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| Mirrors > Home > MPE Home > Th. List > wlkonwlk1l | Structured version Visualization version GIF version | ||
| Description: A walk is a walk from its first vertex to its last vertex. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 22-Mar-2021.) |
| Ref | Expression |
|---|---|
| wlkonwlk1l.w | ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
| Ref | Expression |
|---|---|
| wlkonwlk1l | ⊢ (𝜑 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(lastS‘𝑃))𝑃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wlkonwlk1l.w | . 2 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | |
| 2 | eqidd 2738 | . 2 ⊢ (𝜑 → (𝑃‘0) = (𝑃‘0)) | |
| 3 | wlklenvm1 29640 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) = ((♯‘𝑃) − 1)) | |
| 4 | 3 | fveq2d 6910 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘(♯‘𝐹)) = (𝑃‘((♯‘𝑃) − 1))) |
| 5 | eqid 2737 | . . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 6 | 5 | wlkpwrd 29635 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word (Vtx‘𝐺)) |
| 7 | lsw 14602 | . . . . 5 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (lastS‘𝑃) = (𝑃‘((♯‘𝑃) − 1))) | |
| 8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (lastS‘𝑃) = (𝑃‘((♯‘𝑃) − 1))) |
| 9 | 4, 8 | eqtr4d 2780 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘(♯‘𝐹)) = (lastS‘𝑃)) |
| 10 | 1, 9 | syl 17 | . 2 ⊢ (𝜑 → (𝑃‘(♯‘𝐹)) = (lastS‘𝑃)) |
| 11 | wlkcl 29633 | . . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
| 12 | nn0p1nn 12565 | . . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ0 → ((♯‘𝐹) + 1) ∈ ℕ) | |
| 13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((♯‘𝐹) + 1) ∈ ℕ) |
| 14 | wlklenvp1 29636 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1)) | |
| 15 | 13, 6, 14 | jca32 515 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (((♯‘𝐹) + 1) ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = ((♯‘𝐹) + 1)))) |
| 16 | fstwrdne0 14594 | . . . . . . 7 ⊢ ((((♯‘𝐹) + 1) ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = ((♯‘𝐹) + 1))) → (𝑃‘0) ∈ (Vtx‘𝐺)) | |
| 17 | lswlgt0cl 14607 | . . . . . . 7 ⊢ ((((♯‘𝐹) + 1) ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = ((♯‘𝐹) + 1))) → (lastS‘𝑃) ∈ (Vtx‘𝐺)) | |
| 18 | 16, 17 | jca 511 | . . . . . 6 ⊢ ((((♯‘𝐹) + 1) ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = ((♯‘𝐹) + 1))) → ((𝑃‘0) ∈ (Vtx‘𝐺) ∧ (lastS‘𝑃) ∈ (Vtx‘𝐺))) |
| 19 | 15, 18 | syl 17 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) ∈ (Vtx‘𝐺) ∧ (lastS‘𝑃) ∈ (Vtx‘𝐺))) |
| 20 | eqid 2737 | . . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 21 | 20 | wlkf 29632 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom (iEdg‘𝐺)) |
| 22 | wrdv 14567 | . . . . . 6 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹 ∈ Word V) | |
| 23 | 21, 22 | syl 17 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word V) |
| 24 | 19, 23, 6 | jca32 515 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (((𝑃‘0) ∈ (Vtx‘𝐺) ∧ (lastS‘𝑃) ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word (Vtx‘𝐺)))) |
| 25 | 1, 24 | syl 17 | . . 3 ⊢ (𝜑 → (((𝑃‘0) ∈ (Vtx‘𝐺) ∧ (lastS‘𝑃) ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word (Vtx‘𝐺)))) |
| 26 | 5 | iswlkon 29675 | . . 3 ⊢ ((((𝑃‘0) ∈ (Vtx‘𝐺) ∧ (lastS‘𝑃) ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word (Vtx‘𝐺))) → (𝐹((𝑃‘0)(WalksOn‘𝐺)(lastS‘𝑃))𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘0) ∧ (𝑃‘(♯‘𝐹)) = (lastS‘𝑃)))) |
| 27 | 25, 26 | syl 17 | . 2 ⊢ (𝜑 → (𝐹((𝑃‘0)(WalksOn‘𝐺)(lastS‘𝑃))𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘0) ∧ (𝑃‘(♯‘𝐹)) = (lastS‘𝑃)))) |
| 28 | 1, 2, 10, 27 | mpbir3and 1343 | 1 ⊢ (𝜑 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(lastS‘𝑃))𝑃) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 − cmin 11492 ℕcn 12266 ℕ0cn0 12526 ♯chash 14369 Word cword 14552 lastSclsw 14600 Vtxcvtx 29013 iEdgciedg 29014 Walkscwlks 29614 WalksOncwlkson 29615 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-lsw 14601 df-wlks 29617 df-wlkson 29618 |
| This theorem is referenced by: 3wlkond 30190 |
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