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Theorem wlkiswwlks1 29388
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 29145 . 2 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃 β‰  βˆ…)
2 eqid 2730 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
3 eqid 2730 . . . 4 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
42, 3upgriswlk 29165 . . 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 483 . . . . . 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 14495 . . . . . . . 8 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ 𝑃 ∈ Word (Vtxβ€˜πΊ))
763ad2ant2 1132 . . . . . . 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 723 . . . . . 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 28629 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ UPGraph β†’ 𝐺 ∈ UHGraph)
103uhgrfun 28593 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ UHGraph β†’ Fun (iEdgβ€˜πΊ))
11 funfn 6577 . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . 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 14481 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ (πΉβ€˜π‘–) ∈ dom (iEdgβ€˜πΊ))
1615ad4ant14 748 . . . . . . . . . . . . . . . 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 7081 . . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . . 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 28576 . . . . . . . . . . . . . . 15 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
2018, 19eleqtrrdi 2842 . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . 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 411 . . . . . . . . . . 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 1115 . . . . . . . . 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 406 . . . . . . . 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 14487 . . . . . . . . . . . . . 14 (𝐹 ∈ Word dom (iEdgβ€˜πΊ) β†’ (β™―β€˜πΉ) ∈ β„•0)
30 ffz0hash 14410 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . 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 7418 . . . . . . . . . . . . . . . . 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 12486 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„‚)
34 pncan1 11642 . . . . . . . . . . . . . . . . . 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 2792 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . 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 7427 . . . . . . . . . . 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 3323 . . . . . . . . . 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 1130 . . . . . . . . 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 480 . . . . . . . 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 479 . . . . . 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 2730 . . . . . . 7 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
482, 47iswwlks 29357 . . . . . 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 1341 . . . . 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 411 . . . 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 411 . . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆ…c0 4321  {cpr 4629   class class class wbr 5147  dom cdm 5675  ran crn 5676  Fun wfun 6536   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  0cc0 11112  1c1 11113   + caddc 11115   βˆ’ cmin 11448  β„•0cn0 12476  ...cfz 13488  ..^cfzo 13631  β™―chash 14294  Word cword 14468  Vtxcvtx 28523  iEdgciedg 28524  Edgcedg 28574  UHGraphcuhgr 28583  UPGraphcupgr 28607  Walkscwlks 29120  WWalkscwwlks 29346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-edg 28575  df-uhgr 28585  df-upgr 28609  df-wlks 29123  df-wwlks 29351
This theorem is referenced by:  wlklnwwlkln1  29389  wlkiswwlks  29397  wlkiswwlkupgr  29399  elwspths2spth  29488
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