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

 Description: Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
Hypotheses
Ref Expression
Assertion
Ref Expression
dvhvaddass (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = (𝐹 + (𝐺 + 𝐼)))

StepHypRef Expression
1 coass 5618 . . . 4 (((1st𝐹) ∘ (1st𝐺)) ∘ (1st𝐼)) = ((1st𝐹) ∘ ((1st𝐺) ∘ (1st𝐼)))
2 dvhvaddcl.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
3 dvhvaddcl.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 dvhvaddcl.e . . . . . . . . 9 𝐸 = ((TEndo‘𝐾)‘𝑊)
5 dvhvaddcl.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
6 dvhvaddcl.d . . . . . . . . 9 𝐷 = (Scalar‘𝑈)
7 dvhvaddcl.a . . . . . . . . 9 + = (+g𝑈)
8 dvhvaddcl.p . . . . . . . . 9 = (+g𝐷)
92, 3, 4, 5, 6, 7, 8dvhvadd 35896 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
1093adantr3 1220 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
1110fveq2d 6157 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐹 + 𝐺)) = (1st ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩))
12 fvex 6163 . . . . . . . 8 (1st𝐹) ∈ V
13 fvex 6163 . . . . . . . 8 (1st𝐺) ∈ V
1412, 13coex 7072 . . . . . . 7 ((1st𝐹) ∘ (1st𝐺)) ∈ V
15 ovex 6638 . . . . . . 7 ((2nd𝐹) (2nd𝐺)) ∈ V
1614, 15op1st 7128 . . . . . 6 (1st ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩) = ((1st𝐹) ∘ (1st𝐺))
1711, 16syl6eq 2671 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐹 + 𝐺)) = ((1st𝐹) ∘ (1st𝐺)))
1817coeq1d 5248 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝐹 + 𝐺)) ∘ (1st𝐼)) = (((1st𝐹) ∘ (1st𝐺)) ∘ (1st𝐼)))
192, 3, 4, 5, 6, 7, 8dvhvadd 35896 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩)
20193adantr1 1218 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩)
2120fveq2d 6157 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐺 + 𝐼)) = (1st ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩))
22 fvex 6163 . . . . . . . 8 (1st𝐼) ∈ V
2313, 22coex 7072 . . . . . . 7 ((1st𝐺) ∘ (1st𝐼)) ∈ V
24 ovex 6638 . . . . . . 7 ((2nd𝐺) (2nd𝐼)) ∈ V
2523, 24op1st 7128 . . . . . 6 (1st ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩) = ((1st𝐺) ∘ (1st𝐼))
2621, 25syl6eq 2671 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐺 + 𝐼)) = ((1st𝐺) ∘ (1st𝐼)))
2726coeq2d 5249 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))) = ((1st𝐹) ∘ ((1st𝐺) ∘ (1st𝐼))))
281, 18, 273eqtr4a 2681 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝐹 + 𝐺)) ∘ (1st𝐼)) = ((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))))
29 xp2nd 7151 . . . . . 6 (𝐹 ∈ (𝑇 × 𝐸) → (2nd𝐹) ∈ 𝐸)
30 xp2nd 7151 . . . . . 6 (𝐺 ∈ (𝑇 × 𝐸) → (2nd𝐺) ∈ 𝐸)
31 xp2nd 7151 . . . . . 6 (𝐼 ∈ (𝑇 × 𝐸) → (2nd𝐼) ∈ 𝐸)
3229, 30, 313anim123i 1245 . . . . 5 ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸)) → ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸))
33 eqid 2621 . . . . . . . . . 10 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
342, 33, 5, 6dvhsca 35886 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
352, 33erngdv 35796 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
3634, 35eqeltrd 2698 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
37 drnggrp 18687 . . . . . . . 8 (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
3836, 37syl 17 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Grp)
3938adantr 481 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → 𝐷 ∈ Grp)
40 simpr1 1065 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐹) ∈ 𝐸)
41 eqid 2621 . . . . . . . . 9 (Base‘𝐷) = (Base‘𝐷)
422, 4, 5, 6, 41dvhbase 35887 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
4342adantr 481 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (Base‘𝐷) = 𝐸)
4440, 43eleqtrrd 2701 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐹) ∈ (Base‘𝐷))
45 simpr2 1066 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐺) ∈ 𝐸)
4645, 43eleqtrrd 2701 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐺) ∈ (Base‘𝐷))
47 simpr3 1067 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐼) ∈ 𝐸)
4847, 43eleqtrrd 2701 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐼) ∈ (Base‘𝐷))
4941, 8grpass 17363 . . . . . 6 ((𝐷 ∈ Grp ∧ ((2nd𝐹) ∈ (Base‘𝐷) ∧ (2nd𝐺) ∈ (Base‘𝐷) ∧ (2nd𝐼) ∈ (Base‘𝐷))) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5039, 44, 46, 48, 49syl13anc 1325 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5132, 50sylan2 491 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5210fveq2d 6157 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐹 + 𝐺)) = (2nd ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩))
5314, 15op2nd 7129 . . . . . 6 (2nd ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩) = ((2nd𝐹) (2nd𝐺))
5452, 53syl6eq 2671 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐹 + 𝐺)) = ((2nd𝐹) (2nd𝐺)))
5554oveq1d 6625 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼)) = (((2nd𝐹) (2nd𝐺)) (2nd𝐼)))
5620fveq2d 6157 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐺 + 𝐼)) = (2nd ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩))
5723, 24op2nd 7129 . . . . . 6 (2nd ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩) = ((2nd𝐺) (2nd𝐼))
5856, 57syl6eq 2671 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐺 + 𝐼)) = ((2nd𝐺) (2nd𝐼)))
5958oveq2d 6626 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd𝐹) (2nd ‘(𝐺 + 𝐼))) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
6051, 55, 593eqtr4d 2665 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼)) = ((2nd𝐹) (2nd ‘(𝐺 + 𝐼))))
6128, 60opeq12d 4383 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ⟨((1st ‘(𝐹 + 𝐺)) ∘ (1st𝐼)), ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼))⟩ = ⟨((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))), ((2nd𝐹) (2nd ‘(𝐺 + 𝐼)))⟩)
62 simpl 473 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
632, 3, 4, 5, 6, 8, 7dvhvaddcl 35899 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
64633adantr3 1220 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
65 simpr3 1067 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐼 ∈ (𝑇 × 𝐸))
662, 3, 4, 5, 6, 7, 8dvhvadd 35896 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹 + 𝐺) ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = ⟨((1st ‘(𝐹 + 𝐺)) ∘ (1st𝐼)), ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼))⟩)
6762, 64, 65, 66syl12anc 1321 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = ⟨((1st ‘(𝐹 + 𝐺)) ∘ (1st𝐼)), ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼))⟩)
68 simpr1 1065 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐹 ∈ (𝑇 × 𝐸))
692, 3, 4, 5, 6, 8, 7dvhvaddcl 35899 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
70693adantr1 1218 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
712, 3, 4, 5, 6, 7, 8dvhvadd 35896 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))) → (𝐹 + (𝐺 + 𝐼)) = ⟨((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))), ((2nd𝐹) (2nd ‘(𝐺 + 𝐼)))⟩)
7262, 68, 70, 71syl12anc 1321 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + (𝐺 + 𝐼)) = ⟨((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))), ((2nd𝐹) (2nd ‘(𝐺 + 𝐼)))⟩)
7361, 67, 723eqtr4d 2665 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = (𝐹 + (𝐺 + 𝐼)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ⟨cop 4159   × cxp 5077   ∘ ccom 5083  ‘cfv 5852  (class class class)co 6610  1st c1st 7118  2nd c2nd 7119  Basecbs 15792  +gcplusg 15873  Scalarcsca 15876  Grpcgrp 17354  DivRingcdr 18679  HLchlt 34152  LHypclh 34785  LTrncltrn 34902  TEndoctendo 35555  EDRingcedring 35556  DVecHcdvh 35882 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-riotaBAD 33754 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-tpos 7304  df-undef 7351  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-sca 15889  df-vsca 15890  df-0g 16034  df-preset 16860  df-poset 16878  df-plt 16890  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-p0 16971  df-p1 16972  df-lat 16978  df-clat 17040  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-grp 17357  df-minusg 17358  df-mgp 18422  df-ur 18434  df-ring 18481  df-oppr 18555  df-dvdsr 18573  df-unit 18574  df-invr 18604  df-dvr 18615  df-drng 18681  df-oposet 33978  df-ol 33980  df-oml 33981  df-covers 34068  df-ats 34069  df-atl 34100  df-cvlat 34124  df-hlat 34153  df-llines 34299  df-lplanes 34300  df-lvols 34301  df-lines 34302  df-psubsp 34304  df-pmap 34305  df-padd 34597  df-lhyp 34789  df-laut 34790  df-ldil 34905  df-ltrn 34906  df-trl 34961  df-tendo 35558  df-edring 35560  df-dvech 35883 This theorem is referenced by:  dvhgrp  35911
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