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Mirrors > Home > MPE Home > Th. List > lfgrn1cycl | Structured version Visualization version GIF version |
Description: In a loop-free graph there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.) |
Ref | Expression |
---|---|
lfgrn1cycl.v | ⊢ 𝑉 = (Vtx‘𝐺) |
lfgrn1cycl.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
lfgrn1cycl | ⊢ (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ≠ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cyclprop 27574 | . . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | |
2 | cycliswlk 27579 | . . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
3 | lfgrn1cycl.i | . . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | lfgrn1cycl.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 3, 4 | lfgrwlknloop 27471 | . . . . . . 7 ⊢ ((𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) |
6 | 1nn 11649 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ | |
7 | eleq1 2900 | . . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) = 1 → ((♯‘𝐹) ∈ ℕ ↔ 1 ∈ ℕ)) | |
8 | 6, 7 | mpbiri 260 | . . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 1 → (♯‘𝐹) ∈ ℕ) |
9 | lbfzo0 13078 | . . . . . . . . . . . . 13 ⊢ (0 ∈ (0..^(♯‘𝐹)) ↔ (♯‘𝐹) ∈ ℕ) | |
10 | 8, 9 | sylibr 236 | . . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 1 → 0 ∈ (0..^(♯‘𝐹))) |
11 | fveq2 6670 | . . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0)) | |
12 | fv0p1e1 11761 | . . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1)) | |
13 | 11, 12 | neeq12d 3077 | . . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ (𝑃‘0) ≠ (𝑃‘1))) |
14 | 13 | rspcv 3618 | . . . . . . . . . . . 12 ⊢ (0 ∈ (0..^(♯‘𝐹)) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘0) ≠ (𝑃‘1))) |
15 | 10, 14 | syl 17 | . . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 1 → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘0) ≠ (𝑃‘1))) |
16 | 15 | impcom 410 | . . . . . . . . . 10 ⊢ ((∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ∧ (♯‘𝐹) = 1) → (𝑃‘0) ≠ (𝑃‘1)) |
17 | fveq2 6670 | . . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 1 → (𝑃‘(♯‘𝐹)) = (𝑃‘1)) | |
18 | 17 | neeq2d 3076 | . . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 1 → ((𝑃‘0) ≠ (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) ≠ (𝑃‘1))) |
19 | 18 | adantl 484 | . . . . . . . . . 10 ⊢ ((∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ∧ (♯‘𝐹) = 1) → ((𝑃‘0) ≠ (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) ≠ (𝑃‘1))) |
20 | 16, 19 | mpbird 259 | . . . . . . . . 9 ⊢ ((∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ∧ (♯‘𝐹) = 1) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) |
21 | 20 | ex 415 | . . . . . . . 8 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → ((♯‘𝐹) = 1 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) |
22 | 21 | necon2d 3039 | . . . . . . 7 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (♯‘𝐹) ≠ 1)) |
23 | 5, 22 | syl 17 | . . . . . 6 ⊢ ((𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ∧ 𝐹(Walks‘𝐺)𝑃) → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (♯‘𝐹) ≠ 1)) |
24 | 23 | ex 415 | . . . . 5 ⊢ (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (♯‘𝐹) ≠ 1))) |
25 | 24 | com13 88 | . . . 4 ⊢ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (𝐹(Walks‘𝐺)𝑃 → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → (♯‘𝐹) ≠ 1))) |
26 | 25 | adantl 484 | . . 3 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝐹(Walks‘𝐺)𝑃 → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → (♯‘𝐹) ≠ 1))) |
27 | 1, 2, 26 | sylc 65 | . 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → (♯‘𝐹) ≠ 1)) |
28 | 27 | com12 32 | 1 ⊢ (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ≠ 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 {crab 3142 𝒫 cpw 4539 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 ≤ cle 10676 ℕcn 11638 2c2 11693 ..^cfzo 13034 ♯chash 13691 Vtxcvtx 26781 iEdgciedg 26782 Walkscwlks 27378 Pathscpths 27493 Cyclesccycls 27566 |
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-er 8289 df-map 8408 df-pm 8409 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-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-wlks 27381 df-trls 27474 df-pths 27497 df-cycls 27568 |
This theorem is referenced by: umgrn1cycl 27585 |
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