MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wwlksnredwwlkn Structured version   Visualization version   GIF version
Theorem wwlksnredwwlkn 29417
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 2732 . . 3 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (π‘Š prefix (𝑁 + 1)) = (π‘Š prefix (𝑁 + 1)))
2 eqid 2731 . . . . 5 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
3 wwlksnredwwlkn.e . . . . 5 𝐸 = (Edgβ€˜πΊ)
42, 3wwlknp 29365 . . . 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 768 . . . . . . . . 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 12517 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
7 peano2nn0 12517 . . . . . . . . . . . . 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 12516 . . . . . . . . . . . . . 14 ((𝑁 + 1) ∈ β„•0 β†’ ((𝑁 + 1) + 1) ∈ β„•)
116, 10syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) + 1) ∈ β„•)
12 nn0re 12486 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ ℝ)
13 id 22 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ β†’ 𝑁 ∈ ℝ)
14 peano2re 11392 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ β†’ (𝑁 + 1) ∈ ℝ)
15 peano2re 11392 . . . . . . . . . . . . . . . . 17 ((𝑁 + 1) ∈ ℝ β†’ ((𝑁 + 1) + 1) ∈ ℝ)
1614, 15syl 17 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ β†’ ((𝑁 + 1) + 1) ∈ ℝ)
1713, 14, 163jca 1127 . . . . . . . . . . . . . . 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 12149 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ 𝑁 < (𝑁 + 1))
20 nn0re 12486 . . . . . . . . . . . . . . . 16 ((𝑁 + 1) ∈ β„•0 β†’ (𝑁 + 1) ∈ ℝ)
216, 20syl 17 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ ℝ)
2221ltp1d 12149 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) < ((𝑁 + 1) + 1))
23 lttr 11295 . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . 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 834 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ 𝑁 < ((𝑁 + 1) + 1))
26 elfzo0 13678 . . . . . . . . . . . . 13 (𝑁 ∈ (0..^((𝑁 + 1) + 1)) ↔ (𝑁 ∈ β„•0 ∧ ((𝑁 + 1) + 1) ∈ β„• ∧ 𝑁 < ((𝑁 + 1) + 1)))
279, 11, 25, 26syl3anbrc 1342 . . . . . . . . . . . 12 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ (0..^((𝑁 + 1) + 1)))
28 fz0add1fz1 13707 . . . . . . . . . . . 12 ((((𝑁 + 1) + 1) ∈ β„•0 ∧ 𝑁 ∈ (0..^((𝑁 + 1) + 1))) β†’ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
298, 27, 28syl2anc 583 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
3029adantr 480 . . . . . . . . . 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 7420 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ (1...(β™―β€˜π‘Š)) = (1...((𝑁 + 1) + 1)))
3231eleq2d 2818 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ ((𝑁 + 1) ∈ (1...(β™―β€˜π‘Š)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
3332adantl 481 . . . . . . . . . . 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 481 . . . . . . . . . 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 257 . . . . . . . . 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 511 . . . . . . . 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 1170 . . . . . . 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 14650 . . . . . . 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 14519 . . . . . . . 8 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
41403ad2ant1 1132 . . . . . . 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 481 . . . . . 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 4745 . . . . 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 7419 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (((𝑁 + 1) + 1) βˆ’ 1))
45443ad2ant2 1133 . . . . . . . . . 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 481 . . . . . . . . 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 6895 . . . . . . . 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 4744 . . . . . . 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 12487 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
50 1cnd 11214 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ 1 ∈ β„‚)
5149, 50pncand 11577 . . . . . . . . . 10 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
5251fveq2d 6895 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (π‘Šβ€˜((𝑁 + 1) βˆ’ 1)) = (π‘Šβ€˜π‘))
536nn0cnd 12539 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„‚)
5453, 50pncand 11577 . . . . . . . . . 10 (𝑁 ∈ β„•0 β†’ (((𝑁 + 1) + 1) βˆ’ 1) = (𝑁 + 1))
5554fveq2d 6895 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (π‘Šβ€˜(((𝑁 + 1) + 1) βˆ’ 1)) = (π‘Šβ€˜(𝑁 + 1)))
5652, 55preq12d 4745 . . . . . . . 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 480 . . . . . . 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 2771 . . . . . 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 6891 . . . . . . . . . . . 12 (𝑖 = 𝑁 β†’ (π‘Šβ€˜π‘–) = (π‘Šβ€˜π‘))
60 fvoveq1 7435 . . . . . . . . . . . 12 (𝑖 = 𝑁 β†’ (π‘Šβ€˜(𝑖 + 1)) = (π‘Šβ€˜(𝑁 + 1)))
6159, 60preq12d 4745 . . . . . . . . . . 11 (𝑖 = 𝑁 β†’ {(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} = {(π‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))})
6261eleq1d 2817 . . . . . . . . . 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 13714 . . . . . . . . 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 1134 . . . . . . 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 407 . . . . . 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 2832 . . . . 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 2832 . . . 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 592 . . 3 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)} ∈ 𝐸)
71 wwlksnred 29414 . . . . 5 (𝑁 ∈ β„•0 β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
7271imp 406 . . . 4 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺))
73 eqeq2 2743 . . . . . 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 6891 . . . . . . . 8 (𝑦 = (π‘Š prefix (𝑁 + 1)) β†’ (lastSβ€˜π‘¦) = (lastSβ€˜(π‘Š prefix (𝑁 + 1))))
7574preq1d 4743 . . . . . . 7 (𝑦 = (π‘Š prefix (𝑁 + 1)) β†’ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} = {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)})
7675eleq1d 2817 . . . . . 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 630 . . . . 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 481 . . . 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 696 . 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 412 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069  {cpr 4630   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  β„cr 11113  0cc0 11114  1c1 11115   + caddc 11117   < clt 11253   βˆ’ cmin 11449  β„•cn 12217  β„•0cn0 12477  ...cfz 13489  ..^cfzo 13632  β™―chash 14295  Word cword 14469  lastSclsw 14517   prefix cpfx 14625  Vtxcvtx 28524  Edgcedg 28575   WWalksN cwwlksn 29348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-lsw 14518  df-substr 14596  df-pfx 14626  df-wwlks 29352  df-wwlksn 29353
This theorem is referenced by:  wwlksnredwwlkn0  29418
  Copyright terms: Public domain W3C validator