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Theorem dvhvaddass 36703
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 5692 . . . 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 36698 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
1093adantr3 1242 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
1110fveq2d 6233 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐹 + 𝐺)) = (1st ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩))
12 fvex 6239 . . . . . . . 8 (1st𝐹) ∈ V
13 fvex 6239 . . . . . . . 8 (1st𝐺) ∈ V
1412, 13coex 7160 . . . . . . 7 ((1st𝐹) ∘ (1st𝐺)) ∈ V
15 ovex 6718 . . . . . . 7 ((2nd𝐹) (2nd𝐺)) ∈ V
1614, 15op1st 7218 . . . . . 6 (1st ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩) = ((1st𝐹) ∘ (1st𝐺))
1711, 16syl6eq 2701 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐹 + 𝐺)) = ((1st𝐹) ∘ (1st𝐺)))
1817coeq1d 5316 . . . 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 36698 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩)
20193adantr1 1240 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩)
2120fveq2d 6233 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐺 + 𝐼)) = (1st ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩))
22 fvex 6239 . . . . . . . 8 (1st𝐼) ∈ V
2313, 22coex 7160 . . . . . . 7 ((1st𝐺) ∘ (1st𝐼)) ∈ V
24 ovex 6718 . . . . . . 7 ((2nd𝐺) (2nd𝐼)) ∈ V
2523, 24op1st 7218 . . . . . 6 (1st ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩) = ((1st𝐺) ∘ (1st𝐼))
2621, 25syl6eq 2701 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1st ‘(𝐺 + 𝐼)) = ((1st𝐺) ∘ (1st𝐼)))
2726coeq2d 5317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))) = ((1st𝐹) ∘ ((1st𝐺) ∘ (1st𝐼))))
281, 18, 273eqtr4a 2711 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝐹 + 𝐺)) ∘ (1st𝐼)) = ((1st𝐹) ∘ (1st ‘(𝐺 + 𝐼))))
29 xp2nd 7243 . . . . . 6 (𝐹 ∈ (𝑇 × 𝐸) → (2nd𝐹) ∈ 𝐸)
30 xp2nd 7243 . . . . . 6 (𝐺 ∈ (𝑇 × 𝐸) → (2nd𝐺) ∈ 𝐸)
31 xp2nd 7243 . . . . . 6 (𝐼 ∈ (𝑇 × 𝐸) → (2nd𝐼) ∈ 𝐸)
3229, 30, 313anim123i 1266 . . . . 5 ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸)) → ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸))
33 eqid 2651 . . . . . . . . . 10 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
342, 33, 5, 6dvhsca 36688 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
352, 33erngdv 36598 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
3634, 35eqeltrd 2730 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
37 drnggrp 18803 . . . . . . . 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 1087 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐹) ∈ 𝐸)
41 eqid 2651 . . . . . . . . 9 (Base‘𝐷) = (Base‘𝐷)
422, 4, 5, 6, 41dvhbase 36689 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
4342adantr 480 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (Base‘𝐷) = 𝐸)
4440, 43eleqtrrd 2733 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐹) ∈ (Base‘𝐷))
45 simpr2 1088 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐺) ∈ 𝐸)
4645, 43eleqtrrd 2733 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐺) ∈ (Base‘𝐷))
47 simpr3 1089 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐼) ∈ 𝐸)
4847, 43eleqtrrd 2733 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (2nd𝐼) ∈ (Base‘𝐷))
4941, 8grpass 17478 . . . . . 6 ((𝐷 ∈ Grp ∧ ((2nd𝐹) ∈ (Base‘𝐷) ∧ (2nd𝐺) ∈ (Base‘𝐷) ∧ (2nd𝐼) ∈ (Base‘𝐷))) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5039, 44, 46, 48, 49syl13anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝐹) ∈ 𝐸 ∧ (2nd𝐺) ∈ 𝐸 ∧ (2nd𝐼) ∈ 𝐸)) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5132, 50sylan2 490 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (((2nd𝐹) (2nd𝐺)) (2nd𝐼)) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
5210fveq2d 6233 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐹 + 𝐺)) = (2nd ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩))
5314, 15op2nd 7219 . . . . . 6 (2nd ‘⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩) = ((2nd𝐹) (2nd𝐺))
5452, 53syl6eq 2701 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐹 + 𝐺)) = ((2nd𝐹) (2nd𝐺)))
5554oveq1d 6705 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼)) = (((2nd𝐹) (2nd𝐺)) (2nd𝐼)))
5620fveq2d 6233 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐺 + 𝐼)) = (2nd ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩))
5723, 24op2nd 7219 . . . . . 6 (2nd ‘⟨((1st𝐺) ∘ (1st𝐼)), ((2nd𝐺) (2nd𝐼))⟩) = ((2nd𝐺) (2nd𝐼))
5856, 57syl6eq 2701 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝐺 + 𝐼)) = ((2nd𝐺) (2nd𝐼)))
5958oveq2d 6706 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd𝐹) (2nd ‘(𝐺 + 𝐼))) = ((2nd𝐹) ((2nd𝐺) (2nd𝐼))))
6051, 55, 593eqtr4d 2695 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝐹 + 𝐺)) (2nd𝐼)) = ((2nd𝐹) (2nd ‘(𝐺 + 𝐼))))
6128, 60opeq12d 4441 . 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 472 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
632, 3, 4, 5, 6, 8, 7dvhvaddcl 36701 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
64633adantr3 1242 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
65 simpr3 1089 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐼 ∈ (𝑇 × 𝐸))
662, 3, 4, 5, 6, 7, 8dvhvadd 36698 . . 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 1364 . 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 1087 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐹 ∈ (𝑇 × 𝐸))
692, 3, 4, 5, 6, 8, 7dvhvaddcl 36701 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
70693adantr1 1240 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
712, 3, 4, 5, 6, 7, 8dvhvadd 36698 . . 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 1364 . 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 2695 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 383  w3a 1054   = wceq 1523  wcel 2030  cop 4216   × cxp 5141  ccom 5147  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Basecbs 15904  +gcplusg 15988  Scalarcsca 15991  Grpcgrp 17469  DivRingcdr 18795  HLchlt 34955  LHypclh 35588  LTrncltrn 35705  TEndoctendo 36357  EDRingcedring 36358  DVecHcdvh 36684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-riotaBAD 34557
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-undef 7444  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-0g 16149  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-mgp 18536  df-ur 18548  df-ring 18595  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764  df-tendo 36360  df-edring 36362  df-dvech 36685
This theorem is referenced by:  dvhgrp  36713
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