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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 41806. TODO: combine with hdmap1eu 41806 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 41771 | . 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 41785 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → (𝐼‘〈𝑋, 𝐹, 𝑧〉) = (𝐿‘〈𝑋, 𝐹, 𝑧〉)) |
26 | 25 | oteq2d 4890 | . . . . . . . 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 4191 | . . . . . . . . 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 2734 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
32 | 1, 2, 14 | dvhlmod 41092 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod) |
33 | 32 | ad2antrr 726 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑈 ∈ LMod) |
34 | 17 | eldifad 3974 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
35 | 34 | ad2antrr 726 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑋 ∈ 𝑉) |
36 | 3, 31, 6 | lspsncl 20992 | . . . . . . . . . . 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 20968 | . . . . . . . . 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 21137 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})) |
45 | 44 | necomd 2993 | . . . . . . . . . 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 41709 | . . . . . . . . 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 41785 | . . . . . . . 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 2774 | . . . . . 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 2745 | . . . . 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 3173 | . . 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 3395 | . 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 1536 ∈ wcel 2105 ∀wral 3058 ∃!wreu 3375 Vcvv 3477 ∖ cdif 3959 ∪ cun 3960 ifcif 4530 {csn 4630 〈cotp 4638 ↦ cmpt 5230 ‘cfv 6562 ℩crio 7386 (class class class)co 7430 1st c1st 8010 2nd c2nd 8011 Basecbs 17244 0gc0g 17485 -gcsg 18965 LModclmod 20874 LSubSpclss 20946 LSpanclspn 20986 HLchlt 39331 LHypclh 39966 DVecHcdvh 41060 LCDualclcd 41568 mapdcmpd 41606 HDMap1chdma1 41773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-riotaBAD 38934 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-undef 8296 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17487 df-mre 17630 df-mrc 17631 df-acs 17633 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cntz 19347 df-oppg 19376 df-lsm 19668 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-nzr 20529 df-rlreg 20710 df-domn 20711 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lvec 21119 df-lsatoms 38957 df-lshyp 38958 df-lcv 39000 df-lfl 39039 df-lkr 39067 df-ldual 39105 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-llines 39480 df-lplanes 39481 df-lvols 39482 df-lines 39483 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141 df-tgrp 40725 df-tendo 40737 df-edring 40739 df-dveca 40985 df-disoa 41011 df-dvech 41061 df-dib 41121 df-dic 41155 df-dih 41211 df-doch 41330 df-djh 41377 df-lcdual 41569 df-mapd 41607 df-hdmap1 41775 |
This theorem is referenced by: hdmap1eu 41806 |
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