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Theorem clwwlkf1 29569
Description: Lemma 3 for clwwlkf1o 29571: 𝐹 is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
Hypotheses
Ref Expression
clwwlkf1o.d 𝐷 = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘€) = (π‘€β€˜0)}
clwwlkf1o.f 𝐹 = (𝑑 ∈ 𝐷 ↦ (𝑑 prefix 𝑁))
Assertion
Ref Expression
clwwlkf1 (𝑁 ∈ β„• β†’ 𝐹:𝐷–1-1β†’(𝑁 ClWWalksN 𝐺))
Distinct variable groups:   𝑀,𝐺   𝑀,𝑁   𝑑,𝐷   𝑑,𝐺,𝑀   𝑑,𝑁
Allowed substitution hints:   𝐷(𝑀)   𝐹(𝑀,𝑑)

Proof of Theorem clwwlkf1
Dummy variables 𝑖 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwwlkf1o.d . . 3 𝐷 = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘€) = (π‘€β€˜0)}
2 clwwlkf1o.f . . 3 𝐹 = (𝑑 ∈ 𝐷 ↦ (𝑑 prefix 𝑁))
31, 2clwwlkf 29567 . 2 (𝑁 ∈ β„• β†’ 𝐹:𝐷⟢(𝑁 ClWWalksN 𝐺))
41, 2clwwlkfv 29568 . . . . . 6 (π‘₯ ∈ 𝐷 β†’ (πΉβ€˜π‘₯) = (π‘₯ prefix 𝑁))
51, 2clwwlkfv 29568 . . . . . 6 (𝑦 ∈ 𝐷 β†’ (πΉβ€˜π‘¦) = (𝑦 prefix 𝑁))
64, 5eqeqan12d 2744 . . . . 5 ((π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ↔ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)))
76adantl 480 . . . 4 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ↔ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)))
8 fveq2 6890 . . . . . . . . 9 (𝑀 = π‘₯ β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘₯))
9 fveq1 6889 . . . . . . . . 9 (𝑀 = π‘₯ β†’ (π‘€β€˜0) = (π‘₯β€˜0))
108, 9eqeq12d 2746 . . . . . . . 8 (𝑀 = π‘₯ β†’ ((lastSβ€˜π‘€) = (π‘€β€˜0) ↔ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))
1110, 1elrab2 3685 . . . . . . 7 (π‘₯ ∈ 𝐷 ↔ (π‘₯ ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))
12 fveq2 6890 . . . . . . . . 9 (𝑀 = 𝑦 β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘¦))
13 fveq1 6889 . . . . . . . . 9 (𝑀 = 𝑦 β†’ (π‘€β€˜0) = (π‘¦β€˜0))
1412, 13eqeq12d 2746 . . . . . . . 8 (𝑀 = 𝑦 β†’ ((lastSβ€˜π‘€) = (π‘€β€˜0) ↔ (lastSβ€˜π‘¦) = (π‘¦β€˜0)))
1514, 1elrab2 3685 . . . . . . 7 (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)))
1611, 15anbi12i 625 . . . . . 6 ((π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ↔ ((π‘₯ ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0))))
17 eqid 2730 . . . . . . . . . 10 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
18 eqid 2730 . . . . . . . . . 10 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
1917, 18wwlknp 29364 . . . . . . . . 9 (π‘₯ ∈ (𝑁 WWalksN 𝐺) β†’ (π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘₯β€˜π‘–), (π‘₯β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
2017, 18wwlknp 29364 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑁 WWalksN 𝐺) β†’ (𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
21 simprlr 776 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ (β™―β€˜π‘₯) = (𝑁 + 1))
22 simpllr 772 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ (β™―β€˜π‘¦) = (𝑁 + 1))
2321, 22eqtr4d 2773 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))
2423ad2antlr 723 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) ∧ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)) β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))
25 nncn 12224 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
26 ax-1cn 11170 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ β„‚
27 pncan 11470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
2827eqcomd 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ 𝑁 = ((𝑁 + 1) βˆ’ 1))
2925, 26, 28sylancl 584 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ β„• β†’ 𝑁 = ((𝑁 + 1) βˆ’ 1))
30 oveq1 7418 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜π‘₯) = (𝑁 + 1) β†’ ((β™―β€˜π‘₯) βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
3130eqcomd 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((β™―β€˜π‘₯) = (𝑁 + 1) β†’ ((𝑁 + 1) βˆ’ 1) = ((β™―β€˜π‘₯) βˆ’ 1))
3229, 31sylan9eqr 2792 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((β™―β€˜π‘₯) = (𝑁 + 1) ∧ 𝑁 ∈ β„•) β†’ 𝑁 = ((β™―β€˜π‘₯) βˆ’ 1))
3332oveq2d 7427 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((β™―β€˜π‘₯) = (𝑁 + 1) ∧ 𝑁 ∈ β„•) β†’ (π‘₯ prefix 𝑁) = (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)))
3432oveq2d 7427 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((β™―β€˜π‘₯) = (𝑁 + 1) ∧ 𝑁 ∈ β„•) β†’ (𝑦 prefix 𝑁) = (𝑦 prefix ((β™―β€˜π‘₯) βˆ’ 1)))
3533, 34eqeq12d 2746 . . . . . . . . . . . . . . . . . . . . . . . 24 (((β™―β€˜π‘₯) = (𝑁 + 1) ∧ 𝑁 ∈ β„•) β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) = (𝑦 prefix ((β™―β€˜π‘₯) βˆ’ 1))))
3635ex 411 . . . . . . . . . . . . . . . . . . . . . . 23 ((β™―β€˜π‘₯) = (𝑁 + 1) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) = (𝑦 prefix ((β™―β€˜π‘₯) βˆ’ 1)))))
3736ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) = (𝑦 prefix ((β™―β€˜π‘₯) βˆ’ 1)))))
3837adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) = (𝑦 prefix ((β™―β€˜π‘₯) βˆ’ 1)))))
3938impcom 406 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) = (𝑦 prefix ((β™―β€˜π‘₯) βˆ’ 1))))
4039biimpa 475 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) ∧ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)) β†’ (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) = (𝑦 prefix ((β™―β€˜π‘₯) βˆ’ 1)))
41 simpll 763 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) β†’ 𝑦 ∈ Word (Vtxβ€˜πΊ))
42 simpll 763 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ π‘₯ ∈ Word (Vtxβ€˜πΊ))
4341, 42anim12ci 612 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ (π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ)))
4443adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ (π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ)))
45 nnnn0 12483 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„•0)
46 0nn0 12491 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ β„•0
4745, 46jctil 518 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ β„• β†’ (0 ∈ β„•0 ∧ 𝑁 ∈ β„•0))
4847adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ (0 ∈ β„•0 ∧ 𝑁 ∈ β„•0))
49 nnre 12223 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„• β†’ 𝑁 ∈ ℝ)
5049lep1d 12149 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ β„• β†’ 𝑁 ≀ (𝑁 + 1))
51 breq2 5151 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((β™―β€˜π‘₯) = (𝑁 + 1) β†’ (𝑁 ≀ (β™―β€˜π‘₯) ↔ 𝑁 ≀ (𝑁 + 1)))
5250, 51imbitrrid 245 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β™―β€˜π‘₯) = (𝑁 + 1) β†’ (𝑁 ∈ β„• β†’ 𝑁 ≀ (β™―β€˜π‘₯)))
5352ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ (𝑁 ∈ β„• β†’ 𝑁 ≀ (β™―β€˜π‘₯)))
5453adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ (𝑁 ∈ β„• β†’ 𝑁 ≀ (β™―β€˜π‘₯)))
5554impcom 406 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ 𝑁 ≀ (β™―β€˜π‘₯))
56 breq2 5151 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((β™―β€˜π‘¦) = (𝑁 + 1) β†’ (𝑁 ≀ (β™―β€˜π‘¦) ↔ 𝑁 ≀ (𝑁 + 1)))
5750, 56imbitrrid 245 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β™―β€˜π‘¦) = (𝑁 + 1) β†’ (𝑁 ∈ β„• β†’ 𝑁 ≀ (β™―β€˜π‘¦)))
5857ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) β†’ (𝑁 ∈ β„• β†’ 𝑁 ≀ (β™―β€˜π‘¦)))
5958adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ (𝑁 ∈ β„• β†’ 𝑁 ≀ (β™―β€˜π‘¦)))
6059impcom 406 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ 𝑁 ≀ (β™―β€˜π‘¦))
61 pfxval 14627 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ β„•0) β†’ (π‘₯ prefix 𝑁) = (π‘₯ substr ⟨0, π‘βŸ©))
6261ad2ant2rl 745 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ)) ∧ (0 ∈ β„•0 ∧ 𝑁 ∈ β„•0)) β†’ (π‘₯ prefix 𝑁) = (π‘₯ substr ⟨0, π‘βŸ©))
63 pfxval 14627 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ β„•0) β†’ (𝑦 prefix 𝑁) = (𝑦 substr ⟨0, π‘βŸ©))
6463ad2ant2l 742 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ)) ∧ (0 ∈ β„•0 ∧ 𝑁 ∈ β„•0)) β†’ (𝑦 prefix 𝑁) = (𝑦 substr ⟨0, π‘βŸ©))
6562, 64eqeq12d 2746 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ)) ∧ (0 ∈ β„•0 ∧ 𝑁 ∈ β„•0)) β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (π‘₯ substr ⟨0, π‘βŸ©) = (𝑦 substr ⟨0, π‘βŸ©)))
66653adant3 1130 . . . . . . . . . . . . . . . . . . . . . . . 24 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ)) ∧ (0 ∈ β„•0 ∧ 𝑁 ∈ β„•0) ∧ (𝑁 ≀ (β™―β€˜π‘₯) ∧ 𝑁 ≀ (β™―β€˜π‘¦))) β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (π‘₯ substr ⟨0, π‘βŸ©) = (𝑦 substr ⟨0, π‘βŸ©)))
67 swrdspsleq 14619 . . . . . . . . . . . . . . . . . . . . . . . 24 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ)) ∧ (0 ∈ β„•0 ∧ 𝑁 ∈ β„•0) ∧ (𝑁 ≀ (β™―β€˜π‘₯) ∧ 𝑁 ≀ (β™―β€˜π‘¦))) β†’ ((π‘₯ substr ⟨0, π‘βŸ©) = (𝑦 substr ⟨0, π‘βŸ©) ↔ βˆ€π‘– ∈ (0..^𝑁)(π‘₯β€˜π‘–) = (π‘¦β€˜π‘–)))
6866, 67bitrd 278 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ)) ∧ (0 ∈ β„•0 ∧ 𝑁 ∈ β„•0) ∧ (𝑁 ≀ (β™―β€˜π‘₯) ∧ 𝑁 ≀ (β™―β€˜π‘¦))) β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) ↔ βˆ€π‘– ∈ (0..^𝑁)(π‘₯β€˜π‘–) = (π‘¦β€˜π‘–)))
6944, 48, 55, 60, 68syl112anc 1372 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) ↔ βˆ€π‘– ∈ (0..^𝑁)(π‘₯β€˜π‘–) = (π‘¦β€˜π‘–)))
70 lbfzo0 13676 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ β„•)
7170biimpri 227 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ β„• β†’ 0 ∈ (0..^𝑁))
7271adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ 0 ∈ (0..^𝑁))
73 fveq2 6890 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 0 β†’ (π‘₯β€˜π‘–) = (π‘₯β€˜0))
74 fveq2 6890 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 0 β†’ (π‘¦β€˜π‘–) = (π‘¦β€˜0))
7573, 74eqeq12d 2746 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 0 β†’ ((π‘₯β€˜π‘–) = (π‘¦β€˜π‘–) ↔ (π‘₯β€˜0) = (π‘¦β€˜0)))
7675rspcv 3607 . . . . . . . . . . . . . . . . . . . . . . 23 (0 ∈ (0..^𝑁) β†’ (βˆ€π‘– ∈ (0..^𝑁)(π‘₯β€˜π‘–) = (π‘¦β€˜π‘–) β†’ (π‘₯β€˜0) = (π‘¦β€˜0)))
7772, 76syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ (βˆ€π‘– ∈ (0..^𝑁)(π‘₯β€˜π‘–) = (π‘¦β€˜π‘–) β†’ (π‘₯β€˜0) = (π‘¦β€˜0)))
7869, 77sylbid 239 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ (π‘₯β€˜0) = (π‘¦β€˜0)))
7978imp 405 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) ∧ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)) β†’ (π‘₯β€˜0) = (π‘¦β€˜0))
80 simpr 483 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ (lastSβ€˜π‘₯) = (π‘₯β€˜0))
81 simpr 483 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) β†’ (lastSβ€˜π‘¦) = (π‘¦β€˜0))
8280, 81eqeqan12rd 2745 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ ((lastSβ€˜π‘₯) = (lastSβ€˜π‘¦) ↔ (π‘₯β€˜0) = (π‘¦β€˜0)))
8382ad2antlr 723 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) ∧ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)) β†’ ((lastSβ€˜π‘₯) = (lastSβ€˜π‘¦) ↔ (π‘₯β€˜0) = (π‘¦β€˜0)))
8479, 83mpbird 256 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) ∧ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)) β†’ (lastSβ€˜π‘₯) = (lastSβ€˜π‘¦))
8524, 40, 84jca32 514 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) ∧ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)) β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) ∧ ((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) = (𝑦 prefix ((β™―β€˜π‘₯) βˆ’ 1)) ∧ (lastSβ€˜π‘₯) = (lastSβ€˜π‘¦))))
8642adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ π‘₯ ∈ Word (Vtxβ€˜πΊ))
8786adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ π‘₯ ∈ Word (Vtxβ€˜πΊ))
8841adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ 𝑦 ∈ Word (Vtxβ€˜πΊ))
8988adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ 𝑦 ∈ Word (Vtxβ€˜πΊ))
90 1red 11219 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ β„• β†’ 1 ∈ ℝ)
91 nngt0 12247 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ β„• β†’ 0 < 𝑁)
92 0lt1 11740 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 < 1
9392a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ β„• β†’ 0 < 1)
9449, 90, 91, 93addgt0d 11793 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ β„• β†’ 0 < (𝑁 + 1))
95 breq2 5151 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((β™―β€˜π‘₯) = (𝑁 + 1) β†’ (0 < (β™―β€˜π‘₯) ↔ 0 < (𝑁 + 1)))
9694, 95imbitrrid 245 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜π‘₯) = (𝑁 + 1) β†’ (𝑁 ∈ β„• β†’ 0 < (β™―β€˜π‘₯)))
9796ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ (𝑁 ∈ β„• β†’ 0 < (β™―β€˜π‘₯)))
9897adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ (𝑁 ∈ β„• β†’ 0 < (β™―β€˜π‘₯)))
9998impcom 406 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ 0 < (β™―β€˜π‘₯))
10087, 89, 993jca 1126 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) β†’ (π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ 0 < (β™―β€˜π‘₯)))
101100adantr 479 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) ∧ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)) β†’ (π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ 0 < (β™―β€˜π‘₯)))
102 pfxsuff1eqwrdeq 14653 . . . . . . . . . . . . . . . . . . 19 ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ 0 < (β™―β€˜π‘₯)) β†’ (π‘₯ = 𝑦 ↔ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) ∧ ((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) = (𝑦 prefix ((β™―β€˜π‘₯) βˆ’ 1)) ∧ (lastSβ€˜π‘₯) = (lastSβ€˜π‘¦)))))
103101, 102syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) ∧ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)) β†’ (π‘₯ = 𝑦 ↔ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) ∧ ((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) = (𝑦 prefix ((β™―β€˜π‘₯) βˆ’ 1)) ∧ (lastSβ€˜π‘₯) = (lastSβ€˜π‘¦)))))
10485, 103mpbird 256 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„• ∧ (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))) ∧ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)) β†’ π‘₯ = 𝑦)
105104exp31 418 . . . . . . . . . . . . . . . 16 (𝑁 ∈ β„• β†’ ((((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) ∧ ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0))) β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦)))
106105expdcom 413 . . . . . . . . . . . . . . 15 (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) β†’ (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦))))
107106ex 411 . . . . . . . . . . . . . 14 ((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) β†’ ((lastSβ€˜π‘¦) = (π‘¦β€˜0) β†’ (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦)))))
1081073adant3 1130 . . . . . . . . . . . . 13 ((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) β†’ ((lastSβ€˜π‘¦) = (π‘¦β€˜0) β†’ (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦)))))
10920, 108syl 17 . . . . . . . . . . . 12 (𝑦 ∈ (𝑁 WWalksN 𝐺) β†’ ((lastSβ€˜π‘¦) = (π‘¦β€˜0) β†’ (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦)))))
110109imp 405 . . . . . . . . . . 11 ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) β†’ (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦))))
111110expdcom 413 . . . . . . . . . 10 ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) β†’ ((lastSβ€˜π‘₯) = (π‘₯β€˜0) β†’ ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦)))))
1121113adant3 1130 . . . . . . . . 9 ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘₯β€˜π‘–), (π‘₯β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) β†’ ((lastSβ€˜π‘₯) = (π‘₯β€˜0) β†’ ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦)))))
11319, 112syl 17 . . . . . . . 8 (π‘₯ ∈ (𝑁 WWalksN 𝐺) β†’ ((lastSβ€˜π‘₯) = (π‘₯β€˜0) β†’ ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦)))))
114113imp31 416 . . . . . . 7 (((π‘₯ ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0))) β†’ (𝑁 ∈ β„• β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦)))
115114com12 32 . . . . . 6 (𝑁 ∈ β„• β†’ (((π‘₯ ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0))) β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦)))
11616, 115biimtrid 241 . . . . 5 (𝑁 ∈ β„• β†’ ((π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦)))
117116imp 405 . . . 4 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ ((π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁) β†’ π‘₯ = 𝑦))
1187, 117sylbid 239 . . 3 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) β†’ π‘₯ = 𝑦))
119118ralrimivva 3198 . 2 (𝑁 ∈ β„• β†’ βˆ€π‘₯ ∈ 𝐷 βˆ€π‘¦ ∈ 𝐷 ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) β†’ π‘₯ = 𝑦))
120 dff13 7256 . 2 (𝐹:𝐷–1-1β†’(𝑁 ClWWalksN 𝐺) ↔ (𝐹:𝐷⟢(𝑁 ClWWalksN 𝐺) ∧ βˆ€π‘₯ ∈ 𝐷 βˆ€π‘¦ ∈ 𝐷 ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) β†’ π‘₯ = 𝑦)))
1213, 119, 120sylanbrc 581 1 (𝑁 ∈ β„• β†’ 𝐹:𝐷–1-1β†’(𝑁 ClWWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430  {cpr 4629  βŸ¨cop 4633   class class class wbr 5147   ↦ cmpt 5230  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252   ≀ cle 11253   βˆ’ cmin 11448  β„•cn 12216  β„•0cn0 12476  ..^cfzo 13631  β™―chash 14294  Word cword 14468  lastSclsw 14516   substr csubstr 14594   prefix cpfx 14624  Vtxcvtx 28523  Edgcedg 28574   WWalksN cwwlksn 29347   ClWWalksN cclwwlkn 29544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-lsw 14517  df-s1 14550  df-substr 14595  df-pfx 14625  df-wwlks 29351  df-wwlksn 29352  df-clwwlk 29502  df-clwwlkn 29545
This theorem is referenced by:  clwwlkf1o  29571
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