Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhgrp Structured version   Visualization version   GIF version

Theorem dvhgrp 39573
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 2737 . . . 4 (Baseβ€˜π‘ˆ) = (Baseβ€˜π‘ˆ)
61, 2, 3, 4, 5dvhvbase 39553 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π‘ˆ) = (𝑇 Γ— 𝐸))
76eqcomd 2743 . 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 39561 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 Γ— 𝐸) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑓 + 𝑔) ∈ (𝑇 Γ— 𝐸))
13123impb 1116 . 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 39563 . 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 38616 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ 𝑇)
17 eqid 2737 . . . . . . . 8 ((EDRingβ€˜πΎ)β€˜π‘Š) = ((EDRingβ€˜πΎ)β€˜π‘Š)
181, 17, 4, 10dvhsca 39548 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š))
191, 17erngdv 39459 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((EDRingβ€˜πΎ)β€˜π‘Š) ∈ DivRing)
2018, 19eqeltrd 2838 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ DivRing)
21 drnggrp 20196 . . . . . 6 (𝐷 ∈ DivRing β†’ 𝐷 ∈ Grp)
2220, 21syl 17 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Grp)
23 eqid 2737 . . . . . 6 (Baseβ€˜π·) = (Baseβ€˜π·)
24 dvhgrp.o . . . . . 6 0 = (0gβ€˜π·)
2523, 24grpidcl 18779 . . . . 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 39549 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π·) = 𝐸)
2826, 27eleqtrd 2840 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 ∈ 𝐸)
29 opelxpi 5671 . . 3 ((( I β†Ύ 𝐡) ∈ 𝑇 ∧ 0 ∈ 𝐸) β†’ ⟨( I β†Ύ 𝐡), 0 ⟩ ∈ (𝑇 Γ— 𝐸))
3016, 28, 29syl2anc 585 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ⟨( I β†Ύ 𝐡), 0 ⟩ ∈ (𝑇 Γ— 𝐸))
31 simpl 484 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
3216adantr 482 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ ( I β†Ύ 𝐡) ∈ 𝑇)
3328adantr 482 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ 0 ∈ 𝐸)
34 xp1st 7954 . . . . . 6 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ (1st β€˜π‘“) ∈ 𝑇)
3534adantl 483 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (1st β€˜π‘“) ∈ 𝑇)
36 xp2nd 7955 . . . . . 6 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜π‘“) ∈ 𝐸)
3736adantl 483 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (2nd β€˜π‘“) ∈ 𝐸)
381, 2, 3, 4, 10, 8, 11dvhopvadd 39559 . . . . 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 1380 . . . 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 38590 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (1st β€˜π‘“) ∈ 𝑇) β†’ (1st β€˜π‘“):𝐡–1-1-onto→𝐡)
4135, 40syldan 592 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (1st β€˜π‘“):𝐡–1-1-onto→𝐡)
42 f1of 6785 . . . . . 6 ((1st β€˜π‘“):𝐡–1-1-onto→𝐡 β†’ (1st β€˜π‘“):𝐡⟢𝐡)
43 fcoi2 6718 . . . . . 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 482 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ 𝐷 ∈ Grp)
4627adantr 482 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (Baseβ€˜π·) = 𝐸)
4737, 46eleqtrrd 2841 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (2nd β€˜π‘“) ∈ (Baseβ€˜π·))
4823, 11, 24grplid 18781 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·)) β†’ ( 0 ⨣ (2nd β€˜π‘“)) = (2nd β€˜π‘“))
4945, 47, 48syl2anc 585 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ ( 0 ⨣ (2nd β€˜π‘“)) = (2nd β€˜π‘“))
5044, 49opeq12d 4839 . . . 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 2777 . . 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 7961 . . . . 5 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ 𝑓 = ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)
5352adantl 483 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ 𝑓 = ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)
5453oveq2d 7374 . . 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 2787 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (⟨( I β†Ύ 𝐡), 0 ⟩ + 𝑓) = 𝑓)
561, 2ltrncnv 38612 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (1st β€˜π‘“) ∈ 𝑇) β†’ β—‘(1st β€˜π‘“) ∈ 𝑇)
5735, 56syldan 592 . . 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 18799 . . . . 5 ((𝐷 ∈ Grp ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·)) β†’ (πΌβ€˜(2nd β€˜π‘“)) ∈ (Baseβ€˜π·))
6045, 47, 59syl2anc 585 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (πΌβ€˜(2nd β€˜π‘“)) ∈ (Baseβ€˜π·))
6160, 46eleqtrd 2840 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (πΌβ€˜(2nd β€˜π‘“)) ∈ 𝐸)
62 opelxpi 5671 . . 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 585 . 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 7374 . . 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 39559 . . . . 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 1380 . . . 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 6814 . . . . . 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 18801 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·)) β†’ ((πΌβ€˜(2nd β€˜π‘“)) ⨣ (2nd β€˜π‘“)) = 0 )
7045, 47, 69syl2anc 585 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ ((πΌβ€˜(2nd β€˜π‘“)) ⨣ (2nd β€˜π‘“)) = 0 )
7168, 70opeq12d 4839 . . . 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 2777 . . 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 2777 . 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 18773 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4593   I cid 5531   Γ— cxp 5632  β—‘ccnv 5633   β†Ύ cres 5636   ∘ ccom 5638  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  Basecbs 17084  +gcplusg 17134  Scalarcsca 17137  0gc0g 17322  Grpcgrp 18749  invgcminusg 18750  DivRingcdr 20186  HLchlt 37815  LHypclh 38450  LTrncltrn 38567  TEndoctendo 39218  EDRingcedring 39219  DVecHcdvh 39544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-riotaBAD 37418
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-tpos 8158  df-undef 8205  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8649  df-map 8768  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-nn 12155  df-2 12217  df-3 12218  df-4 12219  df-5 12220  df-6 12221  df-n0 12415  df-z 12501  df-uz 12765  df-fz 13426  df-struct 17020  df-sets 17037  df-slot 17055  df-ndx 17067  df-base 17085  df-ress 17114  df-plusg 17147  df-mulr 17148  df-sca 17150  df-vsca 17151  df-0g 17324  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-p1 18316  df-lat 18322  df-clat 18389  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-grp 18752  df-minusg 18753  df-mgp 19898  df-ur 19915  df-ring 19967  df-oppr 20050  df-dvdsr 20071  df-unit 20072  df-invr 20102  df-dvr 20113  df-drng 20188  df-oposet 37641  df-ol 37643  df-oml 37644  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-llines 37964  df-lplanes 37965  df-lvols 37966  df-lines 37967  df-psubsp 37969  df-pmap 37970  df-padd 38262  df-lhyp 38454  df-laut 38455  df-ldil 38570  df-ltrn 38571  df-trl 38625  df-tendo 39221  df-edring 39223  df-dvech 39545
This theorem is referenced by:  dvhlveclem  39574
  Copyright terms: Public domain W3C validator