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Theorem wlkiswwlks2 29118
Description: A walk as word corresponds to the sequence of vertices in a walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
Assertion
Ref Expression
wlkiswwlks2 (𝐺 ∈ USPGraph β†’ (𝑃 ∈ (WWalksβ€˜πΊ) β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
Distinct variable groups:   𝑓,𝐺   𝑃,𝑓
Proof of Theorem wlkiswwlks2
Dummy variables 𝑖 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
21wwlkbp 29084 . . 3 (𝑃 ∈ (WWalksβ€˜πΊ) β†’ (𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)))
3 eqid 2732 . . . . 5 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
41, 3iswwlks 29079 . . . 4 (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)))
5 ovex 7438 . . . . . . . . . . . . . . 15 (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ V
6 mptexg 7219 . . . . . . . . . . . . . . 15 ((0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ V β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ V)
75, 6mp1i 13 . . . . . . . . . . . . . 14 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ V)
8 simprr 771 . . . . . . . . . . . . . . . . . 18 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ 𝐺 ∈ USPGraph)
9 simplr 767 . . . . . . . . . . . . . . . . . 18 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ 𝑃 ∈ Word (Vtxβ€˜πΊ))
10 hashge1 14345 . . . . . . . . . . . . . . . . . . . 20 ((𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑃 β‰  βˆ…) β†’ 1 ≀ (β™―β€˜π‘ƒ))
1110ancoms 459 . . . . . . . . . . . . . . . . . . 19 ((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ 1 ≀ (β™―β€˜π‘ƒ))
1211adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ 1 ≀ (β™―β€˜π‘ƒ))
138, 9, 123jca 1128 . . . . . . . . . . . . . . . . 17 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ (𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ 1 ≀ (β™―β€˜π‘ƒ)))
1413adantr 481 . . . . . . . . . . . . . . . 16 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ (𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ 1 ≀ (β™―β€˜π‘ƒ)))
15 edgval 28298 . . . . . . . . . . . . . . . . . . . 20 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
1615a1i 11 . . . . . . . . . . . . . . . . . . 19 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ))
1716eleq2d 2819 . . . . . . . . . . . . . . . . . 18 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran (iEdgβ€˜πΊ)))
1817ralbidv 3177 . . . . . . . . . . . . . . . . 17 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 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‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran (iEdgβ€˜πΊ)))
1918biimpd 228 . . . . . . . . . . . . . . . 16 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran (iEdgβ€˜πΊ)))
20 eqid 2732 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))
21 eqid 2732 . . . . . . . . . . . . . . . . 17 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
2220, 21wlkiswwlks2lem6 29117 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran (iEdgβ€˜πΊ) β†’ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
2314, 19, 22sylsyld 61 . . . . . . . . . . . . . . 15 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
24 eleq1 2821 . . . . . . . . . . . . . . . . . 18 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ↔ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ)))
25 fveq2 6888 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (β™―β€˜π‘“) = (β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))
2625oveq2d 7421 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (0...(β™―β€˜π‘“)) = (0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})))))
2726feq2d 6700 . . . . . . . . . . . . . . . . . 18 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ↔ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ)))
2825oveq2d 7421 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (0..^(β™―β€˜π‘“)) = (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})))))
29 fveq1 6887 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (π‘“β€˜π‘–) = ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–))
3029fveqeq2d 6896 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ ((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
3128, 30raleqbidv 3342 . . . . . . . . . . . . . . . . . 18 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
3224, 27, 313anbi123d 1436 . . . . . . . . . . . . . . . . 17 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ ((𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ↔ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
3332imbi2d 340 . . . . . . . . . . . . . . . 16 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) ↔ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))))
3433adantl 482 . . . . . . . . . . . . . . 15 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) ↔ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))))
3523, 34mpbird 256 . . . . . . . . . . . . . 14 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
367, 35spcimedv 3585 . . . . . . . . . . . . 13 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
3736ex 413 . . . . . . . . . . . 12 ((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))))
3837com23 86 . . . . . . . . . . 11 ((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph) β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))))
39383impia 1117 . . . . . . . . . 10 ((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)) β†’ (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph) β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4039expd 416 . . . . . . . . 9 ((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)) β†’ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))))
4140impcom 408 . . . . . . . 8 (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜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))) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4241imp 407 . . . . . . 7 ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜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))) ∧ 𝐺 ∈ USPGraph) β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
43 uspgrupgr 28425 . . . . . . . . . 10 (𝐺 ∈ USPGraph β†’ 𝐺 ∈ UPGraph)
441, 21upgriswlk 28887 . . . . . . . . . 10 (𝐺 ∈ UPGraph β†’ (𝑓(Walksβ€˜πΊ)𝑃 ↔ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4543, 44syl 17 . . . . . . . . 9 (𝐺 ∈ USPGraph β†’ (𝑓(Walksβ€˜πΊ)𝑃 ↔ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4645adantl 482 . . . . . . . 8 ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜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))) ∧ 𝐺 ∈ USPGraph) β†’ (𝑓(Walksβ€˜πΊ)𝑃 ↔ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4746exbidv 1924 . . . . . . 7 ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜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))) ∧ 𝐺 ∈ USPGraph) β†’ (βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃 ↔ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4842, 47mpbird 256 . . . . . 6 ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜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))) ∧ 𝐺 ∈ USPGraph) β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃)
4948ex 413 . . . . 5 (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜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))) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
5049ex 413 . . . 4 ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ ((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)) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃)))
514, 50biimtrid 241 . . 3 ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (𝑃 ∈ (WWalksβ€˜πΊ) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃)))
522, 51mpcom 38 . 2 (𝑃 ∈ (WWalksβ€˜πΊ) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
5352com12 32 1 (𝐺 ∈ USPGraph β†’ (𝑃 ∈ (WWalksβ€˜πΊ) β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  Vcvv 3474  βˆ…c0 4321  {cpr 4629   class class class wbr 5147   ↦ cmpt 5230  β—‘ccnv 5674  dom cdm 5675  ran crn 5676  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  0cc0 11106  1c1 11107   + caddc 11109   ≀ cle 11245   βˆ’ cmin 11440  ...cfz 13480  ..^cfzo 13623  β™―chash 14286  Word cword 14460  Vtxcvtx 28245  iEdgciedg 28246  Edgcedg 28296  UPGraphcupgr 28329  USPGraphcuspgr 28397  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-uspgr 28399  df-wlks 28845  df-wwlks 29073
This theorem is referenced by:  wlkiswwlks  29119  wlklnwwlkln2  29126
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