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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 2822 | . 2 ⊢ (𝜑 → (𝑃‘0) = (𝑃‘0)) | |
3 | wlklenvm1 27403 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) = ((♯‘𝑃) − 1)) | |
4 | 3 | fveq2d 6674 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘(♯‘𝐹)) = (𝑃‘((♯‘𝑃) − 1))) |
5 | eqid 2821 | . . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
6 | 5 | wlkpwrd 27399 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word (Vtx‘𝐺)) |
7 | lsw 13916 | . . . . 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 2859 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘(♯‘𝐹)) = (lastS‘𝑃)) |
10 | 1, 9 | syl 17 | . 2 ⊢ (𝜑 → (𝑃‘(♯‘𝐹)) = (lastS‘𝑃)) |
11 | wlkcl 27397 | . . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
12 | nn0p1nn 11937 | . . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ0 → ((♯‘𝐹) + 1) ∈ ℕ) | |
13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((♯‘𝐹) + 1) ∈ ℕ) |
14 | wlklenvp1 27400 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1)) | |
15 | 13, 6, 14 | jca32 518 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (((♯‘𝐹) + 1) ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = ((♯‘𝐹) + 1)))) |
16 | fstwrdne0 13908 | . . . . . . 7 ⊢ ((((♯‘𝐹) + 1) ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = ((♯‘𝐹) + 1))) → (𝑃‘0) ∈ (Vtx‘𝐺)) | |
17 | lswlgt0cl 13921 | . . . . . . 7 ⊢ ((((♯‘𝐹) + 1) ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = ((♯‘𝐹) + 1))) → (lastS‘𝑃) ∈ (Vtx‘𝐺)) | |
18 | 16, 17 | jca 514 | . . . . . 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 2821 | . . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
21 | 20 | wlkf 27396 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom (iEdg‘𝐺)) |
22 | wrdv 13878 | . . . . . 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 518 | . . . 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 27439 | . . 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 1338 | 1 ⊢ (𝜑 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(lastS‘𝑃))𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 − cmin 10870 ℕcn 11638 ℕ0cn0 11898 ♯chash 13691 Word cword 13862 lastSclsw 13914 Vtxcvtx 26781 iEdgciedg 26782 Walkscwlks 27378 WalksOncwlkson 27379 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-lsw 13915 df-wlks 27381 df-wlkson 27382 |
This theorem is referenced by: 3wlkond 27950 |
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