| 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 3962 | . . 3
⊢ (𝜑 → 𝑋 ∈ 𝑉) | 
| 8 |  | mapdh8ac.y | . . . 4
⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | 
| 9 | 8 | eldifad 3962 | . . 3
⊢ (𝜑 → 𝑌 ∈ 𝑉) | 
| 10 |  | mapdh8ac.z | . . . 4
⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | 
| 11 | 10 | eldifad 3962 | . . 3
⊢ (𝜑 → 𝑍 ∈ 𝑉) | 
| 12 | 1, 2, 3, 4, 5, 7, 9, 11 | dvh3dim2 41451 | . 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 1133 | . . . 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 1133 | . . . 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 1133 | . . . 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 1133 | . . . 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 1133 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) | 
| 31 | 6 | 3ad2ant1 1133 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑋 ∈ (𝑉 ∖ { 0 })) | 
| 32 | 8 | 3ad2ant1 1133 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑌 ∈ (𝑉 ∖ { 0 })) | 
| 33 | 10 | 3ad2ant1 1133 | . . . 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 1133 | . . . 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 1133 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) | 
| 38 |  | eqidd 2737 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝐼‘〈𝑋, 𝐹, 𝑤〉) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) | 
| 39 |  | eqid 2736 | . . . . 5
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) | 
| 40 | 1, 2, 5 | dvhlmod 41113 | . . . . . 6
⊢ (𝜑 → 𝑈 ∈ LMod) | 
| 41 | 40 | 3ad2ant1 1133 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑈 ∈ LMod) | 
| 42 | 3, 39, 4, 40, 7, 9 | lspprcl 20977 | . . . . . 6
⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) | 
| 43 | 42 | 3ad2ant1 1133 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) | 
| 44 |  | simp2 1137 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑤 ∈ 𝑉) | 
| 45 |  | simp3l 1201 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | 
| 46 | 14, 39, 41, 43, 44, 45 | lssneln0 20952 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑤 ∈ (𝑉 ∖ { 0 })) | 
| 47 | 1, 2, 5 | dvhlvec 41112 | . . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ LVec) | 
| 48 | 47 | 3ad2ant1 1133 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑈 ∈ LVec) | 
| 49 | 7 | 3ad2ant1 1133 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑋 ∈ 𝑉) | 
| 50 | 9 | 3ad2ant1 1133 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑌 ∈ 𝑉) | 
| 51 | 3, 4, 48, 44, 49, 50, 45 | lspindpi 21135 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) | 
| 52 | 51 | simprd 495 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})) | 
| 53 | 52 | necomd 2995 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) | 
| 54 |  | simpl1 1191 | . . . . . . 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 1192 | . . . . . 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 484 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) → 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) | 
| 62 |  | prcom 4731 | . . . . . . . 8
⊢ {𝑌, 𝑤} = {𝑤, 𝑌} | 
| 63 | 62 | fveq2i 6908 | . . . . . . 7
⊢ (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌}) | 
| 64 | 61, 63 | eleqtrdi 2850 | . . . . . 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 21132 | . . . . 5
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) → 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | 
| 66 | 45, 65 | mtand 815 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) | 
| 67 | 11 | 3ad2ant1 1133 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → 𝑍 ∈ 𝑉) | 
| 68 |  | simp3r 1202 | . . . . . 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 21135 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑍}))) | 
| 70 | 69 | simprd 495 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑍})) | 
| 71 |  | simpl1 1191 | . . . . . . 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 1192 | . . . . . 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 484 | . . . . . 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 21132 | . . . . 5
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) ∧ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) → 𝑤 ∈ (𝑁‘{𝑋, 𝑍})) | 
| 80 | 68, 79 | mtand 815 | . . . 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 41781 | . . 3
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑍}))) → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 𝑇〉)) | 
| 82 | 81 | rexlimdv3a 3158 | . 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〉)) |