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Theorem wwlksnredwwlkn 29138
Description: For each walk (as word) of length at least 1 there is a shorter walk (as word). (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
wwlksnredwwlkn (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)))
Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑁   𝑦,π‘Š
Proof of Theorem wwlksnredwwlkn
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2733 . . 3 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (π‘Š prefix (𝑁 + 1)) = (π‘Š prefix (𝑁 + 1)))
2 eqid 2732 . . . . 5 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
3 wwlksnredwwlkn.e . . . . 5 𝐸 = (Edgβ€˜πΊ)
42, 3wwlknp 29086 . . . 4 (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))
5 simprl 769 . . . . . . . . 9 ((𝑁 ∈ β„•0 ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1))) β†’ π‘Š ∈ Word (Vtxβ€˜πΊ))
6 peano2nn0 12508 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
7 peano2nn0 12508 . . . . . . . . . . . . 13 ((𝑁 + 1) ∈ β„•0 β†’ ((𝑁 + 1) + 1) ∈ β„•0)
86, 7syl 17 . . . . . . . . . . . 12 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) + 1) ∈ β„•0)
9 id 22 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„•0)
10 nn0p1nn 12507 . . . . . . . . . . . . . 14 ((𝑁 + 1) ∈ β„•0 β†’ ((𝑁 + 1) + 1) ∈ β„•)
116, 10syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) + 1) ∈ β„•)
12 nn0re 12477 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ ℝ)
13 id 22 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ β†’ 𝑁 ∈ ℝ)
14 peano2re 11383 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ β†’ (𝑁 + 1) ∈ ℝ)
15 peano2re 11383 . . . . . . . . . . . . . . . . 17 ((𝑁 + 1) ∈ ℝ β†’ ((𝑁 + 1) + 1) ∈ ℝ)
1614, 15syl 17 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ β†’ ((𝑁 + 1) + 1) ∈ ℝ)
1713, 14, 163jca 1128 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℝ β†’ (𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ))
1812, 17syl 17 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ (𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ))
1912ltp1d 12140 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ 𝑁 < (𝑁 + 1))
20 nn0re 12477 . . . . . . . . . . . . . . . 16 ((𝑁 + 1) ∈ β„•0 β†’ (𝑁 + 1) ∈ ℝ)
216, 20syl 17 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ ℝ)
2221ltp1d 12140 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) < ((𝑁 + 1) + 1))
23 lttr 11286 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ) β†’ ((𝑁 < (𝑁 + 1) ∧ (𝑁 + 1) < ((𝑁 + 1) + 1)) β†’ 𝑁 < ((𝑁 + 1) + 1)))
2423imp 407 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ) ∧ (𝑁 < (𝑁 + 1) ∧ (𝑁 + 1) < ((𝑁 + 1) + 1))) β†’ 𝑁 < ((𝑁 + 1) + 1))
2518, 19, 22, 24syl12anc 835 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ 𝑁 < ((𝑁 + 1) + 1))
26 elfzo0 13669 . . . . . . . . . . . . 13 (𝑁 ∈ (0..^((𝑁 + 1) + 1)) ↔ (𝑁 ∈ β„•0 ∧ ((𝑁 + 1) + 1) ∈ β„• ∧ 𝑁 < ((𝑁 + 1) + 1)))
279, 11, 25, 26syl3anbrc 1343 . . . . . . . . . . . 12 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ (0..^((𝑁 + 1) + 1)))
28 fz0add1fz1 13698 . . . . . . . . . . . 12 ((((𝑁 + 1) + 1) ∈ β„•0 ∧ 𝑁 ∈ (0..^((𝑁 + 1) + 1))) β†’ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
298, 27, 28syl2anc 584 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
3029adantr 481 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1))) β†’ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
31 oveq2 7413 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ (1...(β™―β€˜π‘Š)) = (1...((𝑁 + 1) + 1)))
3231eleq2d 2819 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ ((𝑁 + 1) ∈ (1...(β™―β€˜π‘Š)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
3332adantl 482 . . . . . . . . . . 11 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1)) β†’ ((𝑁 + 1) ∈ (1...(β™―β€˜π‘Š)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
3433adantl 482 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1))) β†’ ((𝑁 + 1) ∈ (1...(β™―β€˜π‘Š)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
3530, 34mpbird 256 . . . . . . . . 9 ((𝑁 ∈ β„•0 ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1))) β†’ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š)))
365, 35jca 512 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1))) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š))))
37363adantr3 1171 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ (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†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š))))
38 pfxfvlsw 14641 . . . . . . 7 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 + 1) ∈ (1...(β™―β€˜π‘Š))) β†’ (lastSβ€˜(π‘Š prefix (𝑁 + 1))) = (π‘Šβ€˜((𝑁 + 1) βˆ’ 1)))
3937, 38syl 17 . . . . . 6 ((𝑁 ∈ β„•0 ∧ (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†’ (lastSβ€˜(π‘Š prefix (𝑁 + 1))) = (π‘Šβ€˜((𝑁 + 1) βˆ’ 1)))
40 lsw 14510 . . . . . . . 8 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
41403ad2ant1 1133 . . . . . . 7 ((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†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
4241adantl 482 . . . . . 6 ((𝑁 ∈ β„•0 ∧ (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†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
4339, 42preq12d 4744 . . . . 5 ((𝑁 ∈ β„•0 ∧ (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†’ {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)} = {(π‘Šβ€˜((𝑁 + 1) βˆ’ 1)), (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1))})
44 oveq1 7412 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (((𝑁 + 1) + 1) βˆ’ 1))
45443ad2ant2 1134 . . . . . . . . . 10 ((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†’ ((β™―β€˜π‘Š) βˆ’ 1) = (((𝑁 + 1) + 1) βˆ’ 1))
4645adantl 482 . . . . . . . . 9 ((𝑁 ∈ β„•0 ∧ (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†’ ((β™―β€˜π‘Š) βˆ’ 1) = (((𝑁 + 1) + 1) βˆ’ 1))
4746fveq2d 6892 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ (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†’ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)) = (π‘Šβ€˜(((𝑁 + 1) + 1) βˆ’ 1)))
4847preq2d 4743 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ (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†’ {(π‘Šβ€˜((𝑁 + 1) βˆ’ 1)), (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1))} = {(π‘Šβ€˜((𝑁 + 1) βˆ’ 1)), (π‘Šβ€˜(((𝑁 + 1) + 1) βˆ’ 1))})
49 nn0cn 12478 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
50 1cnd 11205 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ 1 ∈ β„‚)
5149, 50pncand 11568 . . . . . . . . . 10 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
5251fveq2d 6892 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (π‘Šβ€˜((𝑁 + 1) βˆ’ 1)) = (π‘Šβ€˜π‘))
536nn0cnd 12530 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„‚)
5453, 50pncand 11568 . . . . . . . . . 10 (𝑁 ∈ β„•0 β†’ (((𝑁 + 1) + 1) βˆ’ 1) = (𝑁 + 1))
5554fveq2d 6892 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (π‘Šβ€˜(((𝑁 + 1) + 1) βˆ’ 1)) = (π‘Šβ€˜(𝑁 + 1)))
5652, 55preq12d 4744 . . . . . . . 8 (𝑁 ∈ β„•0 β†’ {(π‘Šβ€˜((𝑁 + 1) βˆ’ 1)), (π‘Šβ€˜(((𝑁 + 1) + 1) βˆ’ 1))} = {(π‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))})
5756adantr 481 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ (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†’ {(π‘Šβ€˜((𝑁 + 1) βˆ’ 1)), (π‘Šβ€˜(((𝑁 + 1) + 1) βˆ’ 1))} = {(π‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))})
5848, 57eqtrd 2772 . . . . . 6 ((𝑁 ∈ β„•0 ∧ (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†’ {(π‘Šβ€˜((𝑁 + 1) βˆ’ 1)), (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1))} = {(π‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))})
59 fveq2 6888 . . . . . . . . . . . 12 (𝑖 = 𝑁 β†’ (π‘Šβ€˜π‘–) = (π‘Šβ€˜π‘))
60 fvoveq1 7428 . . . . . . . . . . . 12 (𝑖 = 𝑁 β†’ (π‘Šβ€˜(𝑖 + 1)) = (π‘Šβ€˜(𝑁 + 1)))
6159, 60preq12d 4744 . . . . . . . . . . 11 (𝑖 = 𝑁 β†’ {(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} = {(π‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))})
6261eleq1d 2818 . . . . . . . . . 10 (𝑖 = 𝑁 β†’ ({(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ↔ {(π‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))} ∈ 𝐸))
6362rspcv 3608 . . . . . . . . 9 (𝑁 ∈ (0..^(𝑁 + 1)) β†’ (βˆ€π‘– ∈ (0..^(𝑁 + 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 β†’ {(π‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))} ∈ 𝐸))
64 fzonn0p1 13705 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ (0..^(𝑁 + 1)))
6563, 64syl11 33 . . . . . . . 8 (βˆ€π‘– ∈ (0..^(𝑁 + 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 β†’ (𝑁 ∈ β„•0 β†’ {(π‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))} ∈ 𝐸))
66653ad2ant3 1135 . . . . . . 7 ((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‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))} ∈ 𝐸))
6766impcom 408 . . . . . 6 ((𝑁 ∈ β„•0 ∧ (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†’ {(π‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))} ∈ 𝐸)
6858, 67eqeltrd 2833 . . . . 5 ((𝑁 ∈ β„•0 ∧ (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†’ {(π‘Šβ€˜((𝑁 + 1) βˆ’ 1)), (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1))} ∈ 𝐸)
6943, 68eqeltrd 2833 . . . 4 ((𝑁 ∈ β„•0 ∧ (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†’ {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)} ∈ 𝐸)
704, 69sylan2 593 . . 3 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)} ∈ 𝐸)
71 wwlksnred 29135 . . . . 5 (𝑁 ∈ β„•0 β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
7271imp 407 . . . 4 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺))
73 eqeq2 2744 . . . . . 6 (𝑦 = (π‘Š prefix (𝑁 + 1)) β†’ ((π‘Š prefix (𝑁 + 1)) = 𝑦 ↔ (π‘Š prefix (𝑁 + 1)) = (π‘Š prefix (𝑁 + 1))))
74 fveq2 6888 . . . . . . . 8 (𝑦 = (π‘Š prefix (𝑁 + 1)) β†’ (lastSβ€˜π‘¦) = (lastSβ€˜(π‘Š prefix (𝑁 + 1))))
7574preq1d 4742 . . . . . . 7 (𝑦 = (π‘Š prefix (𝑁 + 1)) β†’ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} = {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)})
7675eleq1d 2818 . . . . . 6 (𝑦 = (π‘Š prefix (𝑁 + 1)) β†’ ({(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸 ↔ {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)} ∈ 𝐸))
7773, 76anbi12d 631 . . . . 5 (𝑦 = (π‘Š prefix (𝑁 + 1)) β†’ (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) ↔ ((π‘Š prefix (𝑁 + 1)) = (π‘Š prefix (𝑁 + 1)) ∧ {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)} ∈ 𝐸)))
7877adantl 482 . . . 4 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) ∧ 𝑦 = (π‘Š prefix (𝑁 + 1))) β†’ (((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸) ↔ ((π‘Š prefix (𝑁 + 1)) = (π‘Š prefix (𝑁 + 1)) ∧ {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)} ∈ 𝐸)))
7972, 78rspcedv 3605 . . 3 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (((π‘Š prefix (𝑁 + 1)) = (π‘Š prefix (𝑁 + 1)) ∧ {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)} ∈ 𝐸) β†’ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘Š prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} ∈ 𝐸)))
801, 70, 79mp2and 697 . 2 ((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))
8180ex 413 1 (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)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {cpr 4629   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  β„cr 11105  0cc0 11106  1c1 11107   + caddc 11109   < clt 11244   βˆ’ cmin 11440  β„•cn 12208  β„•0cn0 12468  ...cfz 13480  ..^cfzo 13623  β™―chash 14286  Word cword 14460  lastSclsw 14508   prefix cpfx 14616  Vtxcvtx 28245  Edgcedg 28296   WWalksN cwwlksn 29069
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-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-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  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-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-lsw 14509  df-substr 14587  df-pfx 14617  df-wwlks 29073  df-wwlksn 29074
This theorem is referenced by:  wwlksnredwwlkn0  29139
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