23 24 34
|
hdmapval2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ( 𝑆 ‘ 0 ) = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 𝑥 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) , 𝑥 〉 ) , 0 〉 ) ) |
36 |
|
eqid |
⊢ ( LSpan ‘ 𝐶 ) = ( LSpan ‘ 𝐶 ) |
37 |
|
eqid |
⊢ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
38 |
1 2 8 3 4 19 5 20 7 13
|
hvmapcl2 |
⊢ ( 𝜑 → ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) ∈ ( ( Base ‘ 𝐶 ) ∖ { 𝑄 } ) ) |
39 |
38
|
eldifad |
⊢ ( 𝜑 → ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) ∈ ( Base ‘ 𝐶 ) ) |
40 |
39
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) ∈ ( Base ‘ 𝐶 ) ) |
41 |
1 2 8 3 9 4 36 37 20 7 13
|
mapdhvmap |
⊢ ( 𝜑 → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ) = ( ( LSpan ‘ 𝐶 ) ‘ { ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) } ) ) |
42 |
41
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ) = ( ( LSpan ‘ 𝐶 ) ‘ { ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) } ) ) |
43 |
1 2 7
|
dvhlvec |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
44 |
43
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → 𝑈 ∈ LVec ) |
45 |
14
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ∈ ( Base ‘ 𝑈 ) ) |
46 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) |
47 |
8 9 44 24 45 23 46
|
lspindpi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ( ( ( LSpan ‘ 𝑈 ) ‘ { 𝑥 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑥 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 0 } ) ) ) |
48 |
47
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑥 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ) |
49 |
48
|
necomd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑥 } ) ) |
50 |
13
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ∈ ( ( Base ‘ 𝑈 ) ∖ { 0 } ) ) |
51 |
1 2 8 3 9 4 19 36 37 21 22 40 42 49 50 24
|
hdmap1cl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) , 𝑥 〉 ) ∈ ( Base ‘ 𝐶 ) ) |
52 |
1 2 8 3 4 19 5 21 22 51 24
|
hdmap1val0 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 𝑥 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) , 𝑥 〉 ) , 0 〉 ) = 𝑄 ) |
53 |
35 52
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) ) → ( 𝑆 ‘ 0 ) = 𝑄 ) |
54 |
53
|
rexlimdv3a |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( Base ‘ 𝑈 ) ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 0 } ) → ( 𝑆 ‘ 0 ) = 𝑄 ) ) |
55 |
18 54
|
mpd |
⊢ ( 𝜑 → ( 𝑆 ‘ 0 ) = 𝑄 ) |