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Theorem wlkiswwlks1 29110
Description: The sequence of vertices in a walk is a walk as word in a pseudograph. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.)
Assertion
Ref Expression
wlkiswwlks1 (𝐺 ∈ UPGraph β†’ (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃 ∈ (WWalksβ€˜πΊ)))
Proof of Theorem wlkiswwlks1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wlkn0 28867 . 2 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃 β‰  βˆ…)
2 eqid 2732 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
3 eqid 2732 . . . 4 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
42, 3upgriswlk 28887 . . 3 (𝐺 ∈ UPGraph β†’ (𝐹(Walksβ€˜πΊ)𝑃 ↔ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
5 simpr 485 . . . . . 6 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) ∧ 𝑃 β‰  βˆ…) β†’ 𝑃 β‰  βˆ…)
6 ffz0iswrd 14487 . . . . . . . 8 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ 𝑃 ∈ Word (Vtxβ€˜πΊ))
763ad2ant2 1134 . . . . . . 7 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ 𝑃 ∈ Word (Vtxβ€˜πΊ))
87ad2antlr 725 . . . . . 6 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) ∧ 𝑃 β‰  βˆ…) β†’ 𝑃 ∈ Word (Vtxβ€˜πΊ))
9 upgruhgr 28351 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ UPGraph β†’ 𝐺 ∈ UHGraph)
103uhgrfun 28315 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ UHGraph β†’ Fun (iEdgβ€˜πΊ))
11 funfn 6575 . . . . . . . . . . . . . . . . . . 19 (Fun (iEdgβ€˜πΊ) ↔ (iEdgβ€˜πΊ) Fn dom (iEdgβ€˜πΊ))
1211biimpi 215 . . . . . . . . . . . . . . . . . 18 (Fun (iEdgβ€˜πΊ) β†’ (iEdgβ€˜πΊ) Fn dom (iEdgβ€˜πΊ))
139, 10, 123syl 18 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ UPGraph β†’ (iEdgβ€˜πΊ) Fn dom (iEdgβ€˜πΊ))
1413ad2antlr 725 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ (iEdgβ€˜πΊ) Fn dom (iEdgβ€˜πΊ))
15 wrdsymbcl 14473 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ (πΉβ€˜π‘–) ∈ dom (iEdgβ€˜πΊ))
1615ad4ant14 750 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ (πΉβ€˜π‘–) ∈ dom (iEdgβ€˜πΊ))
17 fnfvelrn 7079 . . . . . . . . . . . . . . . 16 (((iEdgβ€˜πΊ) Fn dom (iEdgβ€˜πΊ) ∧ (πΉβ€˜π‘–) ∈ dom (iEdgβ€˜πΊ)) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) ∈ ran (iEdgβ€˜πΊ))
1814, 16, 17syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) ∈ ran (iEdgβ€˜πΊ))
19 edgval 28298 . . . . . . . . . . . . . . 15 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
2018, 19eleqtrrdi 2844 . . . . . . . . . . . . . 14 ((((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) ∈ (Edgβ€˜πΊ))
21 eleq1 2821 . . . . . . . . . . . . . . 15 ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) ∈ (Edgβ€˜πΊ)))
2221eqcoms 2740 . . . . . . . . . . . . . 14 (((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) ∈ (Edgβ€˜πΊ)))
2320, 22syl5ibrcom 246 . . . . . . . . . . . . 13 ((((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
2423ralimdva 3167 . . . . . . . . . . . 12 (((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐺 ∈ UPGraph) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
2524ex 413 . . . . . . . . . . 11 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) β†’ (𝐺 ∈ UPGraph β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))))
2625com23 86 . . . . . . . . . 10 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ (𝐺 ∈ UPGraph β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))))
27263impia 1117 . . . . . . . . 9 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ (𝐺 ∈ UPGraph β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
2827impcom 408 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))
29 lencl 14479 . . . . . . . . . . . . . 14 (𝐹 ∈ Word dom (iEdgβ€˜πΊ) β†’ (β™―β€˜πΉ) ∈ β„•0)
30 ffz0hash 14402 . . . . . . . . . . . . . . . 16 (((β™―β€˜πΉ) ∈ β„•0 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) β†’ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1))
3130ex 413 . . . . . . . . . . . . . . 15 ((β™―β€˜πΉ) ∈ β„•0 β†’ (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)))
32 oveq1 7412 . . . . . . . . . . . . . . . . 17 ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (((β™―β€˜πΉ) + 1) βˆ’ 1))
33 nn0cn 12478 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„‚)
34 pncan1 11634 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜πΉ) ∈ β„‚ β†’ (((β™―β€˜πΉ) + 1) βˆ’ 1) = (β™―β€˜πΉ))
3533, 34syl 17 . . . . . . . . . . . . . . . . 17 ((β™―β€˜πΉ) ∈ β„•0 β†’ (((β™―β€˜πΉ) + 1) βˆ’ 1) = (β™―β€˜πΉ))
3632, 35sylan9eqr 2794 . . . . . . . . . . . . . . . 16 (((β™―β€˜πΉ) ∈ β„•0 ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (β™―β€˜πΉ))
3736ex 413 . . . . . . . . . . . . . . 15 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (β™―β€˜πΉ)))
3831, 37syld 47 . . . . . . . . . . . . . 14 ((β™―β€˜πΉ) ∈ β„•0 β†’ (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (β™―β€˜πΉ)))
3929, 38syl 17 . . . . . . . . . . . . 13 (𝐹 ∈ Word dom (iEdgβ€˜πΊ) β†’ (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (β™―β€˜πΉ)))
4039imp 407 . . . . . . . . . . . 12 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (β™―β€˜πΉ))
4140oveq2d 7421 . . . . . . . . . . 11 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) β†’ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) = (0..^(β™―β€˜πΉ)))
4241raleqdv 3325 . . . . . . . . . 10 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
43423adant3 1132 . . . . . . . . 9 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
4443adantl 482 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
4528, 44mpbird 256 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))
4645adantr 481 . . . . . 6 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) ∧ 𝑃 β‰  βˆ…) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))
47 eqid 2732 . . . . . . 7 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
482, 47iswwlks 29079 . . . . . 6 (𝑃 ∈ (WWalksβ€˜πΊ) ↔ (𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
495, 8, 46, 48syl3anbrc 1343 . . . . 5 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) ∧ 𝑃 β‰  βˆ…) β†’ 𝑃 ∈ (WWalksβ€˜πΊ))
5049ex 413 . . . 4 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) β†’ (𝑃 β‰  βˆ… β†’ 𝑃 ∈ (WWalksβ€˜πΊ)))
5150ex 413 . . 3 (𝐺 ∈ UPGraph β†’ ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ (𝑃 β‰  βˆ… β†’ 𝑃 ∈ (WWalksβ€˜πΊ))))
524, 51sylbid 239 . 2 (𝐺 ∈ UPGraph β†’ (𝐹(Walksβ€˜πΊ)𝑃 β†’ (𝑃 β‰  βˆ… β†’ 𝑃 ∈ (WWalksβ€˜πΊ))))
531, 52mpdi 45 1 (𝐺 ∈ UPGraph β†’ (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃 ∈ (WWalksβ€˜πΊ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆ…c0 4321  {cpr 4629   class class class wbr 5147  dom cdm 5675  ran crn 5676  Fun wfun 6534   Fn wfn 6535  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  0cc0 11106  1c1 11107   + caddc 11109   βˆ’ cmin 11440  β„•0cn0 12468  ...cfz 13480  ..^cfzo 13623  β™―chash 14286  Word cword 14460  Vtxcvtx 28245  iEdgciedg 28246  Edgcedg 28296  UHGraphcuhgr 28305  UPGraphcupgr 28329  Walkscwlks 28842  WWalkscwwlks 29068
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-edg 28297  df-uhgr 28307  df-upgr 28331  df-wlks 28845  df-wwlks 29073
This theorem is referenced by:  wlklnwwlkln1  29111  wlkiswwlks  29119  wlkiswwlkupgr  29121  elwspths2spth  29210
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