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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 39120. TODO: combine with hdmap1eu 39120 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 39085 | . 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 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
22 | 17 | ad2antrr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
23 | 15 | ad2antrr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → 𝐹 ∈ 𝐷) |
24 | simplr 768 | . . . . . . . . . 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 39099 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → (𝐼‘〈𝑋, 𝐹, 𝑧〉) = (𝐿‘〈𝑋, 𝐹, 𝑧〉)) |
26 | 25 | oteq2d 4778 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) → 〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉 = 〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) |
27 | 26 | fveq2d 6649 | . . . . . . 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 4103 | . . . . . . . . 9 ⊢ (𝑧 ∈ (𝑁‘{𝑋}) → 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))) | |
29 | 28 | con3i 157 | . . . . . . . 8 ⊢ (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑋})) |
30 | 14 | ad2antrr 725 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
31 | eqid 2798 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
32 | 1, 2, 14 | dvhlmod 38406 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod) |
33 | 32 | ad2antrr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑈 ∈ LMod) |
34 | 17 | eldifad 3893 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
35 | 34 | ad2antrr 725 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑋 ∈ 𝑉) |
36 | 3, 31, 6 | lspsncl 19742 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
37 | 33, 35, 36 | syl2anc 587 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
38 | simplr 768 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ 𝑉) | |
39 | simpr 488 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → ¬ 𝑧 ∈ (𝑁‘{𝑋})) | |
40 | 5, 31, 33, 37, 38, 39 | lssneln0 19717 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ (𝑉 ∖ { 0 })) |
41 | 15 | ad2antrr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝐹 ∈ 𝐷) |
42 | 16 | ad2antrr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
43 | 17 | ad2antrr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
44 | 3, 6, 33, 38, 35, 39 | lspsnne2 19883 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})) |
45 | 44 | necomd 3042 | . . . . . . . . . 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 39023 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝐿‘〈𝑋, 𝐹, 𝑧〉) ∈ 𝐷) |
47 | 18 | ad2antrr 725 | . . . . . . . . 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 39099 | . . . . . . . 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 595 | . . . . . . 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 2833 | . . . . . 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 2809 | . . . . 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 803 | . . . 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 3161 | . . 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 3342 | . 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 260 | 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 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃!wreu 3108 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ifcif 4425 {csn 4525 〈cotp 4533 ↦ cmpt 5110 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 Basecbs 16475 0gc0g 16705 -gcsg 18097 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 HLchlt 36646 LHypclh 37280 DVecHcdvh 38374 LCDualclcd 38882 mapdcmpd 38920 HDMap1chdma1 39087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-lshyp 36273 df-lcv 36315 df-lfl 36354 df-lkr 36382 df-ldual 36420 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tgrp 38039 df-tendo 38051 df-edring 38053 df-dveca 38299 df-disoa 38325 df-dvech 38375 df-dib 38435 df-dic 38469 df-dih 38525 df-doch 38644 df-djh 38691 df-lcdual 38883 df-mapd 38921 df-hdmap1 39089 |
This theorem is referenced by: hdmap1eu 39120 |
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