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Mirrors > Home > MPE Home > Th. List > wlklenvm1 | Structured version Visualization version GIF version |
Description: The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
Ref | Expression |
---|---|
wlklenvm1 | ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) = ((♯‘𝑃) − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlklenvp1 27394 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1)) | |
2 | oveq1 7157 | . . . 4 ⊢ ((♯‘𝑃) = ((♯‘𝐹) + 1) → ((♯‘𝑃) − 1) = (((♯‘𝐹) + 1) − 1)) | |
3 | wlkcl 27391 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
4 | 3 | nn0cnd 11951 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℂ) |
5 | pncan1 11058 | . . . . 5 ⊢ ((♯‘𝐹) ∈ ℂ → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹)) |
7 | 2, 6 | sylan9eqr 2878 | . . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝑃) = ((♯‘𝐹) + 1)) → ((♯‘𝑃) − 1) = (♯‘𝐹)) |
8 | 7 | eqcomd 2827 | . 2 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝑃) = ((♯‘𝐹) + 1)) → (♯‘𝐹) = ((♯‘𝑃) − 1)) |
9 | 1, 8 | mpdan 685 | 1 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) = ((♯‘𝑃) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 1c1 10532 + caddc 10534 − cmin 10864 ♯chash 13684 Walkscwlks 27372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
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 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-wlks 27375 |
This theorem is referenced by: wlkeq 27409 uspgr2wlkeqi 27423 wlkonwlk1l 27439 wlksoneq1eq2 27440 wlklnwwlkln2lem 27654 wlknewwlksn 27659 wspthsnwspthsnon 27689 wspthsnonn0vne 27690 elwspths2spth 27740 clwlkclwwlkfolem 27779 clwlkclwwlkf1 27782 clwlknf1oclwwlknlem1 27854 clwlknf1oclwwlkn 27857 revwlk 32366 |
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