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Theorem wwlksnredwwlkn0 28938
Description: For each walk (as word) of length at least 1 there is a shorter walk (as word) starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 26-Oct-2022.)
Hypothesis
Ref Expression
wwlksnredwwlkn.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
wwlksnredwwlkn0 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ ((π‘Šβ€˜0) = 𝑃 ↔ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)))
Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑁   𝑦,π‘Š   𝑦,𝑃
Proof of Theorem wwlksnredwwlkn0
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlksnredwwlkn.e . . . . 5 𝐸 = (Edgβ€˜πΊ)
21wwlksnredwwlkn 28937 . . . 4 (𝑁 ∈ β„•0 β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)))
32imp 407 . . 3 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸))
4 simpl 483 . . . . . . . . 9 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) β†’ (π‘Š prefix (𝑁 + 1)) = 𝑦)
54adantl 482 . . . . . . . 8 (((((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) ∧ ((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)) β†’ (π‘Š prefix (𝑁 + 1)) = 𝑦)
6 fveq1 6861 . . . . . . . . . . . . . 14 (𝑦 = (π‘Š prefix (𝑁 + 1)) β†’ (π‘¦β€˜0) = ((π‘Š prefix (𝑁 + 1))β€˜0))
76eqcoms 2739 . . . . . . . . . . . . 13 ((π‘Š prefix (𝑁 + 1)) = 𝑦 β†’ (π‘¦β€˜0) = ((π‘Š prefix (𝑁 + 1))β€˜0))
87adantr 481 . . . . . . . . . . . 12 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺))) β†’ (π‘¦β€˜0) = ((π‘Š prefix (𝑁 + 1))β€˜0))
9 eqid 2731 . . . . . . . . . . . . . . . . . . 19 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
109, 1wwlknp 28885 . . . . . . . . . . . . . . . . . 18 (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1) ∧ βˆ€π‘– ∈ (0..^(𝑁 + 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸))
11 nn0p1nn 12476 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
12 peano2nn0 12477 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
13 nn0re 12446 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ β„•0 β†’ (𝑁 + 1) ∈ ℝ)
14 lep1 12020 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ ℝ β†’ (𝑁 + 1) ≀ ((𝑁 + 1) + 1))
1512, 13, 143syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ≀ ((𝑁 + 1) + 1))
16 peano2nn0 12477 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) ∈ β„•0 β†’ ((𝑁 + 1) + 1) ∈ β„•0)
1716nn0zd 12549 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ β„•0 β†’ ((𝑁 + 1) + 1) ∈ β„€)
18 fznn 13534 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑁 + 1) + 1) ∈ β„€ β†’ ((𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)) ↔ ((𝑁 + 1) ∈ β„• ∧ (𝑁 + 1) ≀ ((𝑁 + 1) + 1))))
1912, 17, 183syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)) ↔ ((𝑁 + 1) ∈ β„• ∧ (𝑁 + 1) ≀ ((𝑁 + 1) + 1))))
2011, 15, 19mpbir2and 711 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
21 oveq2 7385 . . . . . . . . . . . . . . . . . . . . . . 23 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ (1...(β™―β€˜π‘Š)) = (1...((𝑁 + 1) + 1)))
2221eleq2d 2818 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ ((𝑁 + 1) ∈ (1...(β™―β€˜π‘Š)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
2320, 22imbitrrid 245 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š))))
2423adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1)) β†’ (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š))))
25 simpl 483 . . . . . . . . . . . . . . . . . . . 20 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1)) β†’ π‘Š ∈ Word (Vtxβ€˜πΊ))
2624, 25jctild 526 . . . . . . . . . . . . . . . . . . 19 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1)) β†’ (𝑁 ∈ β„•0 β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š)))))
27263adant3 1132 . . . . . . . . . . . . . . . . . 18 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1) ∧ βˆ€π‘– ∈ (0..^(𝑁 + 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸) β†’ (𝑁 ∈ β„•0 β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š)))))
2810, 27syl 17 . . . . . . . . . . . . . . . . 17 (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (𝑁 ∈ β„•0 β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š)))))
2928impcom 408 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š))))
3029adantl 482 . . . . . . . . . . . . . . 15 (((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š))))
3130adantr 481 . . . . . . . . . . . . . 14 ((((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š))))
3231adantl 482 . . . . . . . . . . . . 13 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺))) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š))))
33 pfxfv0 14607 . . . . . . . . . . . . 13 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š))) β†’ ((π‘Š prefix (𝑁 + 1))β€˜0) = (π‘Šβ€˜0))
3432, 33syl 17 . . . . . . . . . . . 12 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺))) β†’ ((π‘Š prefix (𝑁 + 1))β€˜0) = (π‘Šβ€˜0))
35 simprll 777 . . . . . . . . . . . 12 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺))) β†’ (π‘Šβ€˜0) = 𝑃)
368, 34, 353eqtrd 2775 . . . . . . . . . . 11 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺))) β†’ (π‘¦β€˜0) = 𝑃)
3736ex 413 . . . . . . . . . 10 ((π‘Š prefix (𝑁 + 1)) = 𝑦 β†’ ((((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) β†’ (π‘¦β€˜0) = 𝑃))
3837adantr 481 . . . . . . . . 9 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) β†’ ((((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) β†’ (π‘¦β€˜0) = 𝑃))
3938impcom 408 . . . . . . . 8 (((((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) ∧ ((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)) β†’ (π‘¦β€˜0) = 𝑃)
40 simpr 485 . . . . . . . . 9 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) β†’ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)
4140adantl 482 . . . . . . . 8 (((((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) ∧ ((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)) β†’ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)
425, 39, 413jca 1128 . . . . . . 7 (((((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) ∧ ((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)) β†’ ((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸))
4342ex 413 . . . . . 6 ((((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) β†’ (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) β†’ ((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)))
4443reximdva 3167 . . . . 5 (((π‘Šβ€˜0) = 𝑃 ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) β†’ (βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) β†’ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)))
4544ex 413 . . . 4 ((π‘Šβ€˜0) = 𝑃 β†’ ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) β†’ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸))))
4645com13 88 . . 3 (βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) β†’ ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ ((π‘Šβ€˜0) = 𝑃 β†’ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸))))
473, 46mpcom 38 . 2 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ ((π‘Šβ€˜0) = 𝑃 β†’ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)))
4829, 33syl 17 . . . . . . . . 9 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ ((π‘Š prefix (𝑁 + 1))β€˜0) = (π‘Šβ€˜0))
4948eqcomd 2737 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (π‘Šβ€˜0) = ((π‘Š prefix (𝑁 + 1))β€˜0))
5049adantl 482 . . . . . . 7 ((((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃) ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) β†’ (π‘Šβ€˜0) = ((π‘Š prefix (𝑁 + 1))β€˜0))
51 fveq1 6861 . . . . . . . . 9 ((π‘Š prefix (𝑁 + 1)) = 𝑦 β†’ ((π‘Š prefix (𝑁 + 1))β€˜0) = (π‘¦β€˜0))
5251adantr 481 . . . . . . . 8 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃) β†’ ((π‘Š prefix (𝑁 + 1))β€˜0) = (π‘¦β€˜0))
5352adantr 481 . . . . . . 7 ((((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃) ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) β†’ ((π‘Š prefix (𝑁 + 1))β€˜0) = (π‘¦β€˜0))
54 simpr 485 . . . . . . . 8 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃) β†’ (π‘¦β€˜0) = 𝑃)
5554adantr 481 . . . . . . 7 ((((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃) ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) β†’ (π‘¦β€˜0) = 𝑃)
5650, 53, 553eqtrd 2775 . . . . . 6 ((((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃) ∧ (𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))) β†’ (π‘Šβ€˜0) = 𝑃)
5756ex 413 . . . . 5 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃) β†’ ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (π‘Šβ€˜0) = 𝑃))
58573adant3 1132 . . . 4 (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) β†’ ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (π‘Šβ€˜0) = 𝑃))
5958com12 32 . . 3 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) β†’ (π‘Šβ€˜0) = 𝑃))
6059rexlimdvw 3159 . 2 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) β†’ (π‘Šβ€˜0) = 𝑃))
6147, 60impbid 211 1 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ ((π‘Šβ€˜0) = 𝑃 ↔ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3060  βˆƒwrex 3069  {cpr 4608   class class class wbr 5125  β€˜cfv 6516  (class class class)co 7377  β„cr 11074  0cc0 11075  1c1 11076   + caddc 11078   ≀ cle 11214  β„•cn 12177  β„•0cn0 12437  β„€cz 12523  ...cfz 13449  ..^cfzo 13592  β™―chash 14255  Word cword 14429  lastSclsw 14477   prefix cpfx 14585  Vtxcvtx 28044  Edgcedg 28095   WWalksN cwwlksn 28868
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 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-1st 7941  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8670  df-map 8789  df-en 8906  df-dom 8907  df-sdom 8908  df-fin 8909  df-card 9899  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-n0 12438  df-z 12524  df-uz 12788  df-fz 13450  df-fzo 13593  df-hash 14256  df-word 14430  df-lsw 14478  df-substr 14556  df-pfx 14586  df-wwlks 28872  df-wwlksn 28873
This theorem is referenced by:  rusgrnumwwlks  29016
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