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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 28008 (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 27975 | . . . . . 6 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
7 | wlkcl 27383 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
8 | nn0z 11992 | . . . . . . . . . 10 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℤ) | |
9 | 8 | anim1i 616 | . . . . . . . . 9 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 0 < (♯‘𝐹)) → ((♯‘𝐹) ∈ ℤ ∧ 0 < (♯‘𝐹))) |
10 | elnnz 11978 | . . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℤ ∧ 0 < (♯‘𝐹))) | |
11 | 9, 10 | sylibr 236 | . . . . . . . 8 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 0 < (♯‘𝐹)) → (♯‘𝐹) ∈ ℕ) |
12 | 11 | ex 415 | . . . . . . 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 13119 | . . . 4 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹))) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹))) |
18 | 4, 17 | eqeltrd 2913 | . 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 27978 | 1 ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5052 ↾ cres 5543 “ cima 5544 ‘cfv 6341 (class class class)co 7142 0cc0 10523 1c1 10524 < clt 10661 − cmin 10856 ℕcn 11624 ℕ0cn0 11884 ℤcz 11968 ...cfz 12882 ..^cfzo 13023 ♯chash 13680 prefix cpfx 14017 Vtxcvtx 26767 iEdgciedg 26768 Walkscwlks 27364 Circuitsccrcts 27551 EulerPathsceupth 27960 |
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 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-pm 8395 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-n0 11885 df-z 11969 df-uz 12231 df-fz 12883 df-fzo 13024 df-hash 13681 df-word 13852 df-substr 13988 df-pfx 14018 df-wlks 27367 df-trls 27460 df-eupth 27961 |
This theorem is referenced by: eucrct2eupth 28008 |
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