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Theorem clwwlkf1 29566
Description: Lemma 3 for clwwlkf1o 29568: 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 29564 . 2 (𝑁 ∈ β„• β†’ 𝐹:𝐷⟢(𝑁 ClWWalksN 𝐺))
41, 2clwwlkfv 29565 . . . . . 6 (π‘₯ ∈ 𝐷 β†’ (πΉβ€˜π‘₯) = (π‘₯ prefix 𝑁))
51, 2clwwlkfv 29565 . . . . . 6 (𝑦 ∈ 𝐷 β†’ (πΉβ€˜π‘¦) = (𝑦 prefix 𝑁))
64, 5eqeqan12d 2745 . . . . 5 ((π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ↔ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)))
76adantl 481 . . . 4 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ↔ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)))
8 fveq2 6892 . . . . . . . . 9 (𝑀 = π‘₯ β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘₯))
9 fveq1 6891 . . . . . . . . 9 (𝑀 = π‘₯ β†’ (π‘€β€˜0) = (π‘₯β€˜0))
108, 9eqeq12d 2747 . . . . . . . 8 (𝑀 = π‘₯ β†’ ((lastSβ€˜π‘€) = (π‘€β€˜0) ↔ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))
1110, 1elrab2 3687 . . . . . . 7 (π‘₯ ∈ 𝐷 ↔ (π‘₯ ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)))
12 fveq2 6892 . . . . . . . . 9 (𝑀 = 𝑦 β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘¦))
13 fveq1 6891 . . . . . . . . 9 (𝑀 = 𝑦 β†’ (π‘€β€˜0) = (π‘¦β€˜0))
1412, 13eqeq12d 2747 . . . . . . . 8 (𝑀 = 𝑦 β†’ ((lastSβ€˜π‘€) = (π‘€β€˜0) ↔ (lastSβ€˜π‘¦) = (π‘¦β€˜0)))
1514, 1elrab2 3687 . . . . . . 7 (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)))
1611, 15anbi12i 626 . . . . . 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 2731 . . . . . . . . . 10 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
18 eqid 2731 . . . . . . . . . 10 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
1917, 18wwlknp 29361 . . . . . . . . 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 29361 . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . 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 12225 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
26 ax-1cn 11171 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ β„‚
27 pncan 11471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
2827eqcomd 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ 𝑁 = ((𝑁 + 1) βˆ’ 1))
2925, 26, 28sylancl 585 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ β„• β†’ 𝑁 = ((𝑁 + 1) βˆ’ 1))
30 oveq1 7419 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜π‘₯) = (𝑁 + 1) β†’ ((β™―β€˜π‘₯) βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
3130eqcomd 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((β™―β€˜π‘₯) = (𝑁 + 1) β†’ ((𝑁 + 1) βˆ’ 1) = ((β™―β€˜π‘₯) βˆ’ 1))
3229, 31sylan9eqr 2793 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((β™―β€˜π‘₯) = (𝑁 + 1) ∧ 𝑁 ∈ β„•) β†’ 𝑁 = ((β™―β€˜π‘₯) βˆ’ 1))
3332oveq2d 7428 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((β™―β€˜π‘₯) = (𝑁 + 1) ∧ 𝑁 ∈ β„•) β†’ (π‘₯ prefix 𝑁) = (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)))
3432oveq2d 7428 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((β™―β€˜π‘₯) = (𝑁 + 1) ∧ 𝑁 ∈ β„•) β†’ (𝑦 prefix 𝑁) = (𝑦 prefix ((β™―β€˜π‘₯) βˆ’ 1)))
3533, 34eqeq12d 2747 . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) β†’ 𝑦 ∈ Word (Vtxβ€˜πΊ))
42 simpll 764 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ π‘₯ ∈ Word (Vtxβ€˜πΊ))
4341, 42anim12ci 613 . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . 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 12484 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„•0)
46 0nn0 12492 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ β„•0
4745, 46jctil 519 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ β„• β†’ (0 ∈ β„•0 ∧ 𝑁 ∈ β„•0))
4847adantr 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†’ (0 ∈ β„•0 ∧ 𝑁 ∈ β„•0))
49 nnre 12224 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„• β†’ 𝑁 ∈ ℝ)
5049lep1d 12150 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ β„• β†’ 𝑁 ≀ (𝑁 + 1))
51 breq2 5153 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . . . 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 5153 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . . . . 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 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‘¦)))
6059impcom 407 . . . . . . . . . . . . . . . . . . . . . . 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 14628 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ β„•0) β†’ (π‘₯ prefix 𝑁) = (π‘₯ substr ⟨0, π‘βŸ©))
6261ad2ant2rl 746 . . . . . . . . . . . . . . . . . . . . . . . . . 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 14628 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ β„•0) β†’ (𝑦 prefix 𝑁) = (𝑦 substr ⟨0, π‘βŸ©))
6463ad2ant2l 743 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2747 . . . . . . . . . . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . . . . . . . . . . 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 14620 . . . . . . . . . . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . . . . . . . . . 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 13677 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ β„•)
7170biimpri 227 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ β„• β†’ 0 ∈ (0..^𝑁))
7271adantr 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†’ 0 ∈ (0..^𝑁))
73 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 0 β†’ (π‘₯β€˜π‘–) = (π‘₯β€˜0))
74 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 0 β†’ (π‘¦β€˜π‘–) = (π‘¦β€˜0))
7573, 74eqeq12d 2747 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 0 β†’ ((π‘₯β€˜π‘–) = (π‘¦β€˜π‘–) ↔ (π‘₯β€˜0) = (π‘¦β€˜0)))
7675rspcv 3609 . . . . . . . . . . . . . . . . . . . . . . 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 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)))) ∧ (π‘₯ prefix 𝑁) = (𝑦 prefix 𝑁)) β†’ (π‘₯β€˜0) = (π‘¦β€˜0))
80 simpr 484 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘₯) = (𝑁 + 1)) ∧ (lastSβ€˜π‘₯) = (π‘₯β€˜0)) β†’ (lastSβ€˜π‘₯) = (π‘₯β€˜0))
81 simpr 484 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (lastSβ€˜π‘¦) = (π‘¦β€˜0)) β†’ (lastSβ€˜π‘¦) = (π‘¦β€˜0))
8280, 81eqeqan12rd 2746 . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . 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 515 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . 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 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†’ 𝑦 ∈ Word (Vtxβ€˜πΊ))
8988adantl 481 . . . . . . . . . . . . . . . . . . . . 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 11220 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ β„• β†’ 1 ∈ ℝ)
91 nngt0 12248 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ β„• β†’ 0 < 𝑁)
92 0lt1 11741 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 < 1
9392a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ β„• β†’ 0 < 1)
9449, 90, 91, 93addgt0d 11794 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ β„• β†’ 0 < (𝑁 + 1))
95 breq2 5153 . . . . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . 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 14654 . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . 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 406 . . . . . . . . . . 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 414 . . . . . . . . . 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 1131 . . . . . . . . 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 417 . . . . . . 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 406 . . . 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 3199 . 2 (𝑁 ∈ β„• β†’ βˆ€π‘₯ ∈ 𝐷 βˆ€π‘¦ ∈ 𝐷 ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) β†’ π‘₯ = 𝑦))
120 dff13 7257 . 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 582 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  {crab 3431  {cpr 4631  βŸ¨cop 4635   class class class wbr 5149   ↦ cmpt 5232  βŸΆwf 6540  β€“1-1β†’wf1 6541  β€˜cfv 6544  (class class class)co 7412  β„‚cc 11111  0cc0 11113  1c1 11114   + caddc 11116   < clt 11253   ≀ cle 11254   βˆ’ cmin 11449  β„•cn 12217  β„•0cn0 12477  ..^cfzo 13632  β™―chash 14295  Word cword 14469  lastSclsw 14517   substr csubstr 14595   prefix cpfx 14625  Vtxcvtx 28520  Edgcedg 28571   WWalksN cwwlksn 29344   ClWWalksN cclwwlkn 29541
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
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 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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-card 9937  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-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-lsw 14518  df-s1 14551  df-substr 14596  df-pfx 14626  df-wwlks 29348  df-wwlksn 29349  df-clwwlk 29499  df-clwwlkn 29542
This theorem is referenced by:  clwwlkf1o  29568
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