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 | | mapdh8ac.x |
. . . 4
⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
7 | 6 | eldifad 3878 |
. . 3
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
8 | | mapdh8ac.y |
. . . 4
⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
9 | 8 | eldifad 3878 |
. . 3
⊢ (𝜑 → 𝑌 ∈ 𝑉) |
10 | | mapdh8ac.z |
. . . 4
⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
11 | 10 | eldifad 3878 |
. . 3
⊢ (𝜑 → 𝑍 ∈ 𝑉) |
12 | 1, 2, 3, 4, 5, 7, 9, 11 | dvh3dim2 39199 |
. 2
⊢ (𝜑 → ∃𝑤 ∈ 𝑉 (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) |
13 | | mapdh8a.s |
. . . 4
⊢ − =
(-g‘𝑈) |
14 | | mapdh8a.o |
. . . 4
⊢ 0 =
(0g‘𝑈) |
15 | | mapdh8a.c |
. . . 4
⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
16 | | mapdh8a.d |
. . . 4
⊢ 𝐷 = (Base‘𝐶) |
17 | | mapdh8a.r |
. . . 4
⊢ 𝑅 = (-g‘𝐶) |
18 | | mapdh8a.q |
. . . 4
⊢ 𝑄 = (0g‘𝐶) |
19 | | mapdh8a.j |
. . . 4
⊢ 𝐽 = (LSpan‘𝐶) |
20 | | mapdh8a.m |
. . . 4
⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
21 | | 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)}))))) |
22 | 5 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
23 | | mapdh8ac.f |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ 𝐷) |
24 | 23 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝐹 ∈ 𝐷) |
25 | | mapdh8ac.mn |
. . . . 5
⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
26 | 25 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
27 | | mapdh8ac.eg |
. . . . 5
⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
28 | 27 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
29 | | mapdh8ac.ee |
. . . . 5
⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) |
30 | 29 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) |
31 | 6 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
32 | 8 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
33 | 10 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑍 ∈ (𝑉 ∖ { 0 })) |
34 | | mapdh8ac.t |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) |
35 | 34 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑇 ∈ (𝑉 ∖ { 0 })) |
36 | | mapdh8ac.yn |
. . . . 5
⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) |
37 | 36 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) |
38 | | eqidd 2738 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝐼‘〈𝑋, 𝐹, 𝑤〉) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) |
39 | | eqid 2737 |
. . . . 5
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) |
40 | 1, 2, 5 | dvhlmod 38861 |
. . . . . 6
⊢ (𝜑 → 𝑈 ∈ LMod) |
41 | 40 | 3ad2ant1 1135 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑈 ∈ LMod) |
42 | 3, 39, 4, 40, 7, 9 | lspprcl 20015 |
. . . . . 6
⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
43 | 42 | 3ad2ant1 1135 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
44 | | simp2 1139 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑤 ∈ 𝑉) |
45 | | simp3l 1203 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
46 | 14, 39, 41, 43, 44, 45 | lssneln0 19989 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑤 ∈ (𝑉 ∖ { 0 })) |
47 | 1, 2, 5 | dvhlvec 38860 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ LVec) |
48 | 47 | 3ad2ant1 1135 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑈 ∈ LVec) |
49 | 7 | 3ad2ant1 1135 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑋 ∈ 𝑉) |
50 | 9 | 3ad2ant1 1135 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑌 ∈ 𝑉) |
51 | 3, 4, 48, 44, 49, 50, 45 | lspindpi 20169 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
52 | 51 | simprd 499 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})) |
53 | 52 | necomd 2996 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) |
54 | | simpl1 1193 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) → 𝜑) |
55 | 54, 47 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) → 𝑈 ∈ LVec) |
56 | 54, 6 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
57 | | simpl2 1194 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) → 𝑤 ∈ 𝑉) |
58 | 54, 9 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) → 𝑌 ∈ 𝑉) |
59 | | mapdh8ad.xy |
. . . . . . 7
⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
60 | 54, 59 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
61 | | simpr 488 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) → 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) |
62 | | prcom 4648 |
. . . . . . . 8
⊢ {𝑌, 𝑤} = {𝑤, 𝑌} |
63 | 62 | fveq2i 6720 |
. . . . . . 7
⊢ (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌}) |
64 | 61, 63 | eleqtrdi 2848 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) → 𝑋 ∈ (𝑁‘{𝑤, 𝑌})) |
65 | 3, 14, 4, 55, 56, 57, 58, 60, 64 | lspexch 20166 |
. . . . 5
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) → 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
66 | 45, 65 | mtand 816 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) |
67 | 11 | 3ad2ant1 1135 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑍 ∈ 𝑉) |
68 | | simp3r 1204 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍})) |
69 | 3, 4, 48, 44, 49, 67, 68 | lspindpi 20169 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑍}))) |
70 | 69 | simprd 499 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑍})) |
71 | | simpl1 1193 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) → 𝜑) |
72 | 71, 47 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) → 𝑈 ∈ LVec) |
73 | 71, 6 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
74 | | simpl2 1194 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) → 𝑤 ∈ 𝑉) |
75 | 71, 11 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) → 𝑍 ∈ 𝑉) |
76 | | mapdh8ad.xz |
. . . . . . 7
⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
77 | 71, 76 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
78 | | simpr 488 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) → 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) |
79 | 3, 14, 4, 72, 73, 74, 75, 77, 78 | lspexch 20166 |
. . . . 5
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) → 𝑤 ∈ (𝑁‘{𝑋, 𝑍})) |
80 | 68, 79 | mtand 816 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) |
81 | 1, 2, 3, 13, 14, 4, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 30, 31, 32, 33, 35, 37, 38, 46, 53, 66, 70, 80 | mapdh8ac 39529 |
. . 3
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 𝑇〉)) |
82 | 81 | rexlimdv3a 3205 |
. 2
⊢ (𝜑 → (∃𝑤 ∈ 𝑉 (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍})) → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 𝑇〉))) |
83 | 12, 82 | mpd 15 |
1
⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 𝑇〉)) |