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Theorem usg2wlkonot 26410
 Description: A walk of length 2 between two vertices as ordered triple in an undirected simple graph. This theorem would also hold for undirected multigraphs, but to prove this the cases 𝐴 = 𝐵 and/or 𝐵 = 𝐶 must be considered separately. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
Assertion
Ref Expression
usg2wlkonot ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))

Proof of Theorem usg2wlkonot
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 25867 . . 3 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2 el2wlkonotot 26400 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
323adantr2 1214 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
41, 3sylan 487 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
5 vex 3176 . . . . . . . . . . . 12 𝑓 ∈ V
6 vex 3176 . . . . . . . . . . . 12 𝑝 ∈ V
75, 6pm3.2i 470 . . . . . . . . . . 11 (𝑓 ∈ V ∧ 𝑝 ∈ V)
8 is2wlk 26095 . . . . . . . . . . 11 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V)) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ↔ (𝑓:(0..^2)⟶dom 𝐸𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}))))
91, 7, 8sylancl 693 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ↔ (𝑓:(0..^2)⟶dom 𝐸𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}))))
10 preq12 4214 . . . . . . . . . . . . . . . . 17 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1)) → {𝐴, 𝐵} = {(𝑝‘0), (𝑝‘1)})
11103adant3 1074 . . . . . . . . . . . . . . . 16 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → {𝐴, 𝐵} = {(𝑝‘0), (𝑝‘1)})
1211eqeq2d 2620 . . . . . . . . . . . . . . 15 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ↔ (𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)}))
13 preq12 4214 . . . . . . . . . . . . . . . . 17 ((𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → {𝐵, 𝐶} = {(𝑝‘1), (𝑝‘2)})
14133adant1 1072 . . . . . . . . . . . . . . . 16 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → {𝐵, 𝐶} = {(𝑝‘1), (𝑝‘2)})
1514eqeq2d 2620 . . . . . . . . . . . . . . 15 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐸‘(𝑓‘1)) = {𝐵, 𝐶} ↔ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}))
1612, 15anbi12d 743 . . . . . . . . . . . . . 14 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) ↔ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)})))
1716bicomd 212 . . . . . . . . . . . . 13 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}) ↔ ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶})))
18173anbi3d 1397 . . . . . . . . . . . 12 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑓:(0..^2)⟶dom 𝐸𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)})) ↔ (𝑓:(0..^2)⟶dom 𝐸𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}))))
19 usgrafun 25878 . . . . . . . . . . . . . . . . . 18 (𝑉 USGrph 𝐸 → Fun 𝐸)
20 c0ex 9913 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
2120prid1 4241 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ {0, 1}
22 fzo0to2pr 12420 . . . . . . . . . . . . . . . . . . . . . . 23 (0..^2) = {0, 1}
2321, 22eleqtrri 2687 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ (0..^2)
24 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓:(0..^2)⟶dom 𝐸 ∧ 0 ∈ (0..^2)) → (𝑓‘0) ∈ dom 𝐸)
2523, 24mpan2 703 . . . . . . . . . . . . . . . . . . . . 21 (𝑓:(0..^2)⟶dom 𝐸 → (𝑓‘0) ∈ dom 𝐸)
26 fvelrn 6260 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝐸 ∧ (𝑓‘0) ∈ dom 𝐸) → (𝐸‘(𝑓‘0)) ∈ ran 𝐸)
2725, 26sylan2 490 . . . . . . . . . . . . . . . . . . . 20 ((Fun 𝐸𝑓:(0..^2)⟶dom 𝐸) → (𝐸‘(𝑓‘0)) ∈ ran 𝐸)
28 eleq1 2676 . . . . . . . . . . . . . . . . . . . 20 ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} → ((𝐸‘(𝑓‘0)) ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
2927, 28syl5ibcom 234 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐸𝑓:(0..^2)⟶dom 𝐸) → ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} → {𝐴, 𝐵} ∈ ran 𝐸))
30 1ex 9914 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ V
3130prid2 4242 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ {0, 1}
3231, 22eleqtrri 2687 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ (0..^2)
33 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓:(0..^2)⟶dom 𝐸 ∧ 1 ∈ (0..^2)) → (𝑓‘1) ∈ dom 𝐸)
3432, 33mpan2 703 . . . . . . . . . . . . . . . . . . . . 21 (𝑓:(0..^2)⟶dom 𝐸 → (𝑓‘1) ∈ dom 𝐸)
35 fvelrn 6260 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝐸 ∧ (𝑓‘1) ∈ dom 𝐸) → (𝐸‘(𝑓‘1)) ∈ ran 𝐸)
3634, 35sylan2 490 . . . . . . . . . . . . . . . . . . . 20 ((Fun 𝐸𝑓:(0..^2)⟶dom 𝐸) → (𝐸‘(𝑓‘1)) ∈ ran 𝐸)
37 eleq1 2676 . . . . . . . . . . . . . . . . . . . 20 ((𝐸‘(𝑓‘1)) = {𝐵, 𝐶} → ((𝐸‘(𝑓‘1)) ∈ ran 𝐸 ↔ {𝐵, 𝐶} ∈ ran 𝐸))
3836, 37syl5ibcom 234 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐸𝑓:(0..^2)⟶dom 𝐸) → ((𝐸‘(𝑓‘1)) = {𝐵, 𝐶} → {𝐵, 𝐶} ∈ ran 𝐸))
3929, 38anim12d 584 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐸𝑓:(0..^2)⟶dom 𝐸) → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
4019, 39sylan 487 . . . . . . . . . . . . . . . . 17 ((𝑉 USGrph 𝐸𝑓:(0..^2)⟶dom 𝐸) → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
4140a1d 25 . . . . . . . . . . . . . . . 16 ((𝑉 USGrph 𝐸𝑓:(0..^2)⟶dom 𝐸) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
4241expcom 450 . . . . . . . . . . . . . . 15 (𝑓:(0..^2)⟶dom 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
4342com24 93 . . . . . . . . . . . . . 14 (𝑓:(0..^2)⟶dom 𝐸 → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
4443a1d 25 . . . . . . . . . . . . 13 (𝑓:(0..^2)⟶dom 𝐸 → (𝑝:(0...2)⟶𝑉 → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))))
45443imp 1249 . . . . . . . . . . . 12 ((𝑓:(0..^2)⟶dom 𝐸𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶})) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
4618, 45syl6bi 242 . . . . . . . . . . 11 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑓:(0..^2)⟶dom 𝐸𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)})) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
4746com14 94 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → ((𝑓:(0..^2)⟶dom 𝐸𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)})) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
489, 47sylbid 229 . . . . . . . . 9 (𝑉 USGrph 𝐸 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
4948com14 94 . . . . . . . 8 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
5049expdcom 454 . . . . . . 7 (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))))
51503imp 1249 . . . . . 6 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
5251com13 86 . . . . 5 (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉𝐶𝑉) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
5352imp 444 . . . 4 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
5453exlimdvv 1849 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
55 usg2wlk 26145 . . . . 5 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
56553expib 1260 . . . 4 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
5756adantr 480 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
5854, 57impbid 201 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
594, 58bitrd 267 1 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  {cpr 4127  ⟨cotp 4133   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  ...cfz 12197  ..^cfzo 12334  #chash 12979   USGrph cusg 25859   Walks cwalk 26026   2WalksOnOt c2wlkonot 26382 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wlkon 26042  df-2wlkonot 26385 This theorem is referenced by:  usg2spthonot  26415  usg2spthonot0  26416  frg2woteu  26582  frg2woteqm  26586
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