| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | mapdh8a.h | . . . . . 6
⊢ 𝐻 = (LHyp‘𝐾) | 
| 2 |  | mapdh8a.u | . . . . . 6
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | 
| 3 |  | mapdh8a.v | . . . . . 6
⊢ 𝑉 = (Base‘𝑈) | 
| 4 |  | mapdh8a.s | . . . . . 6
⊢  − =
(-g‘𝑈) | 
| 5 |  | mapdh8a.o | . . . . . 6
⊢  0 =
(0g‘𝑈) | 
| 6 |  | mapdh8a.n | . . . . . 6
⊢ 𝑁 = (LSpan‘𝑈) | 
| 7 |  | mapdh8a.c | . . . . . 6
⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | 
| 8 |  | mapdh8a.d | . . . . . 6
⊢ 𝐷 = (Base‘𝐶) | 
| 9 |  | mapdh8a.r | . . . . . 6
⊢ 𝑅 = (-g‘𝐶) | 
| 10 |  | mapdh8a.q | . . . . . 6
⊢ 𝑄 = (0g‘𝐶) | 
| 11 |  | mapdh8a.j | . . . . . 6
⊢ 𝐽 = (LSpan‘𝐶) | 
| 12 |  | mapdh8a.m | . . . . . 6
⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | 
| 13 |  | mapdh8a.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)}))))) | 
| 14 |  | mapdh8a.k | . . . . . . 7
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 15 | 14 | 3ad2ant1 1133 | . . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 16 |  | mapdh8h.f | . . . . . . 7
⊢ (𝜑 → 𝐹 ∈ 𝐷) | 
| 17 | 16 | 3ad2ant1 1133 | . . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → 𝐹 ∈ 𝐷) | 
| 18 |  | mapdh8h.mn | . . . . . . 7
⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | 
| 19 | 18 | 3ad2ant1 1133 | . . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | 
| 20 |  | mapdh9a.x | . . . . . . 7
⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | 
| 21 | 20 | 3ad2ant1 1133 | . . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → 𝑋 ∈ (𝑉 ∖ { 0 })) | 
| 22 |  | eqid 2736 | . . . . . . 7
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) | 
| 23 | 1, 2, 14 | dvhlmod 41113 | . . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ LMod) | 
| 24 | 23 | 3ad2ant1 1133 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → 𝑈 ∈ LMod) | 
| 25 | 20 | eldifad 3962 | . . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ 𝑉) | 
| 26 |  | mapdh9a.t | . . . . . . . . 9
⊢ (𝜑 → 𝑇 ∈ 𝑉) | 
| 27 | 3, 22, 6, 23, 25, 26 | lspprcl 20977 | . . . . . . . 8
⊢ (𝜑 → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈)) | 
| 28 | 27 | 3ad2ant1 1133 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈)) | 
| 29 |  | simp2l 1199 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → 𝑧 ∈ 𝑉) | 
| 30 |  | simp3l 1201 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) | 
| 31 | 5, 22, 24, 28, 29, 30 | lssneln0 20952 | . . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → 𝑧 ∈ (𝑉 ∖ { 0 })) | 
| 32 |  | simp2r 1200 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → 𝑤 ∈ 𝑉) | 
| 33 |  | simp3r 1202 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇})) | 
| 34 | 5, 22, 24, 28, 32, 33 | lssneln0 20952 | . . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → 𝑤 ∈ (𝑉 ∖ { 0 })) | 
| 35 | 1, 2, 14 | dvhlvec 41112 | . . . . . . . . . 10
⊢ (𝜑 → 𝑈 ∈ LVec) | 
| 36 | 35 | 3ad2ant1 1133 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → 𝑈 ∈ LVec) | 
| 37 | 25 | 3ad2ant1 1133 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → 𝑋 ∈ 𝑉) | 
| 38 | 26 | 3ad2ant1 1133 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → 𝑇 ∈ 𝑉) | 
| 39 | 3, 6, 36, 29, 37, 38, 30 | lspindpi 21135 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) | 
| 40 | 39 | simpld 494 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})) | 
| 41 | 40 | necomd 2995 | . . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑧})) | 
| 42 | 3, 6, 36, 32, 37, 38, 33 | lspindpi 21135 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))) | 
| 43 | 42 | simpld 494 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋})) | 
| 44 | 43 | necomd 2995 | . . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤})) | 
| 45 | 39 | simprd 495 | . . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})) | 
| 46 | 42 | simprd 495 | . . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) | 
| 47 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 15, 17, 19, 21, 31, 34, 41, 44, 45, 46, 38 | mapdh8 41791 | . . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐼‘〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉)) | 
| 48 | 47 | 3exp 1119 | . . . 4
⊢ (𝜑 → ((𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐼‘〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉)))) | 
| 49 | 48 | ralrimivv 3199 | . . 3
⊢ (𝜑 → ∀𝑧 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐼‘〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉))) | 
| 50 | 1, 2, 3, 6, 14, 25, 26 | dvh3dim 41449 | . . . . 5
⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) | 
| 51 | 14 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 52 | 16 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝐹 ∈ 𝐷) | 
| 53 | 18 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | 
| 54 | 20 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑋 ∈ (𝑉 ∖ { 0 })) | 
| 55 |  | simplr 768 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ 𝑉) | 
| 56 | 35 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LVec) | 
| 57 | 25 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑋 ∈ 𝑉) | 
| 58 | 26 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑇 ∈ 𝑉) | 
| 59 |  | simpr 484 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) | 
| 60 | 3, 6, 56, 55, 57, 58, 59 | lspindpi 21135 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) | 
| 61 | 60 | simpld 494 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})) | 
| 62 | 61 | necomd 2995 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑧})) | 
| 63 | 10, 13, 1, 12, 2, 3,
4, 5, 6, 7,
8, 9, 11, 51, 52, 53, 54, 55, 62 | mapdhcl 41730 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐼‘〈𝑋, 𝐹, 𝑧〉) ∈ 𝐷) | 
| 64 |  | eqidd 2737 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐼‘〈𝑋, 𝐹, 𝑧〉) = (𝐼‘〈𝑋, 𝐹, 𝑧〉)) | 
| 65 | 23 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LMod) | 
| 66 | 27 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈)) | 
| 67 | 5, 22, 65, 66, 55, 59 | lssneln0 20952 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ (𝑉 ∖ { 0 })) | 
| 68 | 10, 13, 1, 12, 2, 3,
4, 5, 6, 7,
8, 9, 11, 51, 52, 53, 54, 67, 63, 62 | mapdheq 41731 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ((𝐼‘〈𝑋, 𝐹, 𝑧〉) = (𝐼‘〈𝑋, 𝐹, 𝑧〉) ↔ ((𝑀‘(𝑁‘{𝑧})) = (𝐽‘{(𝐼‘〈𝑋, 𝐹, 𝑧〉)}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑧)})) = (𝐽‘{(𝐹𝑅(𝐼‘〈𝑋, 𝐹, 𝑧〉))})))) | 
| 69 | 64, 68 | mpbid 232 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ((𝑀‘(𝑁‘{𝑧})) = (𝐽‘{(𝐼‘〈𝑋, 𝐹, 𝑧〉)}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑧)})) = (𝐽‘{(𝐹𝑅(𝐼‘〈𝑋, 𝐹, 𝑧〉))}))) | 
| 70 | 69 | simpld 494 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑀‘(𝑁‘{𝑧})) = (𝐽‘{(𝐼‘〈𝑋, 𝐹, 𝑧〉)})) | 
| 71 | 60 | simprd 495 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})) | 
| 72 | 10, 13, 1, 12, 2, 3,
4, 5, 6, 7,
8, 9, 11, 51, 63, 70, 67, 58, 71 | mapdhcl 41730 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) ∈ 𝐷) | 
| 73 | 72 | ex 412 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) ∈ 𝐷)) | 
| 74 | 73 | ancld 550 | . . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) ∈ 𝐷))) | 
| 75 | 74 | reximdva 3167 | . . . . 5
⊢ (𝜑 → (∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) ∈ 𝐷))) | 
| 76 | 50, 75 | mpd 15 | . . . 4
⊢ (𝜑 → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) ∈ 𝐷)) | 
| 77 |  | eleq1w 2823 | . . . . . 6
⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ↔ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) | 
| 78 | 77 | notbid 318 | . . . . 5
⊢ (𝑧 = 𝑤 → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ↔ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇}))) | 
| 79 |  | oteq1 4881 | . . . . . . 7
⊢ (𝑧 = 𝑤 → 〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉 = 〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) | 
| 80 |  | oteq3 4883 | . . . . . . . . 9
⊢ (𝑧 = 𝑤 → 〈𝑋, 𝐹, 𝑧〉 = 〈𝑋, 𝐹, 𝑤〉) | 
| 81 | 80 | fveq2d 6909 | . . . . . . . 8
⊢ (𝑧 = 𝑤 → (𝐼‘〈𝑋, 𝐹, 𝑧〉) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) | 
| 82 | 81 | oteq2d 4885 | . . . . . . 7
⊢ (𝑧 = 𝑤 → 〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉 = 〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉) | 
| 83 | 79, 82 | eqtrd 2776 | . . . . . 6
⊢ (𝑧 = 𝑤 → 〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉 = 〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉) | 
| 84 | 83 | fveq2d 6909 | . . . . 5
⊢ (𝑧 = 𝑤 → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐼‘〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉)) | 
| 85 | 78, 84 | reusv3 5404 | . . . 4
⊢
(∃𝑧 ∈
𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) ∈ 𝐷) → (∀𝑧 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐼‘〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉)) ↔ ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)))) | 
| 86 | 76, 85 | syl 17 | . . 3
⊢ (𝜑 → (∀𝑧 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐼‘〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉)) ↔ ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)))) | 
| 87 | 49, 86 | mpbid 232 | . 2
⊢ (𝜑 → ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) | 
| 88 |  | reusv1 5396 | . . 3
⊢
(∃𝑧 ∈
𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → (∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) ↔ ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)))) | 
| 89 | 50, 88 | syl 17 | . 2
⊢ (𝜑 → (∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) ↔ ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)))) | 
| 90 | 87, 89 | mpbird 257 | 1
⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) |