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

Theorem dvhvaddass 41080
Description: Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
Hypotheses
Ref Expression
dvhvaddcl.h 𝐻 = (LHyp‘𝐾)
dvhvaddcl.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhvaddcl.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhvaddcl.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvhvaddcl.d 𝐷 = (Scalar‘𝑈)
dvhvaddcl.p = (+g𝐷)
dvhvaddcl.a + = (+g𝑈)
Assertion
Ref Expression
dvhvaddass (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = (𝐹 + (𝐺 + 𝐼)))

Proof of Theorem dvhvaddass
StepHypRef Expression
1 coass 6287 . . . 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 41075 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
1093adantr3 1170 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
1110fveq2d 6911 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐹 + 𝐺)) = (1st ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩))
12 fvex 6920 . . . . . . . 8 (1st𝐹) ∈ V
13 fvex 6920 . . . . . . . 8 (1st𝐺) ∈ V
1412, 13coex 7953 . . . . . . 7 ((1st𝐹) ∘ (1st𝐺)) ∈ V
15 ovex 7464 . . . . . . 7 ((2nd𝐹) (2nd𝐺)) ∈ V
1614, 15op1st 8021 . . . . . 6 (1st ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩) = ((1st𝐹) ∘ (1st𝐺))
1711, 16eqtrdi 2791 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐹 + 𝐺)) = ((1st𝐹) ∘ (1st𝐺)))
1817coeq1d 5875 . . . 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 41075 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩)
20193adantr1 1168 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩)
2120fveq2d 6911 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐺 + 𝐼)) = (1st ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩))
22 fvex 6920 . . . . . . . 8 (1st𝐼) ∈ V
2313, 22coex 7953 . . . . . . 7 ((1st𝐺) ∘ (1st𝐼)) ∈ V
24 ovex 7464 . . . . . . 7 ((2nd𝐺) (2nd𝐼)) ∈ V
2523, 24op1st 8021 . . . . . 6 (1st ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩) = ((1st𝐺) ∘ (1st𝐼))
2621, 25eqtrdi 2791 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐺 + 𝐼)) = ((1st𝐺) ∘ (1st𝐼)))
2726coeq2d 5876 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))) = ((1st𝐹) ∘ ((1st𝐺) ∘ (1st𝐼))))
281, 18, 273eqtr4a 2801 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝐹 + 𝐺)) ∘ (1st𝐼)) = ((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))))
29 xp2nd 8046 . . . . . 6 (𝐹 ∈ (𝑇 × 𝐸) → (2nd𝐹) ∈ 𝐸)
30 xp2nd 8046 . . . . . 6 (𝐺 ∈ (𝑇 × 𝐸) → (2nd𝐺) ∈ 𝐸)
31 xp2nd 8046 . . . . . 6 (𝐼 ∈ (𝑇 × 𝐸) → (2nd𝐼) ∈ 𝐸)
3229, 30, 313anim123i 1150 . . . . 5 ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸)) → ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸))
33 eqid 2735 . . . . . . . . . 10 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
342, 33, 5, 6dvhsca 41065 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
352, 33erngdv 40976 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
3634, 35eqeltrd 2839 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
37 drnggrp 20756 . . . . . . . 8 (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
3836, 37syl 17 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Grp)
3938adantr 480 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → 𝐷 ∈ Grp)
40 simpr1 1193 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐹) ∈ 𝐸)
41 eqid 2735 . . . . . . . . 9 (Base‘𝐷) = (Base‘𝐷)
422, 4, 5, 6, 41dvhbase 41066 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
4342adantr 480 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (Base‘𝐷) = 𝐸)
4440, 43eleqtrrd 2842 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐹) ∈ (Base‘𝐷))
45 simpr2 1194 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐺) ∈ 𝐸)
4645, 43eleqtrrd 2842 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐺) ∈ (Base‘𝐷))
47 simpr3 1195 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐼) ∈ 𝐸)
4847, 43eleqtrrd 2842 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐼) ∈ (Base‘𝐷))
4941, 8grpass 18973 . . . . . 6 ((𝐷 ∈ Grp ∧ ((2nd𝐹) ∈ (Base‘𝐷) ∧ (2nd𝐺) ∈ (Base‘𝐷) ∧ (2nd𝐼) ∈ (Base‘𝐷))) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5039, 44, 46, 48, 49syl13anc 1371 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5132, 50sylan2 593 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5210fveq2d 6911 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐹 + 𝐺)) = (2nd ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩))
5314, 15op2nd 8022 . . . . . 6 (2nd ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩) = ((2nd𝐹) (2nd𝐺))
5452, 53eqtrdi 2791 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐹 + 𝐺)) = ((2nd𝐹) (2nd𝐺)))
5554oveq1d 7446 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼)) = (((2nd𝐹) (2nd𝐺)) (2nd𝐼)))
5620fveq2d 6911 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐺 + 𝐼)) = (2nd ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩))
5723, 24op2nd 8022 . . . . . 6 (2nd ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩) = ((2nd𝐺) (2nd𝐼))
5856, 57eqtrdi 2791 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐺 + 𝐼)) = ((2nd𝐺) (2nd𝐼)))
5958oveq2d 7447 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd𝐹) (2nd ‘(𝐺 + 𝐼))) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
6051, 55, 593eqtr4d 2785 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼)) = ((2nd𝐹) (2nd ‘(𝐺 + 𝐼))))
6128, 60opeq12d 4886 . 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 482 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
632, 3, 4, 5, 6, 8, 7dvhvaddcl 41078 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
64633adantr3 1170 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
65 simpr3 1195 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐼 ∈ (𝑇 × 𝐸))
662, 3, 4, 5, 6, 7, 8dvhvadd 41075 . . 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 837 . 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 1193 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐹 ∈ (𝑇 × 𝐸))
692, 3, 4, 5, 6, 8, 7dvhvaddcl 41078 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
70693adantr1 1168 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
712, 3, 4, 5, 6, 7, 8dvhvadd 41075 . . 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 837 . 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 2785 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 395  w3a 1086   = wceq 1537  wcel 2106  cop 4637   × cxp 5687  ccom 5693  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  +gcplusg 17298  Scalarcsca 17301  Grpcgrp 18964  DivRingcdr 20746  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  TEndoctendo 40735  EDRingcedring 40736  DVecHcdvh 41061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-riotaBAD 38935
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-undef 8297  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-0g 17488  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-drng 20748  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142  df-tendo 40738  df-edring 40740  df-dvech 41062
This theorem is referenced by:  dvhgrp  41090
  Copyright terms: Public domain W3C validator