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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh8d | Structured version Visualization version GIF version |
Description: Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-May-2015.) |
Ref | Expression |
---|---|
mapdh8a.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdh8a.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdh8a.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdh8a.s | ⊢ − = (-g‘𝑈) |
mapdh8a.o | ⊢ 0 = (0g‘𝑈) |
mapdh8a.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdh8a.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdh8a.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdh8a.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdh8a.q | ⊢ 𝑄 = (0g‘𝐶) |
mapdh8a.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdh8a.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdh8a.i | ⊢ 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)}))))) |
mapdh8a.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdh8d.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh8d.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdh8b.eg | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
mapdh8d.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdh8d.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdh8d.xt | ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) |
mapdh8d.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) |
mapdh8d.w | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
mapdh8d.wt | ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) |
mapdh8d.ut | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) |
mapdh8d.vw | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) |
mapdh8d.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) |
Ref | Expression |
---|---|
mapdh8d | ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh8a.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdh8a.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | mapdh8a.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
4 | mapdh8a.s | . . . 4 ⊢ − = (-g‘𝑈) | |
5 | mapdh8a.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
6 | mapdh8a.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | mapdh8a.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | mapdh8a.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
9 | mapdh8a.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
10 | mapdh8a.q | . . . 4 ⊢ 𝑄 = (0g‘𝐶) | |
11 | mapdh8a.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
12 | mapdh8a.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
13 | mapdh8a.i | . . . 4 ⊢ 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 | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | 14 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
16 | mapdh8b.eg | . . . . . 6 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | |
17 | mapdh8d.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
18 | mapdh8d.mn | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
19 | mapdh8d.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
20 | mapdh8d.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
21 | 20 | eldifad 3733 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
22 | 1, 2, 14 | dvhlvec 36912 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LVec) |
23 | 19 | eldifad 3733 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
24 | mapdh8d.w | . . . . . . . . . 10 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
25 | 24 | eldifad 3733 | . . . . . . . . 9 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
26 | mapdh8d.xn | . . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) | |
27 | 3, 6, 22, 23, 21, 25, 26 | lspindpi 19345 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤}))) |
28 | 27 | simpld 476 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
29 | 10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 17, 18, 19, 21, 28 | mapdhcl 37530 | . . . . . 6 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) |
30 | 16, 29 | eqeltrrd 2850 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐷) |
31 | 30 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝐺 ∈ 𝐷) |
32 | 10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 17, 18, 19, 20, 30, 28 | mapdheq 37531 | . . . . . . 7 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))) |
33 | 16, 32 | mpbid 222 | . . . . . 6 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) |
34 | 33 | simpld 476 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) |
35 | 34 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) |
36 | mapdh8d.vw | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) | |
37 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 16, 19, 20, 36, 24, 26 | mapdh8a 37578 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑤〉) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) |
38 | 37 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘〈𝑌, 𝐺, 𝑤〉) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) |
39 | 20 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
40 | 24 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑤 ∈ (𝑉 ∖ { 0 })) |
41 | mapdh8d.wt | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) | |
42 | 41 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) |
43 | mapdh8d.xt | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) | |
44 | 43 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑇 ∈ (𝑉 ∖ { 0 })) |
45 | 36 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) |
46 | simpr 471 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) | |
47 | 26 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) |
48 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 31, 35, 38, 39, 40, 42, 44, 45, 46, 47 | mapdh8b 37583 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉) = (𝐼‘〈𝑌, 𝐺, 𝑇〉)) |
49 | 17 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝐹 ∈ 𝐷) |
50 | 18 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
51 | eqidd 2771 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘〈𝑋, 𝐹, 𝑤〉) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) | |
52 | 19 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
53 | mapdh8d.yz | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) | |
54 | 53 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) |
55 | mapdh8d.ut | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) | |
56 | 55 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) |
57 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 49, 50, 51, 52, 39, 44, 54, 40, 42, 56, 45, 46, 47 | mapdh8c 37584 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) |
58 | 48, 57 | eqtr3d 2806 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) |
59 | 14 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
60 | 17 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝐹 ∈ 𝐷) |
61 | 18 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
62 | 16 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
63 | 19 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
64 | 20 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
65 | 53 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) |
66 | 43 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑇 ∈ (𝑉 ∖ { 0 })) |
67 | simpr 471 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) | |
68 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 59, 60, 61, 62, 63, 64, 65, 66, 67 | mapdh8a 37578 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) |
69 | 58, 68 | pm2.61dan 796 | 1 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 = wceq 1630 ∈ wcel 2144 ≠ wne 2942 Vcvv 3349 ∖ cdif 3718 ifcif 4223 {csn 4314 {cpr 4316 〈cotp 4322 ↦ cmpt 4861 ‘cfv 6031 ℩crio 6752 (class class class)co 6792 1st c1st 7312 2nd c2nd 7313 Basecbs 16063 0gc0g 16307 -gcsg 17631 LSpanclspn 19183 HLchlt 35152 LHypclh 35785 DVecHcdvh 36881 LCDualclcd 37389 mapdcmpd 37427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-riotaBAD 34754 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-fal 1636 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-ot 4323 df-uni 4573 df-int 4610 df-iun 4654 df-iin 4655 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-of 7043 df-om 7212 df-1st 7314 df-2nd 7315 df-tpos 7503 df-undef 7550 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-n0 11494 df-z 11579 df-uz 11888 df-fz 12533 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-sca 16164 df-vsca 16165 df-0g 16309 df-mre 16453 df-mrc 16454 df-acs 16456 df-preset 17135 df-poset 17153 df-plt 17165 df-lub 17181 df-glb 17182 df-join 17183 df-meet 17184 df-p0 17246 df-p1 17247 df-lat 17253 df-clat 17315 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-grp 17632 df-minusg 17633 df-sbg 17634 df-subg 17798 df-cntz 17956 df-oppg 17982 df-lsm 18257 df-cmn 18401 df-abl 18402 df-mgp 18697 df-ur 18709 df-ring 18756 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-invr 18879 df-dvr 18890 df-drng 18958 df-lmod 19074 df-lss 19142 df-lsp 19184 df-lvec 19315 df-lsatoms 34778 df-lshyp 34779 df-lcv 34821 df-lfl 34860 df-lkr 34888 df-ldual 34926 df-oposet 34978 df-ol 34980 df-oml 34981 df-covers 35068 df-ats 35069 df-atl 35100 df-cvlat 35124 df-hlat 35153 df-llines 35299 df-lplanes 35300 df-lvols 35301 df-lines 35302 df-psubsp 35304 df-pmap 35305 df-padd 35597 df-lhyp 35789 df-laut 35790 df-ldil 35905 df-ltrn 35906 df-trl 35961 df-tgrp 36545 df-tendo 36557 df-edring 36559 df-dveca 36805 df-disoa 36832 df-dvech 36882 df-dib 36942 df-dic 36976 df-dih 37032 df-doch 37151 df-djh 37198 df-lcdual 37390 df-mapd 37428 |
This theorem is referenced by: mapdh8e 37587 |
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