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Mirrors > Home > MPE Home > Th. List > eucrct2eupth1 | Structured version Visualization version GIF version |
Description: Removing one edge (𝐼‘(𝐹‘𝑁)) from a nonempty graph 𝐺 with an Eulerian circuit 〈𝐹, 𝑃〉 results in a graph 𝑆 with an Eulerian path 〈𝐻, 𝑄〉. This is the special case of eucrct2eupth 27792 (with 𝐽 = (𝑁 − 1)) where the last segment/edge of the circuit is removed. (Contributed by AV, 11-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
Ref | Expression |
---|---|
eucrct2eupth1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
eucrct2eupth1.i | ⊢ 𝐼 = (iEdg‘𝐺) |
eucrct2eupth1.d | ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) |
eucrct2eupth1.c | ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) |
eucrct2eupth1.s | ⊢ (Vtx‘𝑆) = 𝑉 |
eucrct2eupth1.g | ⊢ (𝜑 → 0 < (♯‘𝐹)) |
eucrct2eupth1.n | ⊢ (𝜑 → 𝑁 = ((♯‘𝐹) − 1)) |
eucrct2eupth1.e | ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
eucrct2eupth1.h | ⊢ 𝐻 = (𝐹 prefix 𝑁) |
eucrct2eupth1.q | ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) |
Ref | Expression |
---|---|
eucrct2eupth1 | ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eucrct2eupth1.v | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | eucrct2eupth1.i | . 2 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | eucrct2eupth1.d | . 2 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) | |
4 | eucrct2eupth1.n | . . 3 ⊢ (𝜑 → 𝑁 = ((♯‘𝐹) − 1)) | |
5 | eucrct2eupth1.g | . . . . 5 ⊢ (𝜑 → 0 < (♯‘𝐹)) | |
6 | eupthiswlk 27756 | . . . . . 6 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
7 | wlkcl 27115 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
8 | nn0z 11816 | . . . . . . . . . 10 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℤ) | |
9 | 8 | anim1i 606 | . . . . . . . . 9 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 0 < (♯‘𝐹)) → ((♯‘𝐹) ∈ ℤ ∧ 0 < (♯‘𝐹))) |
10 | elnnz 11801 | . . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℤ ∧ 0 < (♯‘𝐹))) | |
11 | 9, 10 | sylibr 226 | . . . . . . . 8 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 0 < (♯‘𝐹)) → (♯‘𝐹) ∈ ℕ) |
12 | 11 | ex 405 | . . . . . . 7 ⊢ ((♯‘𝐹) ∈ ℕ0 → (0 < (♯‘𝐹) → (♯‘𝐹) ∈ ℕ)) |
13 | 7, 12 | syl 17 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0 < (♯‘𝐹) → (♯‘𝐹) ∈ ℕ)) |
14 | 3, 6, 13 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (0 < (♯‘𝐹) → (♯‘𝐹) ∈ ℕ)) |
15 | 5, 14 | mpd 15 | . . . 4 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ) |
16 | fzo0end 12942 | . . . 4 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹))) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹))) |
18 | 4, 17 | eqeltrd 2859 | . 2 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
19 | eucrct2eupth1.e | . 2 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
20 | eucrct2eupth1.h | . 2 ⊢ 𝐻 = (𝐹 prefix 𝑁) | |
21 | eucrct2eupth1.q | . 2 ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) | |
22 | eucrct2eupth1.s | . 2 ⊢ (Vtx‘𝑆) = 𝑉 | |
23 | 1, 2, 3, 18, 19, 20, 21, 22 | eupthres 27760 | 1 ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 class class class wbr 4925 ↾ cres 5405 “ cima 5406 ‘cfv 6185 (class class class)co 6974 0cc0 10333 1c1 10334 < clt 10472 − cmin 10668 ℕcn 11437 ℕ0cn0 11705 ℤcz 11791 ...cfz 12706 ..^cfzo 12847 ♯chash 13503 prefix cpfx 13850 Vtxcvtx 26499 iEdgciedg 26500 Walkscwlks 27096 Circuitsccrcts 27288 EulerPathsceupth 27741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-ifp 1045 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 df-fzo 12848 df-hash 13504 df-word 13671 df-substr 13802 df-pfx 13851 df-wlks 27099 df-trls 27195 df-eupth 27742 |
This theorem is referenced by: eucrct2eupth 27792 |
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