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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 40999 | . . 3 ⊢ (𝐹(CycleS‘𝐺)𝑃 → (𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) | |
2 | cyclisWlk 41004 | . . 3 ⊢ (𝐹(CycleS‘𝐺)𝑃 → 𝐹(1Walks‘𝐺)𝑃) | |
3 | lfgrn1cycl.i | . . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | lfgrn1cycl.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 3, 4 | lfgr1wlknloop 40898 | . . . . . . 7 ⊢ ((𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} ∧ 𝐹(1Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) |
6 | 1nn 10908 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ | |
7 | eleq1 2676 | . . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 1 → ((#‘𝐹) ∈ ℕ ↔ 1 ∈ ℕ)) | |
8 | 6, 7 | mpbiri 247 | . . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 1 → (#‘𝐹) ∈ ℕ) |
9 | lbfzo0 12375 | . . . . . . . . . . . . 13 ⊢ (0 ∈ (0..^(#‘𝐹)) ↔ (#‘𝐹) ∈ ℕ) | |
10 | 8, 9 | sylibr 223 | . . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 1 → 0 ∈ (0..^(#‘𝐹))) |
11 | fveq2 6103 | . . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0)) | |
12 | oveq1 6556 | . . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (𝑘 + 1) = (0 + 1)) | |
13 | 0p1e1 11009 | . . . . . . . . . . . . . . . 16 ⊢ (0 + 1) = 1 | |
14 | 12, 13 | syl6eq 2660 | . . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (𝑘 + 1) = 1) |
15 | 14 | fveq2d 6107 | . . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1)) |
16 | 11, 15 | neeq12d 2843 | . . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ (𝑃‘0) ≠ (𝑃‘1))) |
17 | 16 | rspcv 3278 | . . . . . . . . . . . 12 ⊢ (0 ∈ (0..^(#‘𝐹)) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘0) ≠ (𝑃‘1))) |
18 | 10, 17 | syl 17 | . . . . . . . . . . 11 ⊢ ((#‘𝐹) = 1 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘0) ≠ (𝑃‘1))) |
19 | 18 | impcom 445 | . . . . . . . . . 10 ⊢ ((∀𝑘 ∈ (0..^(#‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ∧ (#‘𝐹) = 1) → (𝑃‘0) ≠ (𝑃‘1)) |
20 | fveq2 6103 | . . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 1 → (𝑃‘(#‘𝐹)) = (𝑃‘1)) | |
21 | 20 | neeq2d 2842 | . . . . . . . . . . 11 ⊢ ((#‘𝐹) = 1 → ((𝑃‘0) ≠ (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) ≠ (𝑃‘1))) |
22 | 21 | adantl 481 | . . . . . . . . . 10 ⊢ ((∀𝑘 ∈ (0..^(#‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ∧ (#‘𝐹) = 1) → ((𝑃‘0) ≠ (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) ≠ (𝑃‘1))) |
23 | 19, 22 | mpbird 246 | . . . . . . . . 9 ⊢ ((∀𝑘 ∈ (0..^(#‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ∧ (#‘𝐹) = 1) → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) |
24 | 23 | ex 449 | . . . . . . . 8 ⊢ (∀𝑘 ∈ (0..^(#‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → ((#‘𝐹) = 1 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))) |
25 | 24 | necon2d 2805 | . . . . . . 7 ⊢ (∀𝑘 ∈ (0..^(#‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (#‘𝐹) ≠ 1)) |
26 | 5, 25 | syl 17 | . . . . . 6 ⊢ ((𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} ∧ 𝐹(1Walks‘𝐺)𝑃) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (#‘𝐹) ≠ 1)) |
27 | 26 | ex 449 | . . . . 5 ⊢ (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → (𝐹(1Walks‘𝐺)𝑃 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (#‘𝐹) ≠ 1))) |
28 | 27 | com13 86 | . . . 4 ⊢ ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝐹(1Walks‘𝐺)𝑃 → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → (#‘𝐹) ≠ 1))) |
29 | 28 | adantl 481 | . . 3 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝐹(1Walks‘𝐺)𝑃 → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → (#‘𝐹) ≠ 1))) |
30 | 1, 2, 29 | sylc 63 | . 2 ⊢ (𝐹(CycleS‘𝐺)𝑃 → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → (#‘𝐹) ≠ 1)) |
31 | 30 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 𝒫 cpw 4108 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 ℕcn 10897 2c2 10947 ..^cfzo 12334 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 1Walksc1wlks 40796 PathScpths 40919 CycleSccycls 40991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-1wlks 40800 df-trls 40901 df-pths 40923 df-cycls 40993 |
This theorem is referenced by: umgrn1cycl 41010 |
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