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Theorem wwlksnredwwlkn 29149
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 2734 . . 3 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (π‘Š prefix (𝑁 + 1)) = (π‘Š prefix (𝑁 + 1)))
2 eqid 2733 . . . . 5 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
3 wwlksnredwwlkn.e . . . . 5 𝐸 = (Edgβ€˜πΊ)
42, 3wwlknp 29097 . . . 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 770 . . . . . . . . 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 12512 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
7 peano2nn0 12512 . . . . . . . . . . . . 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 12511 . . . . . . . . . . . . . 14 ((𝑁 + 1) ∈ β„•0 β†’ ((𝑁 + 1) + 1) ∈ β„•)
116, 10syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) + 1) ∈ β„•)
12 nn0re 12481 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ ℝ)
13 id 22 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ β†’ 𝑁 ∈ ℝ)
14 peano2re 11387 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ β†’ (𝑁 + 1) ∈ ℝ)
15 peano2re 11387 . . . . . . . . . . . . . . . . 17 ((𝑁 + 1) ∈ ℝ β†’ ((𝑁 + 1) + 1) ∈ ℝ)
1614, 15syl 17 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ β†’ ((𝑁 + 1) + 1) ∈ ℝ)
1713, 14, 163jca 1129 . . . . . . . . . . . . . . 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 12144 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ 𝑁 < (𝑁 + 1))
20 nn0re 12481 . . . . . . . . . . . . . . . 16 ((𝑁 + 1) ∈ β„•0 β†’ (𝑁 + 1) ∈ ℝ)
216, 20syl 17 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ ℝ)
2221ltp1d 12144 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) < ((𝑁 + 1) + 1))
23 lttr 11290 . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ 𝑁 < ((𝑁 + 1) + 1))
26 elfzo0 13673 . . . . . . . . . . . . 13 (𝑁 ∈ (0..^((𝑁 + 1) + 1)) ↔ (𝑁 ∈ β„•0 ∧ ((𝑁 + 1) + 1) ∈ β„• ∧ 𝑁 < ((𝑁 + 1) + 1)))
279, 11, 25, 26syl3anbrc 1344 . . . . . . . . . . . 12 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ (0..^((𝑁 + 1) + 1)))
28 fz0add1fz1 13702 . . . . . . . . . . . 12 ((((𝑁 + 1) + 1) ∈ β„•0 ∧ 𝑁 ∈ (0..^((𝑁 + 1) + 1))) β†’ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
298, 27, 28syl2anc 585 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
3029adantr 482 . . . . . . . . . 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 7417 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ (1...(β™―β€˜π‘Š)) = (1...((𝑁 + 1) + 1)))
3231eleq2d 2820 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ ((𝑁 + 1) ∈ (1...(β™―β€˜π‘Š)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
3332adantl 483 . . . . . . . . . . 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 483 . . . . . . . . . 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 513 . . . . . . . 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 1172 . . . . . . 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 14645 . . . . . . 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 14514 . . . . . . . 8 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
41403ad2ant1 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†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
4241adantl 483 . . . . . 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 4746 . . . . 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 7416 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (((𝑁 + 1) + 1) βˆ’ 1))
45443ad2ant2 1135 . . . . . . . . . 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 483 . . . . . . . . 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 6896 . . . . . . . 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 4745 . . . . . . 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 12482 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
50 1cnd 11209 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ 1 ∈ β„‚)
5149, 50pncand 11572 . . . . . . . . . 10 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
5251fveq2d 6896 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (π‘Šβ€˜((𝑁 + 1) βˆ’ 1)) = (π‘Šβ€˜π‘))
536nn0cnd 12534 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„‚)
5453, 50pncand 11572 . . . . . . . . . 10 (𝑁 ∈ β„•0 β†’ (((𝑁 + 1) + 1) βˆ’ 1) = (𝑁 + 1))
5554fveq2d 6896 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (π‘Šβ€˜(((𝑁 + 1) + 1) βˆ’ 1)) = (π‘Šβ€˜(𝑁 + 1)))
5652, 55preq12d 4746 . . . . . . . 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 482 . . . . . . 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 2773 . . . . . 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 6892 . . . . . . . . . . . 12 (𝑖 = 𝑁 β†’ (π‘Šβ€˜π‘–) = (π‘Šβ€˜π‘))
60 fvoveq1 7432 . . . . . . . . . . . 12 (𝑖 = 𝑁 β†’ (π‘Šβ€˜(𝑖 + 1)) = (π‘Šβ€˜(𝑁 + 1)))
6159, 60preq12d 4746 . . . . . . . . . . 11 (𝑖 = 𝑁 β†’ {(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} = {(π‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))})
6261eleq1d 2819 . . . . . . . . . 10 (𝑖 = 𝑁 β†’ ({(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ↔ {(π‘Šβ€˜π‘), (π‘Šβ€˜(𝑁 + 1))} ∈ 𝐸))
6362rspcv 3609 . . . . . . . . 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 13709 . . . . . . . . 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 1136 . . . . . . 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 409 . . . . . 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 2834 . . . . 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 2834 . . . 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 594 . . 3 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)} ∈ 𝐸)
71 wwlksnred 29146 . . . . 5 (𝑁 ∈ β„•0 β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
7271imp 408 . . . 4 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺))
73 eqeq2 2745 . . . . . 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 6892 . . . . . . . 8 (𝑦 = (π‘Š prefix (𝑁 + 1)) β†’ (lastSβ€˜π‘¦) = (lastSβ€˜(π‘Š prefix (𝑁 + 1))))
7574preq1d 4744 . . . . . . 7 (𝑦 = (π‘Š prefix (𝑁 + 1)) β†’ {(lastSβ€˜π‘¦), (lastSβ€˜π‘Š)} = {(lastSβ€˜(π‘Š prefix (𝑁 + 1))), (lastSβ€˜π‘Š)})
7675eleq1d 2819 . . . . . 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 632 . . . . 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 483 . . . 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 3606 . . 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 698 . 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 414 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  {cpr 4631   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113   < clt 11248   βˆ’ cmin 11444  β„•cn 12212  β„•0cn0 12472  ...cfz 13484  ..^cfzo 13627  β™―chash 14290  Word cword 14464  lastSclsw 14512   prefix cpfx 14620  Vtxcvtx 28256  Edgcedg 28307   WWalksN cwwlksn 29080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-lsw 14513  df-substr 14591  df-pfx 14621  df-wwlks 29084  df-wwlksn 29085
This theorem is referenced by:  wwlksnredwwlkn0  29150
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