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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 41429. TODO: combine with hdmap1eu 41429 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 41394 | . 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 724 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
22 | 17 | ad2antrr 724 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
23 | 15 | ad2antrr 724 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → 𝐹 ∈ 𝐷) |
24 | simplr 767 | . . . . . . . . . 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 41408 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → (𝐼‘〈𝑋, 𝐹, 𝑧〉) = (𝐿‘〈𝑋, 𝐹, 𝑧〉)) |
26 | 25 | oteq2d 4888 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → 〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉 = 〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) |
27 | 26 | fveq2d 6900 | . . . . . . 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 4174 | . . . . . . . . 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 724 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
31 | eqid 2725 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
32 | 1, 2, 14 | dvhlmod 40715 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod) |
33 | 32 | ad2antrr 724 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑈 ∈ LMod) |
34 | 17 | eldifad 3956 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
35 | 34 | ad2antrr 724 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑋 ∈ 𝑉) |
36 | 3, 31, 6 | lspsncl 20878 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
37 | 33, 35, 36 | syl2anc 582 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
38 | simplr 767 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ 𝑉) | |
39 | simpr 483 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → ¬ 𝑧 ∈ (𝑁‘{𝑋})) | |
40 | 5, 31, 33, 37, 38, 39 | lssneln0 20854 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ (𝑉 ∖ { 0 })) |
41 | 15 | ad2antrr 724 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝐹 ∈ 𝐷) |
42 | 16 | ad2antrr 724 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
43 | 17 | ad2antrr 724 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
44 | 3, 6, 33, 38, 35, 39 | lspsnne2 21023 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})) |
45 | 44 | necomd 2985 | . . . . . . . . . 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 41332 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝐿‘〈𝑋, 𝐹, 𝑧〉) ∈ 𝐷) |
47 | 18 | ad2antrr 724 | . . . . . . . . 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 41408 | . . . . . . . 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 591 | . . . . . . 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 2765 | . . . . . 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 2736 | . . . . 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 802 | . . . 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 3165 | . . 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 3381 | . 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 256 | 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 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃!wreu 3361 Vcvv 3461 ∖ cdif 3941 ∪ cun 3942 ifcif 4530 {csn 4630 〈cotp 4638 ↦ cmpt 5232 ‘cfv 6549 ℩crio 7374 (class class class)co 7419 1st c1st 7992 2nd c2nd 7993 Basecbs 17188 0gc0g 17429 -gcsg 18905 LModclmod 20760 LSubSpclss 20832 LSpanclspn 20872 HLchlt 38954 LHypclh 39589 DVecHcdvh 40683 LCDualclcd 41191 mapdcmpd 41229 HDMap1chdma1 41396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-riotaBAD 38557 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-sca 17257 df-vsca 17258 df-0g 17431 df-mre 17574 df-mrc 17575 df-acs 17577 df-proset 18295 df-poset 18313 df-plt 18330 df-lub 18346 df-glb 18347 df-join 18348 df-meet 18349 df-p0 18425 df-p1 18426 df-lat 18432 df-clat 18499 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18749 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19091 df-cntz 19285 df-oppg 19314 df-lsm 19608 df-cmn 19754 df-abl 19755 df-mgp 20092 df-rng 20110 df-ur 20139 df-ring 20192 df-oppr 20290 df-dvdsr 20313 df-unit 20314 df-invr 20344 df-dvr 20357 df-drng 20643 df-lmod 20762 df-lss 20833 df-lsp 20873 df-lvec 21005 df-lsatoms 38580 df-lshyp 38581 df-lcv 38623 df-lfl 38662 df-lkr 38690 df-ldual 38728 df-oposet 38780 df-ol 38782 df-oml 38783 df-covers 38870 df-ats 38871 df-atl 38902 df-cvlat 38926 df-hlat 38955 df-llines 39103 df-lplanes 39104 df-lvols 39105 df-lines 39106 df-psubsp 39108 df-pmap 39109 df-padd 39401 df-lhyp 39593 df-laut 39594 df-ldil 39709 df-ltrn 39710 df-trl 39764 df-tgrp 40348 df-tendo 40360 df-edring 40362 df-dveca 40608 df-disoa 40634 df-dvech 40684 df-dib 40744 df-dic 40778 df-dih 40834 df-doch 40953 df-djh 41000 df-lcdual 41192 df-mapd 41230 df-hdmap1 41398 |
This theorem is referenced by: hdmap1eu 41429 |
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