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Theorem umgr2wlk 27727
Description: In a multigraph, there is a walk of length 2 for each pair of adjacent edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
Hypothesis
Ref Expression
umgr2wlk.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
umgr2wlk ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
Distinct variable groups:   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝐶,𝑓,𝑝   𝑓,𝐺,𝑝
Allowed substitution hints:   𝐸(𝑓,𝑝)

Proof of Theorem umgr2wlk
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 umgruhgr 26888 . . . . . 6 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
2 umgr2wlk.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
32eleq2i 2904 . . . . . . 7 ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ (Edg‘𝐺))
4 eqid 2821 . . . . . . . 8 (iEdg‘𝐺) = (iEdg‘𝐺)
54uhgredgiedgb 26910 . . . . . . 7 (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
63, 5syl5bb 285 . . . . . 6 (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
71, 6syl 17 . . . . 5 (𝐺 ∈ UMGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
87biimpd 231 . . . 4 (𝐺 ∈ UMGraph → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
98a1d 25 . . 3 (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))))
1093imp 1107 . 2 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))
112eleq2i 2904 . . . . . . 7 ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ (Edg‘𝐺))
124uhgredgiedgb 26910 . . . . . . 7 (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ (Edg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
1311, 12syl5bb 285 . . . . . 6 (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
141, 13syl 17 . . . . 5 (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
1514biimpd 231 . . . 4 (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
1615a1dd 50 . . 3 (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))))
17163imp 1107 . 2 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))
18 s2cli 14241 . . . . . . . . . 10 ⟨“𝑗𝑖”⟩ ∈ Word V
19 s3cli 14242 . . . . . . . . . 10 ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
2018, 19pm3.2i 473 . . . . . . . . 9 (⟨“𝑗𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V)
21 eqid 2821 . . . . . . . . . 10 ⟨“𝑗𝑖”⟩ = ⟨“𝑗𝑖”⟩
22 eqid 2821 . . . . . . . . . 10 ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩
23 simpl1 1187 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → 𝐺 ∈ UMGraph)
24 3simpc 1146 . . . . . . . . . . 11 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
2524adantr 483 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
26 simpl 485 . . . . . . . . . . . 12 (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))
2726eqcomd 2827 . . . . . . . . . . 11 (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵})
2827adantl 484 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵})
29 simpr 487 . . . . . . . . . . . 12 (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))
3029eqcomd 2827 . . . . . . . . . . 11 (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐺)‘𝑖) = {𝐵, 𝐶})
3130adantl 484 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ((iEdg‘𝐺)‘𝑖) = {𝐵, 𝐶})
322, 4, 21, 22, 23, 25, 28, 31umgr2adedgwlk 27723 . . . . . . . . 9 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → (⟨“𝑗𝑖”⟩(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑗𝑖”⟩) = 2 ∧ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
33 breq12 5070 . . . . . . . . . . 11 ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → (𝑓(Walks‘𝐺)𝑝 ↔ ⟨“𝑗𝑖”⟩(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
34 fveqeq2 6678 . . . . . . . . . . . 12 (𝑓 = ⟨“𝑗𝑖”⟩ → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑗𝑖”⟩) = 2))
3534adantr 483 . . . . . . . . . . 11 ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑗𝑖”⟩) = 2))
36 fveq1 6668 . . . . . . . . . . . . . 14 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
3736eqeq2d 2832 . . . . . . . . . . . . 13 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐴 = (𝑝‘0) ↔ 𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0)))
38 fveq1 6668 . . . . . . . . . . . . . 14 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
3938eqeq2d 2832 . . . . . . . . . . . . 13 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐵 = (𝑝‘1) ↔ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1)))
40 fveq1 6668 . . . . . . . . . . . . . 14 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
4140eqeq2d 2832 . . . . . . . . . . . . 13 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐶 = (𝑝‘2) ↔ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2)))
4237, 39, 413anbi123d 1432 . . . . . . . . . . . 12 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
4342adantl 484 . . . . . . . . . . 11 ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
4433, 35, 433anbi123d 1432 . . . . . . . . . 10 ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ (⟨“𝑗𝑖”⟩(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑗𝑖”⟩) = 2 ∧ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2)))))
4544spc2egv 3599 . . . . . . . . 9 ((⟨“𝑗𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V) → ((⟨“𝑗𝑖”⟩(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑗𝑖”⟩) = 2 ∧ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
4620, 32, 45mpsyl 68 . . . . . . . 8 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
4746exp32 423 . . . . . . 7 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
4847com12 32 . . . . . 6 ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
4948rexlimivw 3282 . . . . 5 (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
5049com13 88 . . . 4 ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
5150rexlimivw 3282 . . 3 (∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
5251com12 32 . 2 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → (∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
5310, 17, 52mp2d 49 1 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wex 1776  wcel 2110  wrex 3139  Vcvv 3494  {cpr 4568   class class class wbr 5065  dom cdm 5554  cfv 6354  0cc0 10536  1c1 10537  2c2 11691  chash 13689  Word cword 13860  ⟨“cs2 14202  ⟨“cs3 14203  iEdgciedg 26781  Edgcedg 26831  UHGraphcuhgr 26840  UMGraphcumgr 26865  Walkscwlks 27377
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-dju 9329  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12892  df-fzo 13033  df-hash 13690  df-word 13861  df-concat 13922  df-s1 13949  df-s2 14209  df-s3 14210  df-edg 26832  df-uhgr 26842  df-upgr 26866  df-umgr 26867  df-wlks 27380
This theorem is referenced by:  umgr2wlkon  27728  umgrwwlks2on  27735
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