Proof of Theorem mapdh6dN
Step | Hyp | Ref
| Expression |
1 | | mapdh.h |
. . . . . 6
⊢ 𝐻 = (LHyp‘𝐾) |
2 | | mapdh.c |
. . . . . 6
⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
3 | | mapdh.k |
. . . . . 6
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
4 | 1, 2, 3 | lcdlmod 39153 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | | mapdh.q |
. . . . . 6
⊢ 𝑄 = (0g‘𝐶) |
6 | | mapdh.i |
. . . . . 6
⊢ 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)}))))) |
7 | | mapdh.m |
. . . . . 6
⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
8 | | mapdh.u |
. . . . . 6
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
9 | | mapdh.v |
. . . . . 6
⊢ 𝑉 = (Base‘𝑈) |
10 | | mapdh.s |
. . . . . 6
⊢ − =
(-g‘𝑈) |
11 | | mapdhc.o |
. . . . . 6
⊢ 0 =
(0g‘𝑈) |
12 | | mapdh.n |
. . . . . 6
⊢ 𝑁 = (LSpan‘𝑈) |
13 | | mapdh.d |
. . . . . 6
⊢ 𝐷 = (Base‘𝐶) |
14 | | mapdh.r |
. . . . . 6
⊢ 𝑅 = (-g‘𝐶) |
15 | | mapdh.j |
. . . . . 6
⊢ 𝐽 = (LSpan‘𝐶) |
16 | | mapdhc.f |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ 𝐷) |
17 | | mapdh.mn |
. . . . . 6
⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
18 | | mapdhcl.x |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
19 | | mapdh6d.w |
. . . . . . 7
⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
20 | 19 | eldifad 3866 |
. . . . . 6
⊢ (𝜑 → 𝑤 ∈ 𝑉) |
21 | 1, 8, 3 | dvhlvec 38670 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ LVec) |
22 | 18 | eldifad 3866 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
23 | | mapdh6d.y |
. . . . . . . . . 10
⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
24 | 23 | eldifad 3866 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌 ∈ 𝑉) |
25 | | mapdh6d.wn |
. . . . . . . . 9
⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
26 | 9, 12, 21, 20, 22, 24, 25 | lspindpi 19957 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
27 | 26 | simpld 499 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋})) |
28 | 27 | necomd 3004 |
. . . . . 6
⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤})) |
29 | 5, 6, 1, 7, 8, 9, 10, 11, 12, 2, 13, 14, 15, 3, 16, 17, 18, 20, 28 | mapdhcl 39288 |
. . . . 5
⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑤〉) ∈ 𝐷) |
30 | | mapdh.a |
. . . . . 6
⊢ ✚ =
(+g‘𝐶) |
31 | 13, 30, 5 | lmod0vrid 19718 |
. . . . 5
⊢ ((𝐶 ∈ LMod ∧ (𝐼‘〈𝑋, 𝐹, 𝑤〉) ∈ 𝐷) → ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ 𝑄) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) |
32 | 4, 29, 31 | syl2anc 588 |
. . . 4
⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ 𝑄) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) |
33 | 32 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ 𝑄) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) |
34 | | oteq3 4767 |
. . . . . 6
⊢ ((𝑌 + 𝑍) = 0 → 〈𝑋, 𝐹, (𝑌 + 𝑍)〉 = 〈𝑋, 𝐹, 0 〉) |
35 | 34 | fveq2d 6655 |
. . . . 5
⊢ ((𝑌 + 𝑍) = 0 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = (𝐼‘〈𝑋, 𝐹, 0 〉)) |
36 | 5, 6, 11, 18, 16 | mapdhval0 39286 |
. . . . 5
⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 0 〉) = 𝑄) |
37 | 35, 36 | sylan9eqr 2816 |
. . . 4
⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = 𝑄) |
38 | 37 | oveq2d 7159 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ 𝑄)) |
39 | | oveq2 7151 |
. . . . . 6
⊢ ((𝑌 + 𝑍) = 0 → (𝑤 + (𝑌 + 𝑍)) = (𝑤 + 0 )) |
40 | 1, 8, 3 | dvhlmod 38671 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ∈ LMod) |
41 | | mapdh.p |
. . . . . . . 8
⊢ + =
(+g‘𝑈) |
42 | 9, 41, 11 | lmod0vrid 19718 |
. . . . . . 7
⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉) → (𝑤 + 0 ) = 𝑤) |
43 | 40, 20, 42 | syl2anc 588 |
. . . . . 6
⊢ (𝜑 → (𝑤 + 0 ) = 𝑤) |
44 | 39, 43 | sylan9eqr 2816 |
. . . . 5
⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑤 + (𝑌 + 𝑍)) = 𝑤) |
45 | 44 | oteq3d 4770 |
. . . 4
⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → 〈𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))〉 = 〈𝑋, 𝐹, 𝑤〉) |
46 | 45 | fveq2d 6655 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝐼‘〈𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))〉) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) |
47 | 33, 38, 46 | 3eqtr4rd 2805 |
. 2
⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝐼‘〈𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))〉) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉))) |
48 | 3 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
49 | 16 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝐹 ∈ 𝐷) |
50 | 17 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
51 | 18 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
52 | 19 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑤 ∈ (𝑉 ∖ { 0 })) |
53 | | mapdh6d.z |
. . . . . . 7
⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
54 | 53 | eldifad 3866 |
. . . . . 6
⊢ (𝜑 → 𝑍 ∈ 𝑉) |
55 | 9, 41 | lmodvacl 19701 |
. . . . . 6
⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉) |
56 | 40, 24, 54, 55 | syl3anc 1369 |
. . . . 5
⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉) |
57 | 56 | anim1i 618 |
. . . 4
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 )) |
58 | | eldifsn 4670 |
. . . 4
⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 )) |
59 | 57, 58 | sylibr 237 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 })) |
60 | | mapdh6d.yz |
. . . . . . 7
⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
61 | | mapdh6d.xn |
. . . . . . . . 9
⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
62 | 9, 12, 21, 22, 24, 54, 61 | lspindpi 19957 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
63 | 62 | simpld 499 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
64 | 9, 41, 11, 12, 21, 18, 23, 53, 19, 60, 63, 25 | mapdindp1 39281 |
. . . . . 6
⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
65 | 9, 41, 11, 12, 21, 18, 23, 53, 19, 60, 63, 25 | mapdindp2 39282 |
. . . . . 6
⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)})) |
66 | 9, 11, 12, 21, 18, 56, 20, 64, 65 | lspindp1 19958 |
. . . . 5
⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑤, (𝑌 + 𝑍)}))) |
67 | 66 | simprd 500 |
. . . 4
⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, (𝑌 + 𝑍)})) |
68 | 67 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑋 ∈ (𝑁‘{𝑤, (𝑌 + 𝑍)})) |
69 | 26 | simprd 500 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})) |
70 | 9, 11, 12, 21, 19, 24, 69 | lspsnne1 19942 |
. . . . . . . 8
⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌})) |
71 | | eqid 2759 |
. . . . . . . . . 10
⊢
(LSSum‘𝑈) =
(LSSum‘𝑈) |
72 | 9, 12, 71, 40, 24, 54 | lsmpr 19914 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍}))) |
73 | 60 | oveq2d 7159 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑌})) = ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍}))) |
74 | | eqid 2759 |
. . . . . . . . . . . . 13
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) |
75 | 9, 74, 12 | lspsncl 19802 |
. . . . . . . . . . . 12
⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
76 | 40, 24, 75 | syl2anc 588 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
77 | 74 | lsssubg 19782 |
. . . . . . . . . . 11
⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑈)) |
78 | 40, 76, 77 | syl2anc 588 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑈)) |
79 | 71 | lsmidm 18840 |
. . . . . . . . . 10
⊢ ((𝑁‘{𝑌}) ∈ (SubGrp‘𝑈) → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘{𝑌})) |
80 | 78, 79 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘{𝑌})) |
81 | 72, 73, 80 | 3eqtr2d 2800 |
. . . . . . . 8
⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌})) |
82 | 70, 81 | neleqtrrd 2873 |
. . . . . . 7
⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑍})) |
83 | 9, 41, 12, 40, 24, 54, 20, 82 | lspindp4 19962 |
. . . . . 6
⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌, (𝑌 + 𝑍)})) |
84 | 9, 12, 21, 20, 24, 56, 83 | lspindpi 19957 |
. . . . 5
⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)}))) |
85 | 84 | simprd 500 |
. . . 4
⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
86 | 85 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
87 | | eqidd 2760 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝐼‘〈𝑋, 𝐹, 𝑤〉) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) |
88 | | eqidd 2760 |
. . 3
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉)) |
89 | 5, 6, 1, 7, 8, 9, 10, 11, 12, 2, 13, 14, 15, 48, 49, 50, 51, 41, 30, 52, 59, 68, 86, 87, 88 | mapdh6aN 39296 |
. 2
⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝐼‘〈𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))〉) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉))) |
90 | 47, 89 | pm2.61dane 3036 |
1
⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))〉) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉))) |