Step | Hyp | Ref
| Expression |
1 | | mapd0.h |
. . 3
⊢ 𝐻 = (LHyp‘𝐾) |
2 | | mapd0.u |
. . 3
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
3 | | eqid 2737 |
. . 3
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) |
4 | | eqid 2737 |
. . 3
⊢
(LFnl‘𝑈) =
(LFnl‘𝑈) |
5 | | eqid 2737 |
. . 3
⊢
(LKer‘𝑈) =
(LKer‘𝑈) |
6 | | eqid 2737 |
. . 3
⊢
((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) |
7 | | mapd0.m |
. . 3
⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
8 | | mapd0.k |
. . 3
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
9 | 1, 2, 8 | dvhlmod 38861 |
. . . 4
⊢ (𝜑 → 𝑈 ∈ LMod) |
10 | | mapd0.o |
. . . . 5
⊢ 𝑂 = (0g‘𝑈) |
11 | 10, 3 | lsssn0 19984 |
. . . 4
⊢ (𝑈 ∈ LMod → {𝑂} ∈ (LSubSp‘𝑈)) |
12 | 9, 11 | syl 17 |
. . 3
⊢ (𝜑 → {𝑂} ∈ (LSubSp‘𝑈)) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 12 | mapdval 39379 |
. 2
⊢ (𝜑 → (𝑀‘{𝑂}) = {𝑓 ∈ (LFnl‘𝑈) ∣ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂})}) |
14 | | simprrr 782 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}) |
15 | 9 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → 𝑈 ∈ LMod) |
16 | 8 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
17 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝑈) =
(Base‘𝑈) |
18 | | simprl 771 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → 𝑔 ∈ (LFnl‘𝑈)) |
19 | 17, 4, 5, 15, 18 | lkrssv 36847 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → ((LKer‘𝑈)‘𝑔) ⊆ (Base‘𝑈)) |
20 | 1, 2, 17, 3, 6 | dochlss 39105 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((LKer‘𝑈)‘𝑔) ⊆ (Base‘𝑈)) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ∈ (LSubSp‘𝑈)) |
21 | 16, 19, 20 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ∈ (LSubSp‘𝑈)) |
22 | 10, 3 | lssle0 19986 |
. . . . . . . . . . . 12
⊢ ((𝑈 ∈ LMod ∧
(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ∈ (LSubSp‘𝑈)) → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = {𝑂})) |
23 | 15, 21, 22 | syl2anc 587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = {𝑂})) |
24 | 14, 23 | mpbid 235 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = {𝑂}) |
25 | 24 | fveq2d 6721 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = (((ocH‘𝐾)‘𝑊)‘{𝑂})) |
26 | | simprrl 781 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔)) |
27 | 1, 2, 6, 17, 10 | doch0 39109 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈)) |
28 | 8, 27 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈)) |
29 | 28 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈)) |
30 | 25, 26, 29 | 3eqtr3d 2785 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → ((LKer‘𝑈)‘𝑔) = (Base‘𝑈)) |
31 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Scalar‘𝑈) =
(Scalar‘𝑈) |
32 | | eqid 2737 |
. . . . . . . . . 10
⊢
(0g‘(Scalar‘𝑈)) =
(0g‘(Scalar‘𝑈)) |
33 | 31, 32, 17, 4, 5 | lkr0f 36845 |
. . . . . . . . 9
⊢ ((𝑈 ∈ LMod ∧ 𝑔 ∈ (LFnl‘𝑈)) → (((LKer‘𝑈)‘𝑔) = (Base‘𝑈) ↔ 𝑔 = ((Base‘𝑈) ×
{(0g‘(Scalar‘𝑈))}))) |
34 | 15, 18, 33 | syl2anc 587 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((LKer‘𝑈)‘𝑔) = (Base‘𝑈) ↔ 𝑔 = ((Base‘𝑈) ×
{(0g‘(Scalar‘𝑈))}))) |
35 | 30, 34 | mpbid 235 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → 𝑔 = ((Base‘𝑈) ×
{(0g‘(Scalar‘𝑈))})) |
36 | | mapd0.c |
. . . . . . . . 9
⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
37 | | mapd0.z |
. . . . . . . . 9
⊢ 0 =
(0g‘𝐶) |
38 | 1, 2, 17, 31, 32, 36, 37, 8 | lcd0v 39362 |
. . . . . . . 8
⊢ (𝜑 → 0 = ((Base‘𝑈) ×
{(0g‘(Scalar‘𝑈))})) |
39 | 38 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → 0 = ((Base‘𝑈) ×
{(0g‘(Scalar‘𝑈))})) |
40 | 35, 39 | eqtr4d 2780 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → 𝑔 = 0 ) |
41 | 40 | ex 416 |
. . . . 5
⊢ (𝜑 → ((𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂})) → 𝑔 = 0 )) |
42 | | eqid 2737 |
. . . . . . . 8
⊢
(Base‘𝐶) =
(Base‘𝐶) |
43 | 1, 36, 42, 37, 8 | lcd0vcl 39365 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈ (Base‘𝐶)) |
44 | 1, 36, 42, 2, 4, 8,
43 | lcdvbaselfl 39346 |
. . . . . . 7
⊢ (𝜑 → 0 ∈ (LFnl‘𝑈)) |
45 | 31, 32, 17, 4, 5 | lkr0f 36845 |
. . . . . . . . . . . . 13
⊢ ((𝑈 ∈ LMod ∧ 0 ∈
(LFnl‘𝑈)) →
(((LKer‘𝑈)‘
0 ) =
(Base‘𝑈) ↔ 0 =
((Base‘𝑈) ×
{(0g‘(Scalar‘𝑈))}))) |
46 | 9, 44, 45 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((LKer‘𝑈)‘ 0 ) = (Base‘𝑈) ↔ 0 = ((Base‘𝑈) ×
{(0g‘(Scalar‘𝑈))}))) |
47 | 38, 46 | mpbird 260 |
. . . . . . . . . . 11
⊢ (𝜑 → ((LKer‘𝑈)‘ 0 ) = (Base‘𝑈)) |
48 | 47 | fveq2d 6721 |
. . . . . . . . . 10
⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈))) |
49 | 48 | fveq2d 6721 |
. . . . . . . . 9
⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)))) |
50 | 1, 2, 6, 17, 8 | dochoc1 39112 |
. . . . . . . . 9
⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(Base‘𝑈))) = (Base‘𝑈)) |
51 | 49, 50 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = (Base‘𝑈)) |
52 | 51, 47 | eqtr4d 2780 |
. . . . . . 7
⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 )) |
53 | 1, 2, 6, 17, 10 | doch1 39110 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)) = {𝑂}) |
54 | 8, 53 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)) = {𝑂}) |
55 | 48, 54 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = {𝑂}) |
56 | | eqimss 3957 |
. . . . . . . 8
⊢
((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = {𝑂} → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂}) |
57 | 55, 56 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂}) |
58 | 44, 52, 57 | jca32 519 |
. . . . . 6
⊢ (𝜑 → ( 0 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂}))) |
59 | | eleq1 2825 |
. . . . . . 7
⊢ (𝑔 = 0 → (𝑔 ∈ (LFnl‘𝑈) ↔ 0 ∈ (LFnl‘𝑈))) |
60 | | 2fveq3 6722 |
. . . . . . . . . 10
⊢ (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) |
61 | 60 | fveq2d 6721 |
. . . . . . . . 9
⊢ (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )))) |
62 | | fveq2 6717 |
. . . . . . . . 9
⊢ (𝑔 = 0 → ((LKer‘𝑈)‘𝑔) = ((LKer‘𝑈)‘ 0 )) |
63 | 61, 62 | eqeq12d 2753 |
. . . . . . . 8
⊢ (𝑔 = 0 → ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ↔ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ))) |
64 | 60 | sseq1d 3932 |
. . . . . . . 8
⊢ (𝑔 = 0 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂})) |
65 | 63, 64 | anbi12d 634 |
. . . . . . 7
⊢ (𝑔 = 0 →
(((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}) ↔ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂}))) |
66 | 59, 65 | anbi12d 634 |
. . . . . 6
⊢ (𝑔 = 0 → ((𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂})) ↔ ( 0 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂})))) |
67 | 58, 66 | syl5ibrcom 250 |
. . . . 5
⊢ (𝜑 → (𝑔 = 0 → (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂})))) |
68 | 41, 67 | impbid 215 |
. . . 4
⊢ (𝜑 → ((𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂})) ↔ 𝑔 = 0 )) |
69 | | 2fveq3 6722 |
. . . . . . . 8
⊢ (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) |
70 | 69 | fveq2d 6721 |
. . . . . . 7
⊢ (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)))) |
71 | | fveq2 6717 |
. . . . . . 7
⊢ (𝑓 = 𝑔 → ((LKer‘𝑈)‘𝑓) = ((LKer‘𝑈)‘𝑔)) |
72 | 70, 71 | eqeq12d 2753 |
. . . . . 6
⊢ (𝑓 = 𝑔 → ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ↔ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔))) |
73 | 69 | sseq1d 3932 |
. . . . . 6
⊢ (𝑓 = 𝑔 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂})) |
74 | 72, 73 | anbi12d 634 |
. . . . 5
⊢ (𝑓 = 𝑔 → (((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂}) ↔ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) |
75 | 74 | elrab 3602 |
. . . 4
⊢ (𝑔 ∈ {𝑓 ∈ (LFnl‘𝑈) ∣ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂})} ↔ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) |
76 | | velsn 4557 |
. . . 4
⊢ (𝑔 ∈ { 0 } ↔ 𝑔 = 0 ) |
77 | 68, 75, 76 | 3bitr4g 317 |
. . 3
⊢ (𝜑 → (𝑔 ∈ {𝑓 ∈ (LFnl‘𝑈) ∣ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂})} ↔ 𝑔 ∈ { 0 })) |
78 | 77 | eqrdv 2735 |
. 2
⊢ (𝜑 → {𝑓 ∈ (LFnl‘𝑈) ∣ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂})} = { 0 }) |
79 | 13, 78 | eqtrd 2777 |
1
⊢ (𝜑 → (𝑀‘{𝑂}) = { 0 }) |