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Theorem dvhgrp 40489
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 2726 . . . 4 (Baseβ€˜π‘ˆ) = (Baseβ€˜π‘ˆ)
61, 2, 3, 4, 5dvhvbase 40469 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π‘ˆ) = (𝑇 Γ— 𝐸))
76eqcomd 2732 . 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 40477 . . 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 40479 . 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 39532 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ 𝑇)
17 eqid 2726 . . . . . . . 8 ((EDRingβ€˜πΎ)β€˜π‘Š) = ((EDRingβ€˜πΎ)β€˜π‘Š)
181, 17, 4, 10dvhsca 40464 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š))
191, 17erngdv 40375 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((EDRingβ€˜πΎ)β€˜π‘Š) ∈ DivRing)
2018, 19eqeltrd 2827 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ DivRing)
21 drnggrp 20595 . . . . . 6 (𝐷 ∈ DivRing β†’ 𝐷 ∈ Grp)
2220, 21syl 17 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Grp)
23 eqid 2726 . . . . . 6 (Baseβ€˜π·) = (Baseβ€˜π·)
24 dvhgrp.o . . . . . 6 0 = (0gβ€˜π·)
2523, 24grpidcl 18893 . . . . 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 40465 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π·) = 𝐸)
2826, 27eleqtrd 2829 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 ∈ 𝐸)
29 opelxpi 5706 . . 3 ((( I β†Ύ 𝐡) ∈ 𝑇 ∧ 0 ∈ 𝐸) β†’ ⟨( I β†Ύ 𝐡), 0 ⟩ ∈ (𝑇 Γ— 𝐸))
3016, 28, 29syl2anc 583 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ⟨( I β†Ύ 𝐡), 0 ⟩ ∈ (𝑇 Γ— 𝐸))
31 simpl 482 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
3216adantr 480 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ ( I β†Ύ 𝐡) ∈ 𝑇)
3328adantr 480 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ 0 ∈ 𝐸)
34 xp1st 8003 . . . . . 6 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ (1st β€˜π‘“) ∈ 𝑇)
3534adantl 481 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (1st β€˜π‘“) ∈ 𝑇)
36 xp2nd 8004 . . . . . 6 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜π‘“) ∈ 𝐸)
3736adantl 481 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (2nd β€˜π‘“) ∈ 𝐸)
381, 2, 3, 4, 10, 8, 11dvhopvadd 40475 . . . . 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 39506 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (1st β€˜π‘“) ∈ 𝑇) β†’ (1st β€˜π‘“):𝐡–1-1-onto→𝐡)
4135, 40syldan 590 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (1st β€˜π‘“):𝐡–1-1-onto→𝐡)
42 f1of 6826 . . . . . 6 ((1st β€˜π‘“):𝐡–1-1-onto→𝐡 β†’ (1st β€˜π‘“):𝐡⟢𝐡)
43 fcoi2 6759 . . . . . 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 480 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ 𝐷 ∈ Grp)
4627adantr 480 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (Baseβ€˜π·) = 𝐸)
4737, 46eleqtrrd 2830 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (2nd β€˜π‘“) ∈ (Baseβ€˜π·))
4823, 11, 24grplid 18895 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·)) β†’ ( 0 ⨣ (2nd β€˜π‘“)) = (2nd β€˜π‘“))
4945, 47, 48syl2anc 583 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ ( 0 ⨣ (2nd β€˜π‘“)) = (2nd β€˜π‘“))
5044, 49opeq12d 4876 . . . 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 2766 . . 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 8010 . . . . 5 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ 𝑓 = ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)
5352adantl 481 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ 𝑓 = ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)
5453oveq2d 7420 . . 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 2776 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (⟨( I β†Ύ 𝐡), 0 ⟩ + 𝑓) = 𝑓)
561, 2ltrncnv 39528 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (1st β€˜π‘“) ∈ 𝑇) β†’ β—‘(1st β€˜π‘“) ∈ 𝑇)
5735, 56syldan 590 . . 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 18915 . . . . 5 ((𝐷 ∈ Grp ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·)) β†’ (πΌβ€˜(2nd β€˜π‘“)) ∈ (Baseβ€˜π·))
6045, 47, 59syl2anc 583 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (πΌβ€˜(2nd β€˜π‘“)) ∈ (Baseβ€˜π·))
6160, 46eleqtrd 2829 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ (πΌβ€˜(2nd β€˜π‘“)) ∈ 𝐸)
62 opelxpi 5706 . . 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 583 . 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 7420 . . 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 40475 . . . . 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 6855 . . . . . 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 18917 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·)) β†’ ((πΌβ€˜(2nd β€˜π‘“)) ⨣ (2nd β€˜π‘“)) = 0 )
7045, 47, 69syl2anc 583 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸)) β†’ ((πΌβ€˜(2nd β€˜π‘“)) ⨣ (2nd β€˜π‘“)) = 0 )
7168, 70opeq12d 4876 . . . 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 2766 . . 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 2766 . 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 18886 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4629   I cid 5566   Γ— cxp 5667  β—‘ccnv 5668   β†Ύ cres 5671   ∘ ccom 5673  βŸΆwf 6532  β€“1-1-ontoβ†’wf1o 6535  β€˜cfv 6536  (class class class)co 7404  1st c1st 7969  2nd c2nd 7970  Basecbs 17151  +gcplusg 17204  Scalarcsca 17207  0gc0g 17392  Grpcgrp 18861  invgcminusg 18862  DivRingcdr 20585  HLchlt 38731  LHypclh 39366  LTrncltrn 39483  TEndoctendo 40134  EDRingcedring 40135  DVecHcdvh 40460
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-riotaBAD 38334
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-tpos 8209  df-undef 8256  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-n0 12474  df-z 12560  df-uz 12824  df-fz 13488  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-sca 17220  df-vsca 17221  df-0g 17394  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-p1 18389  df-lat 18395  df-clat 18462  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-minusg 18865  df-cmn 19700  df-abl 19701  df-mgp 20038  df-rng 20056  df-ur 20085  df-ring 20138  df-oppr 20234  df-dvdsr 20257  df-unit 20258  df-invr 20288  df-dvr 20301  df-drng 20587  df-oposet 38557  df-ol 38559  df-oml 38560  df-covers 38647  df-ats 38648  df-atl 38679  df-cvlat 38703  df-hlat 38732  df-llines 38880  df-lplanes 38881  df-lvols 38882  df-lines 38883  df-psubsp 38885  df-pmap 38886  df-padd 39178  df-lhyp 39370  df-laut 39371  df-ldil 39486  df-ltrn 39487  df-trl 39541  df-tendo 40137  df-edring 40139  df-dvech 40461
This theorem is referenced by:  dvhlveclem  40490
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