![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2826 | . 2 ⊢ (𝜑 → (𝑃‘0) = (𝑃‘0)) | |
3 | wlklenvm1 26919 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) = ((♯‘𝑃) − 1)) | |
4 | 3 | fveq2d 6437 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘(♯‘𝐹)) = (𝑃‘((♯‘𝑃) − 1))) |
5 | eqid 2825 | . . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
6 | 5 | wlkpwrd 26915 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word (Vtx‘𝐺)) |
7 | lsw 13624 | . . . . 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 2864 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘(♯‘𝐹)) = (lastS‘𝑃)) |
10 | 1, 9 | syl 17 | . 2 ⊢ (𝜑 → (𝑃‘(♯‘𝐹)) = (lastS‘𝑃)) |
11 | wlkcl 26913 | . . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
12 | nn0p1nn 11659 | . . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ0 → ((♯‘𝐹) + 1) ∈ ℕ) | |
13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((♯‘𝐹) + 1) ∈ ℕ) |
14 | wlklenvp1 26916 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1)) | |
15 | 13, 6, 14 | jca32 513 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (((♯‘𝐹) + 1) ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = ((♯‘𝐹) + 1)))) |
16 | fstwrdne0 13616 | . . . . . . 7 ⊢ ((((♯‘𝐹) + 1) ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = ((♯‘𝐹) + 1))) → (𝑃‘0) ∈ (Vtx‘𝐺)) | |
17 | lswlgt0cl 13629 | . . . . . . 7 ⊢ ((((♯‘𝐹) + 1) ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = ((♯‘𝐹) + 1))) → (lastS‘𝑃) ∈ (Vtx‘𝐺)) | |
18 | 16, 17 | jca 509 | . . . . . 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 2825 | . . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
21 | 20 | wlkf 26912 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom (iEdg‘𝐺)) |
22 | wrdv 13589 | . . . . . 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 513 | . . . 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 26954 | . . 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 1448 | 1 ⊢ (𝜑 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(lastS‘𝑃))𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 Vcvv 3414 class class class wbr 4873 dom cdm 5342 ‘cfv 6123 (class class class)co 6905 0cc0 10252 1c1 10253 + caddc 10255 − cmin 10585 ℕcn 11350 ℕ0cn0 11618 ♯chash 13410 Word cword 13574 lastSclsw 13622 Vtxcvtx 26294 iEdgciedg 26295 Walkscwlks 26894 WalksOncwlkson 26895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ifp 1092 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-hash 13411 df-word 13575 df-lsw 13623 df-wlks 26897 df-wlkson 26898 |
This theorem is referenced by: 3wlkond 27547 |
Copyright terms: Public domain | W3C validator |