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Mirrors > Home > MPE Home > Th. List > clwwlknon1le1 | Structured version Visualization version GIF version |
Description: There is at most one (closed) walk on vertex 𝑋 of length 1 as word over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknon1le1 | ⊢ (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2733 | . . . . 5 ⊢ (ClWWalksNOn‘𝐺) = (ClWWalksNOn‘𝐺) | |
3 | eqid 2733 | . . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 1, 2, 3 | clwwlknon1loop 29091 | . . . 4 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∈ (Edg‘𝐺)) → (𝑋(ClWWalksNOn‘𝐺)1) = {⟨“𝑋”⟩}) |
5 | fveq2 6846 | . . . . . 6 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {⟨“𝑋”⟩} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘{⟨“𝑋”⟩})) | |
6 | s1cli 14502 | . . . . . . 7 ⊢ ⟨“𝑋”⟩ ∈ Word V | |
7 | hashsng 14278 | . . . . . . 7 ⊢ (⟨“𝑋”⟩ ∈ Word V → (♯‘{⟨“𝑋”⟩}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{⟨“𝑋”⟩}) = 1 |
9 | 5, 8 | eqtrdi 2789 | . . . . 5 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {⟨“𝑋”⟩} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 1) |
10 | 1le1 11791 | . . . . 5 ⊢ 1 ≤ 1 | |
11 | 9, 10 | eqbrtrdi 5148 | . . . 4 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {⟨“𝑋”⟩} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
12 | 4, 11 | syl 17 | . . 3 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∈ (Edg‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
13 | 1, 2, 3 | clwwlknon1nloop 29092 | . . . . 5 ⊢ ({𝑋} ∉ (Edg‘𝐺) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
14 | 13 | adantl 483 | . . . 4 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∉ (Edg‘𝐺)) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
15 | fveq2 6846 | . . . . . 6 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘∅)) | |
16 | hash0 14276 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
17 | 15, 16 | eqtrdi 2789 | . . . . 5 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 0) |
18 | 0le1 11686 | . . . . 5 ⊢ 0 ≤ 1 | |
19 | 17, 18 | eqbrtrdi 5148 | . . . 4 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
20 | 14, 19 | syl 17 | . . 3 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∉ (Edg‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
21 | 12, 20 | pm2.61danel 3060 | . 2 ⊢ (𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
22 | id 22 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ 𝑋 ∈ (Vtx‘𝐺)) | |
23 | 22 | intnanrd 491 | . . . . . 6 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 1 ∈ ℕ)) |
24 | clwwlknon0 29086 | . . . . . 6 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 1 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) | |
25 | 23, 24 | syl 17 | . . . . 5 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
26 | 25 | fveq2d 6850 | . . . 4 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘∅)) |
27 | 26, 16 | eqtrdi 2789 | . . 3 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 0) |
28 | 27, 18 | eqbrtrdi 5148 | . 2 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
29 | 21, 28 | pm2.61i 182 | 1 ⊢ (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∉ wnel 3046 Vcvv 3447 ∅c0 4286 {csn 4590 class class class wbr 5109 ‘cfv 6500 (class class class)co 7361 0cc0 11059 1c1 11060 ≤ cle 11198 ℕcn 12161 ♯chash 14239 Word cword 14411 ⟨“cs1 14492 Vtxcvtx 27996 Edgcedg 28047 ClWWalksNOncclwwlknon 29080 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-xnn0 12494 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 df-word 14412 df-lsw 14460 df-s1 14493 df-clwwlk 28975 df-clwwlkn 29018 df-clwwlknon 29081 |
This theorem is referenced by: (None) |
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