Step | Hyp | Ref
| Expression |
1 | | mapdh8a.h |
. . 3
⊢ 𝐻 = (LHyp‘𝐾) |
2 | | mapdh8a.u |
. . 3
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
3 | | mapdh8a.v |
. . 3
⊢ 𝑉 = (Base‘𝑈) |
4 | | mapdh8a.n |
. . 3
⊢ 𝑁 = (LSpan‘𝑈) |
5 | | mapdh8a.k |
. . 3
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
6 | | mapdh8e.x |
. . . 4
⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
7 | 6 | eldifad 3899 |
. . 3
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
8 | | mapdh8e.y |
. . . 4
⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
9 | 8 | eldifad 3899 |
. . 3
⊢ (𝜑 → 𝑌 ∈ 𝑉) |
10 | 1, 2, 3, 4, 5, 7, 9 | dvh3dim 39460 |
. 2
⊢ (𝜑 → ∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
11 | | mapdh8a.s |
. . . 4
⊢ − =
(-g‘𝑈) |
12 | | mapdh8a.o |
. . . 4
⊢ 0 =
(0g‘𝑈) |
13 | | mapdh8a.c |
. . . 4
⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
14 | | mapdh8a.d |
. . . 4
⊢ 𝐷 = (Base‘𝐶) |
15 | | mapdh8a.r |
. . . 4
⊢ 𝑅 = (-g‘𝐶) |
16 | | mapdh8a.q |
. . . 4
⊢ 𝑄 = (0g‘𝐶) |
17 | | mapdh8a.j |
. . . 4
⊢ 𝐽 = (LSpan‘𝐶) |
18 | | mapdh8a.m |
. . . 4
⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
19 | | mapdh8a.i |
. . . 4
⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd
‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st
‘(1st ‘𝑥)) − (2nd
‘𝑥))})) = (𝐽‘{((2nd
‘(1st ‘𝑥))𝑅ℎ)}))))) |
20 | 5 | 3ad2ant1 1132 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
21 | | mapdh8e.f |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ 𝐷) |
22 | 21 | 3ad2ant1 1132 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝐹 ∈ 𝐷) |
23 | | mapdh8e.mn |
. . . . 5
⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
24 | 23 | 3ad2ant1 1132 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
25 | | mapdh8e.eg |
. . . . 5
⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
26 | 25 | 3ad2ant1 1132 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
27 | 6 | 3ad2ant1 1132 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
28 | 8 | 3ad2ant1 1132 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
29 | | mapdh8e.t |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) |
30 | 29 | 3ad2ant1 1132 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑇 ∈ (𝑉 ∖ { 0 })) |
31 | | mapdh8e.yt |
. . . . 5
⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) |
32 | 31 | 3ad2ant1 1132 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) |
33 | | eqid 2738 |
. . . . 5
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) |
34 | 1, 2, 5 | dvhlmod 39124 |
. . . . . 6
⊢ (𝜑 → 𝑈 ∈ LMod) |
35 | 34 | 3ad2ant1 1132 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LMod) |
36 | 3, 33, 4, 34, 7, 9 | lspprcl 20240 |
. . . . . 6
⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
37 | 36 | 3ad2ant1 1132 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
38 | | simp2 1136 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ 𝑉) |
39 | | simp3 1137 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
40 | 12, 33, 35, 37, 38, 39 | lssneln0 20214 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ (𝑉 ∖ { 0 })) |
41 | 1, 2, 5 | dvhlvec 39123 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ LVec) |
42 | 29 | eldifad 3899 |
. . . . . . . . 9
⊢ (𝜑 → 𝑇 ∈ 𝑉) |
43 | | mapdh8e.xy |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
44 | | mapdh8e.e |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) |
45 | | prcom 4668 |
. . . . . . . . . . 11
⊢ {𝑌, 𝑇} = {𝑇, 𝑌} |
46 | 45 | fveq2i 6777 |
. . . . . . . . . 10
⊢ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑇, 𝑌}) |
47 | 44, 46 | eleqtrdi 2849 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑇, 𝑌})) |
48 | 3, 12, 4, 41, 6, 42, 9, 43, 47 | lspexch 20391 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝑋, 𝑌})) |
49 | 33, 4, 34, 36, 48 | lspsnel5a 20258 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑋, 𝑌})) |
50 | 49 | 3ad2ant1 1132 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑋, 𝑌})) |
51 | 34 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → 𝑈 ∈ LMod) |
52 | 36 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
53 | | simpr 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → 𝑤 ∈ 𝑉) |
54 | 3, 33, 4, 51, 52, 53 | lspsnel5 20257 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → (𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌}))) |
55 | 54 | biimprd 247 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → ((𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌}) → 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))) |
56 | 55 | con3d 152 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) → ¬ (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌}))) |
57 | 56 | 3impia 1116 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌})) |
58 | | nssne2 3982 |
. . . . . 6
⊢ (((𝑁‘{𝑇}) ⊆ (𝑁‘{𝑋, 𝑌}) ∧ ¬ (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑇}) ≠ (𝑁‘{𝑤})) |
59 | 50, 57, 58 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑇}) ≠ (𝑁‘{𝑤})) |
60 | 59 | necomd 2999 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) |
61 | | mapdh8e.xt |
. . . . 5
⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) |
62 | 61 | 3ad2ant1 1132 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) |
63 | 41 | 3ad2ant1 1132 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LVec) |
64 | 7 | 3ad2ant1 1132 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑋 ∈ 𝑉) |
65 | 9 | 3ad2ant1 1132 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑌 ∈ 𝑉) |
66 | 3, 4, 63, 38, 64, 65, 39 | lspindpi 20394 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
67 | 66 | simprd 496 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})) |
68 | 67 | necomd 2999 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) |
69 | 43 | 3ad2ant1 1132 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
70 | 3, 12, 4, 63, 27, 65, 38, 69, 39 | lspindp2l 20396 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ((𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))) |
71 | 70 | simprd 496 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) |
72 | 1, 2, 3, 11, 12, 4, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 27, 28, 30, 32, 40, 60, 62, 68, 71 | mapdh8d 39797 |
. . 3
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) |
73 | 72 | rexlimdv3a 3215 |
. 2
⊢ (𝜑 → (∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉))) |
74 | 10, 73 | mpd 15 |
1
⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) |