 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 5797 . . . 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 36902 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
1093adantr3 1176 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
1110fveq2d 6337 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐹 + 𝐺)) = (1st ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩))
12 fvex 6344 . . . . . . . 8 (1st𝐹) ∈ V
13 fvex 6344 . . . . . . . 8 (1st𝐺) ∈ V
1412, 13coex 7268 . . . . . . 7 ((1st𝐹) ∘ (1st𝐺)) ∈ V
15 ovex 6826 . . . . . . 7 ((2nd𝐹) (2nd𝐺)) ∈ V
1614, 15op1st 7326 . . . . . 6 (1st ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩) = ((1st𝐹) ∘ (1st𝐺))
1711, 16syl6eq 2821 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐹 + 𝐺)) = ((1st𝐹) ∘ (1st𝐺)))
1817coeq1d 5421 . . . 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 36902 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩)
20193adantr1 1174 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩)
2120fveq2d 6337 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐺 + 𝐼)) = (1st ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩))
22 fvex 6344 . . . . . . . 8 (1st𝐼) ∈ V
2313, 22coex 7268 . . . . . . 7 ((1st𝐺) ∘ (1st𝐼)) ∈ V
24 ovex 6826 . . . . . . 7 ((2nd𝐺) (2nd𝐼)) ∈ V
2523, 24op1st 7326 . . . . . 6 (1st ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩) = ((1st𝐺) ∘ (1st𝐼))
2621, 25syl6eq 2821 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐺 + 𝐼)) = ((1st𝐺) ∘ (1st𝐼)))
2726coeq2d 5422 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))) = ((1st𝐹) ∘ ((1st𝐺) ∘ (1st𝐼))))
281, 18, 273eqtr4a 2831 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝐹 + 𝐺)) ∘ (1st𝐼)) = ((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))))
29 xp2nd 7351 . . . . . 6 (𝐹 ∈ (𝑇 × 𝐸) → (2nd𝐹) ∈ 𝐸)
30 xp2nd 7351 . . . . . 6 (𝐺 ∈ (𝑇 × 𝐸) → (2nd𝐺) ∈ 𝐸)
31 xp2nd 7351 . . . . . 6 (𝐼 ∈ (𝑇 × 𝐸) → (2nd𝐼) ∈ 𝐸)
3229, 30, 313anim123i 1154 . . . . 5 ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸)) → ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸))
33 eqid 2771 . . . . . . . . . 10 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
342, 33, 5, 6dvhsca 36892 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
352, 33erngdv 36802 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
3634, 35eqeltrd 2850 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
37 drnggrp 18964 . . . . . . . 8 (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
3836, 37syl 17 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Grp)
3938adantr 466 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → 𝐷 ∈ Grp)
40 simpr1 1233 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐹) ∈ 𝐸)
41 eqid 2771 . . . . . . . . 9 (Base‘𝐷) = (Base‘𝐷)
422, 4, 5, 6, 41dvhbase 36893 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
4342adantr 466 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (Base‘𝐷) = 𝐸)
4440, 43eleqtrrd 2853 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐹) ∈ (Base‘𝐷))
45 simpr2 1235 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐺) ∈ 𝐸)
4645, 43eleqtrrd 2853 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐺) ∈ (Base‘𝐷))
47 simpr3 1237 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐼) ∈ 𝐸)
4847, 43eleqtrrd 2853 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐼) ∈ (Base‘𝐷))
4941, 8grpass 17638 . . . . . 6 ((𝐷 ∈ Grp ∧ ((2nd𝐹) ∈ (Base‘𝐷) ∧ (2nd𝐺) ∈ (Base‘𝐷) ∧ (2nd𝐼) ∈ (Base‘𝐷))) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5039, 44, 46, 48, 49syl13anc 1478 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5132, 50sylan2 580 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5210fveq2d 6337 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐹 + 𝐺)) = (2nd ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩))
5314, 15op2nd 7327 . . . . . 6 (2nd ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩) = ((2nd𝐹) (2nd𝐺))
5452, 53syl6eq 2821 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐹 + 𝐺)) = ((2nd𝐹) (2nd𝐺)))
5554oveq1d 6810 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼)) = (((2nd𝐹) (2nd𝐺)) (2nd𝐼)))
5620fveq2d 6337 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐺 + 𝐼)) = (2nd ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩))
5723, 24op2nd 7327 . . . . . 6 (2nd ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩) = ((2nd𝐺) (2nd𝐼))
5856, 57syl6eq 2821 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐺 + 𝐼)) = ((2nd𝐺) (2nd𝐼)))
5958oveq2d 6811 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd𝐹) (2nd ‘(𝐺 + 𝐼))) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
6051, 55, 593eqtr4d 2815 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼)) = ((2nd𝐹) (2nd ‘(𝐺 + 𝐼))))
6128, 60opeq12d 4548 . 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 468 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
632, 3, 4, 5, 6, 8, 7dvhvaddcl 36905 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
64633adantr3 1176 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
65 simpr3 1237 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐼 ∈ (𝑇 × 𝐸))
662, 3, 4, 5, 6, 7, 8dvhvadd 36902 . . 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 1474 . 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 1233 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐹 ∈ (𝑇 × 𝐸))
692, 3, 4, 5, 6, 8, 7dvhvaddcl 36905 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
70693adantr1 1174 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
712, 3, 4, 5, 6, 7, 8dvhvadd 36902 . . 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 1474 . 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 2815 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 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ⟨cop 4323   × cxp 5248   ∘ ccom 5254  ‘cfv 6030  (class class class)co 6795  1st c1st 7316  2nd c2nd 7317  Basecbs 16063  +gcplusg 16148  Scalarcsca 16151  Grpcgrp 17629  DivRingcdr 18956  HLchlt 35158  LHypclh 35792  LTrncltrn 35909  TEndoctendo 36561  EDRingcedring 36562  DVecHcdvh 36888 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218  ax-riotaBAD 34760 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-1st 7318  df-2nd 7319  df-tpos 7507  df-undef 7554  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-oadd 7720  df-er 7899  df-map 8014  df-en 8113  df-dom 8114  df-sdom 8115  df-fin 8116  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-nn 11226  df-2 11284  df-3 11285  df-4 11286  df-5 11287  df-6 11288  df-n0 11499  df-z 11584  df-uz 11893  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-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-grp 17632  df-minusg 17633  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-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-llines 35306  df-lplanes 35307  df-lvols 35308  df-lines 35309  df-psubsp 35311  df-pmap 35312  df-padd 35604  df-lhyp 35796  df-laut 35797  df-ldil 35912  df-ltrn 35913  df-trl 35968  df-tendo 36564  df-edring 36566  df-dvech 36889 This theorem is referenced by:  dvhgrp  36917
 Copyright terms: Public domain W3C validator