| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1eulem | Structured version Visualization version GIF version | ||
| Description: Lemma for hdmap1eu 41826. TODO: combine with hdmap1eu 41826 or at least share some hypotheses. (Contributed by NM, 15-May-2015.) |
| Ref | Expression |
|---|---|
| hdmap1eulem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| hdmap1eulem.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
| hdmap1eulem.v | ⊢ 𝑉 = (Base‘𝑈) |
| hdmap1eulem.s | ⊢ − = (-g‘𝑈) |
| hdmap1eulem.o | ⊢ 0 = (0g‘𝑈) |
| hdmap1eulem.n | ⊢ 𝑁 = (LSpan‘𝑈) |
| hdmap1eulem.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
| hdmap1eulem.d | ⊢ 𝐷 = (Base‘𝐶) |
| hdmap1eulem.r | ⊢ 𝑅 = (-g‘𝐶) |
| hdmap1eulem.q | ⊢ 𝑄 = (0g‘𝐶) |
| hdmap1eulem.j | ⊢ 𝐽 = (LSpan‘𝐶) |
| hdmap1eulem.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
| hdmap1eulem.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
| hdmap1eulem.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| hdmap1eulem.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
| hdmap1eulem.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
| hdmap1eulem.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
| hdmap1eulem.y | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
| hdmap1eulem.l | ⊢ 𝐿 = (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)}))))) |
| Ref | Expression |
|---|---|
| hdmap1eulem | ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hdmap1eulem.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 2 | hdmap1eulem.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 3 | hdmap1eulem.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
| 4 | hdmap1eulem.s | . . 3 ⊢ − = (-g‘𝑈) | |
| 5 | hdmap1eulem.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
| 6 | hdmap1eulem.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
| 7 | hdmap1eulem.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
| 8 | hdmap1eulem.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
| 9 | hdmap1eulem.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
| 10 | hdmap1eulem.q | . . 3 ⊢ 𝑄 = (0g‘𝐶) | |
| 11 | hdmap1eulem.j | . . 3 ⊢ 𝐽 = (LSpan‘𝐶) | |
| 12 | hdmap1eulem.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
| 13 | hdmap1eulem.l | . . 3 ⊢ 𝐿 = (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 | hdmap1eulem.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 15 | hdmap1eulem.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
| 16 | hdmap1eulem.mn | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
| 17 | hdmap1eulem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
| 18 | hdmap1eulem.y | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
| 19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | mapdh9a 41791 | . 2 ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) |
| 20 | hdmap1eulem.i | . . . . . . . . . 10 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
| 21 | 14 | ad2antrr 726 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 22 | 17 | ad2antrr 726 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
| 23 | 15 | ad2antrr 726 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → 𝐹 ∈ 𝐷) |
| 24 | simplr 769 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → 𝑧 ∈ 𝑉) | |
| 25 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 21, 22, 23, 24, 13 | hdmap1valc 41805 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → (𝐼‘〈𝑋, 𝐹, 𝑧〉) = (𝐿‘〈𝑋, 𝐹, 𝑧〉)) |
| 26 | 25 | oteq2d 4886 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → 〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉 = 〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) |
| 27 | 26 | fveq2d 6910 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐼‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) |
| 28 | elun1 4182 | . . . . . . . . 9 ⊢ (𝑧 ∈ (𝑁‘{𝑋}) → 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) | |
| 29 | 28 | con3i 154 | . . . . . . . 8 ⊢ (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑋})) |
| 30 | 14 | ad2antrr 726 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 31 | eqid 2737 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
| 32 | 1, 2, 14 | dvhlmod 41112 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 33 | 32 | ad2antrr 726 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑈 ∈ LMod) |
| 34 | 17 | eldifad 3963 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| 35 | 34 | ad2antrr 726 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑋 ∈ 𝑉) |
| 36 | 3, 31, 6 | lspsncl 20975 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
| 37 | 33, 35, 36 | syl2anc 584 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
| 38 | simplr 769 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ 𝑉) | |
| 39 | simpr 484 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → ¬ 𝑧 ∈ (𝑁‘{𝑋})) | |
| 40 | 5, 31, 33, 37, 38, 39 | lssneln0 20951 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ (𝑉 ∖ { 0 })) |
| 41 | 15 | ad2antrr 726 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝐹 ∈ 𝐷) |
| 42 | 16 | ad2antrr 726 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
| 43 | 17 | ad2antrr 726 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
| 44 | 3, 6, 33, 38, 35, 39 | lspsnne2 21120 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})) |
| 45 | 44 | necomd 2996 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑧})) |
| 46 | 10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 30, 41, 42, 43, 38, 45 | mapdhcl 41729 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝐿‘〈𝑋, 𝐹, 𝑧〉) ∈ 𝐷) |
| 47 | 18 | ad2antrr 726 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑇 ∈ 𝑉) |
| 48 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 30, 40, 46, 47, 13 | hdmap1valc 41805 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝐼‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) |
| 49 | 29, 48 | sylan2 593 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → (𝐼‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) |
| 50 | 27, 49 | eqtrd 2777 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) |
| 51 | 50 | eqeq2d 2748 | . . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → (𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) ↔ 𝑦 = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) |
| 52 | 51 | pm5.74da 804 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) ↔ (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)))) |
| 53 | 52 | ralbidva 3176 | . . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)))) |
| 54 | 53 | reubidv 3398 | . 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) ↔ ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)))) |
| 55 | 19, 54 | mpbird 257 | 1 ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃!wreu 3378 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ifcif 4525 {csn 4626 〈cotp 4634 ↦ cmpt 5225 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 Basecbs 17247 0gc0g 17484 -gcsg 18953 LModclmod 20858 LSubSpclss 20929 LSpanclspn 20969 HLchlt 39351 LHypclh 39986 DVecHcdvh 41080 LCDualclcd 41588 mapdcmpd 41626 HDMap1chdma1 41793 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-mre 17629 df-mrc 17630 df-acs 17632 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-nzr 20513 df-rlreg 20694 df-domn 20695 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-lshyp 38978 df-lcv 39020 df-lfl 39059 df-lkr 39087 df-ldual 39125 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tgrp 40745 df-tendo 40757 df-edring 40759 df-dveca 41005 df-disoa 41031 df-dvech 41081 df-dib 41141 df-dic 41175 df-dih 41231 df-doch 41350 df-djh 41397 df-lcdual 41589 df-mapd 41627 df-hdmap1 41795 |
| This theorem is referenced by: hdmap1eu 41826 |
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