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Theorem dvhvaddass 38273
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 6091 . . . 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 38268 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
1093adantr3 1168 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
1110fveq2d 6647 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐹 + 𝐺)) = (1st ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩))
12 fvex 6656 . . . . . . . 8 (1st𝐹) ∈ V
13 fvex 6656 . . . . . . . 8 (1st𝐺) ∈ V
1412, 13coex 7610 . . . . . . 7 ((1st𝐹) ∘ (1st𝐺)) ∈ V
15 ovex 7163 . . . . . . 7 ((2nd𝐹) (2nd𝐺)) ∈ V
1614, 15op1st 7672 . . . . . 6 (1st ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩) = ((1st𝐹) ∘ (1st𝐺))
1711, 16syl6eq 2872 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐹 + 𝐺)) = ((1st𝐹) ∘ (1st𝐺)))
1817coeq1d 5705 . . . 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 38268 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩)
20193adantr1 1166 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩)
2120fveq2d 6647 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐺 + 𝐼)) = (1st ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩))
22 fvex 6656 . . . . . . . 8 (1st𝐼) ∈ V
2313, 22coex 7610 . . . . . . 7 ((1st𝐺) ∘ (1st𝐼)) ∈ V
24 ovex 7163 . . . . . . 7 ((2nd𝐺) (2nd𝐼)) ∈ V
2523, 24op1st 7672 . . . . . 6 (1st ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩) = ((1st𝐺) ∘ (1st𝐼))
2621, 25syl6eq 2872 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐺 + 𝐼)) = ((1st𝐺) ∘ (1st𝐼)))
2726coeq2d 5706 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))) = ((1st𝐹) ∘ ((1st𝐺) ∘ (1st𝐼))))
281, 18, 273eqtr4a 2882 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝐹 + 𝐺)) ∘ (1st𝐼)) = ((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))))
29 xp2nd 7697 . . . . . 6 (𝐹 ∈ (𝑇 × 𝐸) → (2nd𝐹) ∈ 𝐸)
30 xp2nd 7697 . . . . . 6 (𝐺 ∈ (𝑇 × 𝐸) → (2nd𝐺) ∈ 𝐸)
31 xp2nd 7697 . . . . . 6 (𝐼 ∈ (𝑇 × 𝐸) → (2nd𝐼) ∈ 𝐸)
3229, 30, 313anim123i 1148 . . . . 5 ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸)) → ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸))
33 eqid 2821 . . . . . . . . . 10 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
342, 33, 5, 6dvhsca 38258 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
352, 33erngdv 38169 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
3634, 35eqeltrd 2912 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
37 drnggrp 19485 . . . . . . . 8 (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
3836, 37syl 17 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Grp)
3938adantr 484 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → 𝐷 ∈ Grp)
40 simpr1 1191 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐹) ∈ 𝐸)
41 eqid 2821 . . . . . . . . 9 (Base‘𝐷) = (Base‘𝐷)
422, 4, 5, 6, 41dvhbase 38259 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
4342adantr 484 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (Base‘𝐷) = 𝐸)
4440, 43eleqtrrd 2915 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐹) ∈ (Base‘𝐷))
45 simpr2 1192 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐺) ∈ 𝐸)
4645, 43eleqtrrd 2915 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐺) ∈ (Base‘𝐷))
47 simpr3 1193 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐼) ∈ 𝐸)
4847, 43eleqtrrd 2915 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐼) ∈ (Base‘𝐷))
4941, 8grpass 18090 . . . . . 6 ((𝐷 ∈ Grp ∧ ((2nd𝐹) ∈ (Base‘𝐷) ∧ (2nd𝐺) ∈ (Base‘𝐷) ∧ (2nd𝐼) ∈ (Base‘𝐷))) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5039, 44, 46, 48, 49syl13anc 1369 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5132, 50sylan2 595 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5210fveq2d 6647 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐹 + 𝐺)) = (2nd ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩))
5314, 15op2nd 7673 . . . . . 6 (2nd ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩) = ((2nd𝐹) (2nd𝐺))
5452, 53syl6eq 2872 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐹 + 𝐺)) = ((2nd𝐹) (2nd𝐺)))
5554oveq1d 7145 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼)) = (((2nd𝐹) (2nd𝐺)) (2nd𝐼)))
5620fveq2d 6647 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐺 + 𝐼)) = (2nd ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩))
5723, 24op2nd 7673 . . . . . 6 (2nd ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩) = ((2nd𝐺) (2nd𝐼))
5856, 57syl6eq 2872 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐺 + 𝐼)) = ((2nd𝐺) (2nd𝐼)))
5958oveq2d 7146 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd𝐹) (2nd ‘(𝐺 + 𝐼))) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
6051, 55, 593eqtr4d 2866 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼)) = ((2nd𝐹) (2nd ‘(𝐺 + 𝐼))))
6128, 60opeq12d 4784 . 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 486 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
632, 3, 4, 5, 6, 8, 7dvhvaddcl 38271 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
64633adantr3 1168 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
65 simpr3 1193 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐼 ∈ (𝑇 × 𝐸))
662, 3, 4, 5, 6, 7, 8dvhvadd 38268 . . 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 835 . 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 1191 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐹 ∈ (𝑇 × 𝐸))
692, 3, 4, 5, 6, 8, 7dvhvaddcl 38271 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
70693adantr1 1166 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
712, 3, 4, 5, 6, 7, 8dvhvadd 38268 . . 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 835 . 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 2866 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 399  w3a 1084   = wceq 1538  wcel 2115  cop 4546   × cxp 5526  ccom 5532  cfv 6328  (class class class)co 7130  1st c1st 7662  2nd c2nd 7663  Basecbs 16461  +gcplusg 16543  Scalarcsca 16546  Grpcgrp 18081  DivRingcdr 19477  HLchlt 36526  LHypclh 37160  LTrncltrn 37277  TEndoctendo 37928  EDRingcedring 37929  DVecHcdvh 38254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-riotaBAD 36129
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-tpos 7867  df-undef 7914  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-map 8383  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-n0 11876  df-z 11960  df-uz 12222  df-fz 12876  df-struct 16463  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-ress 16469  df-plusg 16556  df-mulr 16557  df-sca 16559  df-vsca 16560  df-0g 16693  df-proset 17516  df-poset 17534  df-plt 17546  df-lub 17562  df-glb 17563  df-join 17564  df-meet 17565  df-p0 17627  df-p1 17628  df-lat 17634  df-clat 17696  df-mgm 17830  df-sgrp 17879  df-mnd 17890  df-grp 18084  df-minusg 18085  df-mgp 19218  df-ur 19230  df-ring 19277  df-oppr 19351  df-dvdsr 19369  df-unit 19370  df-invr 19400  df-dvr 19411  df-drng 19479  df-oposet 36352  df-ol 36354  df-oml 36355  df-covers 36442  df-ats 36443  df-atl 36474  df-cvlat 36498  df-hlat 36527  df-llines 36674  df-lplanes 36675  df-lvols 36676  df-lines 36677  df-psubsp 36679  df-pmap 36680  df-padd 36972  df-lhyp 37164  df-laut 37165  df-ldil 37280  df-ltrn 37281  df-trl 37335  df-tendo 37931  df-edring 37933  df-dvech 38255
This theorem is referenced by:  dvhgrp  38283
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