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Theorem dvhgrp 40612
Description: The full vector space π‘ˆ constructed from a Hilbert lattice 𝐾 (given a fiducial hyperplane π‘Š) is a group. (Contributed by NM, 19-Oct-2013.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
dvhgrp.b 𝐡 = (Baseβ€˜πΎ)
dvhgrp.h 𝐻 = (LHypβ€˜πΎ)
dvhgrp.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dvhgrp.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dvhgrp.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
dvhgrp.d 𝐷 = (Scalarβ€˜π‘ˆ)
dvhgrp.p ⨣ = (+gβ€˜π·)
dvhgrp.a + = (+gβ€˜π‘ˆ)
dvhgrp.o 0 = (0gβ€˜π·)
dvhgrp.i 𝐼 = (invgβ€˜π·)
Assertion
Ref Expression
dvhgrp ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ Grp)
Proof of Theorem dvhgrp
Dummy variables 𝑓 𝑔 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvhgrp.h . . . 4 𝐻 = (LHypβ€˜πΎ)
2 dvhgrp.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
3 dvhgrp.e . . . 4 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
4 dvhgrp.u . . . 4 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
5 eqid 2728 . . . 4 (Baseβ€˜π‘ˆ) = (Baseβ€˜π‘ˆ)
61, 2, 3, 4, 5dvhvbase 40592 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π‘ˆ) = (𝑇 Γ— 𝐸))
76eqcomd 2734 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝑇 Γ— 𝐸) = (Baseβ€˜π‘ˆ))
8 dvhgrp.a . . 3 + = (+gβ€˜π‘ˆ)
98a1i 11 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ + = (+gβ€˜π‘ˆ))
10 dvhgrp.d . . . 4 𝐷 = (Scalarβ€˜π‘ˆ)
11 dvhgrp.p . . . 4 ⨣ = (+gβ€˜π·)
121, 2, 3, 4, 10, 11, 8dvhvaddcl 40600 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 Γ— 𝐸) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑓 + 𝑔) ∈ (𝑇 Γ— 𝐸))
13123impb 1112 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ (𝑓 + 𝑔) ∈ (𝑇 Γ— 𝐸))
141, 2, 3, 4, 10, 11, 8dvhvaddass 40602 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 Γ— 𝐸) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸) ∧ β„Ž ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑓 + 𝑔) + β„Ž) = (𝑓 + (𝑔 + β„Ž)))
15 dvhgrp.b . . . 4 𝐡 = (Baseβ€˜πΎ)
1615, 1, 2idltrn 39655 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ 𝑇)
17 eqid 2728 . . . . . . . 8 ((EDRingβ€˜πΎ)β€˜π‘Š) = ((EDRingβ€˜πΎ)β€˜π‘Š)
181, 17, 4, 10dvhsca 40587 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š))
191, 17erngdv 40498 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((EDRingβ€˜πΎ)β€˜π‘Š) ∈ DivRing)
2018, 19eqeltrd 2829 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ DivRing)
21 drnggrp 20641 . . . . . 6 (𝐷 ∈ DivRing β†’ 𝐷 ∈ Grp)
2220, 21syl 17 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Grp)
23 eqid 2728 . . . . . 6 (Baseβ€˜π·) = (Baseβ€˜π·)
24 dvhgrp.o . . . . . 6 0 = (0gβ€˜π·)
2523, 24grpidcl 18929 . . . . 5 (𝐷 ∈ Grp β†’ 0 ∈ (Baseβ€˜π·))
2622, 25syl 17 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 ∈ (Baseβ€˜π·))
271, 3, 4, 10, 23dvhbase 40588 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π·) = 𝐸)
2826, 27eleqtrd 2831 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 ∈ 𝐸)
29 opelxpi 5719 . . 3 ((( I β†Ύ 𝐡) ∈ 𝑇 ∧ 0 ∈ 𝐸) β†’ ⟨( I β†Ύ 𝐡), 0 ⟩ ∈ (𝑇 Γ— 𝐸))
3016, 28, 29syl2anc 582 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ⟨( I β†Ύ 𝐡), 0 ⟩ ∈ (𝑇 Γ— 𝐸))
31 simpl 481 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
3216adantr 479 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ ( I β†Ύ 𝐡) ∈ 𝑇)
3328adantr 479 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ 0 ∈ 𝐸)
34 xp1st 8031 . . . . . 6 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ (1st β€˜π‘“) ∈ 𝑇)
3534adantl 480 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (1st β€˜π‘“) ∈ 𝑇)
36 xp2nd 8032 . . . . . 6 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜π‘“) ∈ 𝐸)
3736adantl 480 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (2nd β€˜π‘“) ∈ 𝐸)
381, 2, 3, 4, 10, 8, 11dvhopvadd 40598 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (( I β†Ύ 𝐡) ∈ 𝑇 ∧ 0 ∈ 𝐸) ∧ ((1st β€˜π‘“) ∈ 𝑇 ∧ (2nd β€˜π‘“) ∈ 𝐸)) β†’ (⟨( I β†Ύ 𝐡), 0 ⟩ + ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩) = ⟨(( I β†Ύ 𝐡) ∘ (1st β€˜π‘“)), ( 0 ⨣ (2nd β€˜π‘“))⟩)
3931, 32, 33, 35, 37, 38syl122anc 1376 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (⟨( I β†Ύ 𝐡), 0 ⟩ + ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩) = ⟨(( I β†Ύ 𝐡) ∘ (1st β€˜π‘“)), ( 0 ⨣ (2nd β€˜π‘“))⟩)
4015, 1, 2ltrn1o 39629 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (1st β€˜π‘“) ∈ 𝑇) β†’ (1st β€˜π‘“):𝐡–1-1-onto→𝐡)
4135, 40syldan 589 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (1st β€˜π‘“):𝐡–1-1-onto→𝐡)
42 f1of 6844 . . . . . 6 ((1st β€˜π‘“):𝐡–1-1-onto→𝐡 β†’ (1st β€˜π‘“):𝐡⟢𝐡)
43 fcoi2 6777 . . . . . 6 ((1st β€˜π‘“):𝐡⟢𝐡 β†’ (( I β†Ύ 𝐡) ∘ (1st β€˜π‘“)) = (1st β€˜π‘“))
4441, 42, 433syl 18 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (( I β†Ύ 𝐡) ∘ (1st β€˜π‘“)) = (1st β€˜π‘“))
4522adantr 479 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ 𝐷 ∈ Grp)
4627adantr 479 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (Baseβ€˜π·) = 𝐸)
4737, 46eleqtrrd 2832 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (2nd β€˜π‘“) ∈ (Baseβ€˜π·))
4823, 11, 24grplid 18931 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·)) β†’ ( 0 ⨣ (2nd β€˜π‘“)) = (2nd β€˜π‘“))
4945, 47, 48syl2anc 582 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ ( 0 ⨣ (2nd β€˜π‘“)) = (2nd β€˜π‘“))
5044, 49opeq12d 4886 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ ⟨(( I β†Ύ 𝐡) ∘ (1st β€˜π‘“)), ( 0 ⨣ (2nd β€˜π‘“))⟩ = ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)
5139, 50eqtrd 2768 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (⟨( I β†Ύ 𝐡), 0 ⟩ + ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩) = ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)
52 1st2nd2 8038 . . . . 5 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ 𝑓 = ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)
5352adantl 480 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ 𝑓 = ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)
5453oveq2d 7442 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (⟨( I β†Ύ 𝐡), 0 ⟩ + 𝑓) = (⟨( I β†Ύ 𝐡), 0 ⟩ + ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩))
5551, 54, 533eqtr4d 2778 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (⟨( I β†Ύ 𝐡), 0 ⟩ + 𝑓) = 𝑓)
561, 2ltrncnv 39651 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (1st β€˜π‘“) ∈ 𝑇) β†’ β—‘(1st β€˜π‘“) ∈ 𝑇)
5735, 56syldan 589 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ β—‘(1st β€˜π‘“) ∈ 𝑇)
58 dvhgrp.i . . . . . 6 𝐼 = (invgβ€˜π·)
5923, 58grpinvcl 18951 . . . . 5 ((𝐷 ∈ Grp ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·)) β†’ (πΌβ€˜(2nd β€˜π‘“)) ∈ (Baseβ€˜π·))
6045, 47, 59syl2anc 582 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (πΌβ€˜(2nd β€˜π‘“)) ∈ (Baseβ€˜π·))
6160, 46eleqtrd 2831 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (πΌβ€˜(2nd β€˜π‘“)) ∈ 𝐸)
62 opelxpi 5719 . . 3 ((β—‘(1st β€˜π‘“) ∈ 𝑇 ∧ (πΌβ€˜(2nd β€˜π‘“)) ∈ 𝐸) β†’ βŸ¨β—‘(1st β€˜π‘“), (πΌβ€˜(2nd β€˜π‘“))⟩ ∈ (𝑇 Γ— 𝐸))
6357, 61, 62syl2anc 582 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ βŸ¨β—‘(1st β€˜π‘“), (πΌβ€˜(2nd β€˜π‘“))⟩ ∈ (𝑇 Γ— 𝐸))
6453oveq2d 7442 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (βŸ¨β—‘(1st β€˜π‘“), (πΌβ€˜(2nd β€˜π‘“))⟩ + 𝑓) = (βŸ¨β—‘(1st β€˜π‘“), (πΌβ€˜(2nd β€˜π‘“))⟩ + ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩))
651, 2, 3, 4, 10, 8, 11dvhopvadd 40598 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (β—‘(1st β€˜π‘“) ∈ 𝑇 ∧ (πΌβ€˜(2nd β€˜π‘“)) ∈ 𝐸) ∧ ((1st β€˜π‘“) ∈ 𝑇 ∧ (2nd β€˜π‘“) ∈ 𝐸)) β†’ (βŸ¨β—‘(1st β€˜π‘“), (πΌβ€˜(2nd β€˜π‘“))⟩ + ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩) = ⟨(β—‘(1st β€˜π‘“) ∘ (1st β€˜π‘“)), ((πΌβ€˜(2nd β€˜π‘“)) ⨣ (2nd β€˜π‘“))⟩)
6631, 57, 61, 35, 37, 65syl122anc 1376 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (βŸ¨β—‘(1st β€˜π‘“), (πΌβ€˜(2nd β€˜π‘“))⟩ + ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩) = ⟨(β—‘(1st β€˜π‘“) ∘ (1st β€˜π‘“)), ((πΌβ€˜(2nd β€˜π‘“)) ⨣ (2nd β€˜π‘“))⟩)
67 f1ococnv1 6873 . . . . . 6 ((1st β€˜π‘“):𝐡–1-1-onto→𝐡 β†’ (β—‘(1st β€˜π‘“) ∘ (1st β€˜π‘“)) = ( I β†Ύ 𝐡))
6841, 67syl 17 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (β—‘(1st β€˜π‘“) ∘ (1st β€˜π‘“)) = ( I β†Ύ 𝐡))
6923, 11, 24, 58grplinv 18953 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·)) β†’ ((πΌβ€˜(2nd β€˜π‘“)) ⨣ (2nd β€˜π‘“)) = 0 )
7045, 47, 69syl2anc 582 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ ((πΌβ€˜(2nd β€˜π‘“)) ⨣ (2nd β€˜π‘“)) = 0 )
7168, 70opeq12d 4886 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ ⟨(β—‘(1st β€˜π‘“) ∘ (1st β€˜π‘“)), ((πΌβ€˜(2nd β€˜π‘“)) ⨣ (2nd β€˜π‘“))⟩ = ⟨( I β†Ύ 𝐡), 0 ⟩)
7266, 71eqtrd 2768 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (βŸ¨β—‘(1st β€˜π‘“), (πΌβ€˜(2nd β€˜π‘“))⟩ + ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩) = ⟨( I β†Ύ 𝐡), 0 ⟩)
7364, 72eqtrd 2768 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (βŸ¨β—‘(1st β€˜π‘“), (πΌβ€˜(2nd β€˜π‘“))⟩ + 𝑓) = ⟨( I β†Ύ 𝐡), 0 ⟩)
747, 9, 13, 14, 30, 55, 63, 73isgrpd 18922 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4638   I cid 5579   Γ— cxp 5680  β—‘ccnv 5681   β†Ύ cres 5684   ∘ ccom 5686  βŸΆwf 6549  β€“1-1-ontoβ†’wf1o 6552  β€˜cfv 6553  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  Basecbs 17187  +gcplusg 17240  Scalarcsca 17243  0gc0g 17428  Grpcgrp 18897  invgcminusg 18898  DivRingcdr 20631  HLchlt 38854  LHypclh 39489  LTrncltrn 39606  TEndoctendo 40257  EDRingcedring 40258  DVecHcdvh 40583
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-riotaBAD 38457
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-tpos 8238  df-undef 8285  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  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 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-0g 17430  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-minusg 18901  df-cmn 19744  df-abl 19745  df-mgp 20082  df-rng 20100  df-ur 20129  df-ring 20182  df-oppr 20280  df-dvdsr 20303  df-unit 20304  df-invr 20334  df-dvr 20347  df-drng 20633  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-llines 39003  df-lplanes 39004  df-lvols 39005  df-lines 39006  df-psubsp 39008  df-pmap 39009  df-padd 39301  df-lhyp 39493  df-laut 39494  df-ldil 39609  df-ltrn 39610  df-trl 39664  df-tendo 40260  df-edring 40262  df-dvech 40584
This theorem is referenced by:  dvhlveclem  40613
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