|   | Mathbox for BTernaryTau | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > acycgr1v | Structured version Visualization version GIF version | ||
| Description: A multigraph with one vertex is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.) | 
| Ref | Expression | 
|---|---|
| acycgrv.1 | ⊢ 𝑉 = (Vtx‘𝐺) | 
| Ref | Expression | 
|---|---|
| acycgr1v | ⊢ ((𝐺 ∈ UMGraph ∧ (♯‘𝑉) = 1) → 𝐺 ∈ AcyclicGraph) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cyclispth 29817 | . . . . . . . . . . . 12 ⊢ (𝑓(Cycles‘𝐺)𝑝 → 𝑓(Paths‘𝐺)𝑝) | |
| 2 | acycgrv.1 | . . . . . . . . . . . . 13 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 3 | 2 | pthhashvtx 35133 | . . . . . . . . . . . 12 ⊢ (𝑓(Paths‘𝐺)𝑝 → (♯‘𝑓) ≤ (♯‘𝑉)) | 
| 4 | 1, 3 | syl 17 | . . . . . . . . . . 11 ⊢ (𝑓(Cycles‘𝐺)𝑝 → (♯‘𝑓) ≤ (♯‘𝑉)) | 
| 5 | 4 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑉) = 1) → (♯‘𝑓) ≤ (♯‘𝑉)) | 
| 6 | breq2 5147 | . . . . . . . . . . 11 ⊢ ((♯‘𝑉) = 1 → ((♯‘𝑓) ≤ (♯‘𝑉) ↔ (♯‘𝑓) ≤ 1)) | |
| 7 | 6 | adantl 481 | . . . . . . . . . 10 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑉) = 1) → ((♯‘𝑓) ≤ (♯‘𝑉) ↔ (♯‘𝑓) ≤ 1)) | 
| 8 | 5, 7 | mpbid 232 | . . . . . . . . 9 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑉) = 1) → (♯‘𝑓) ≤ 1) | 
| 9 | 8 | 3adant1 1131 | . . . . . . . 8 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑉) = 1) → (♯‘𝑓) ≤ 1) | 
| 10 | umgrn1cycl 29827 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑓(Cycles‘𝐺)𝑝) → (♯‘𝑓) ≠ 1) | |
| 11 | 10 | 3adant3 1133 | . . . . . . . . 9 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑉) = 1) → (♯‘𝑓) ≠ 1) | 
| 12 | 11 | necomd 2996 | . . . . . . . 8 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑉) = 1) → 1 ≠ (♯‘𝑓)) | 
| 13 | cycliswlk 29818 | . . . . . . . . . 10 ⊢ (𝑓(Cycles‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝) | |
| 14 | wlkcl 29633 | . . . . . . . . . . . 12 ⊢ (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑓) ∈ ℕ0) | |
| 15 | 14 | nn0red 12588 | . . . . . . . . . . 11 ⊢ (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑓) ∈ ℝ) | 
| 16 | 1red 11262 | . . . . . . . . . . 11 ⊢ (𝑓(Walks‘𝐺)𝑝 → 1 ∈ ℝ) | |
| 17 | 15, 16 | ltlend 11406 | . . . . . . . . . 10 ⊢ (𝑓(Walks‘𝐺)𝑝 → ((♯‘𝑓) < 1 ↔ ((♯‘𝑓) ≤ 1 ∧ 1 ≠ (♯‘𝑓)))) | 
| 18 | 13, 17 | syl 17 | . . . . . . . . 9 ⊢ (𝑓(Cycles‘𝐺)𝑝 → ((♯‘𝑓) < 1 ↔ ((♯‘𝑓) ≤ 1 ∧ 1 ≠ (♯‘𝑓)))) | 
| 19 | 18 | 3ad2ant2 1135 | . . . . . . . 8 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑉) = 1) → ((♯‘𝑓) < 1 ↔ ((♯‘𝑓) ≤ 1 ∧ 1 ≠ (♯‘𝑓)))) | 
| 20 | 9, 12, 19 | mpbir2and 713 | . . . . . . 7 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑉) = 1) → (♯‘𝑓) < 1) | 
| 21 | nn0lt10b 12680 | . . . . . . . . 9 ⊢ ((♯‘𝑓) ∈ ℕ0 → ((♯‘𝑓) < 1 ↔ (♯‘𝑓) = 0)) | |
| 22 | 13, 14, 21 | 3syl 18 | . . . . . . . 8 ⊢ (𝑓(Cycles‘𝐺)𝑝 → ((♯‘𝑓) < 1 ↔ (♯‘𝑓) = 0)) | 
| 23 | 22 | 3ad2ant2 1135 | . . . . . . 7 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑉) = 1) → ((♯‘𝑓) < 1 ↔ (♯‘𝑓) = 0)) | 
| 24 | 20, 23 | mpbid 232 | . . . . . 6 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑉) = 1) → (♯‘𝑓) = 0) | 
| 25 | hasheq0 14402 | . . . . . . 7 ⊢ (𝑓 ∈ V → ((♯‘𝑓) = 0 ↔ 𝑓 = ∅)) | |
| 26 | 25 | elv 3485 | . . . . . 6 ⊢ ((♯‘𝑓) = 0 ↔ 𝑓 = ∅) | 
| 27 | 24, 26 | sylib 218 | . . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑉) = 1) → 𝑓 = ∅) | 
| 28 | 27 | 3com23 1127 | . . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ (♯‘𝑉) = 1 ∧ 𝑓(Cycles‘𝐺)𝑝) → 𝑓 = ∅) | 
| 29 | 28 | 3expia 1122 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ (♯‘𝑉) = 1) → (𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)) | 
| 30 | 29 | alrimivv 1928 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ (♯‘𝑉) = 1) → ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)) | 
| 31 | isacycgr1 35151 | . . 3 ⊢ (𝐺 ∈ UMGraph → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅))) | |
| 32 | 31 | adantr 480 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ (♯‘𝑉) = 1) → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅))) | 
| 33 | 30, 32 | mpbird 257 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ (♯‘𝑉) = 1) → 𝐺 ∈ AcyclicGraph) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 0cc0 11155 1c1 11156 < clt 11295 ≤ cle 11296 ℕ0cn0 12526 ♯chash 14369 Vtxcvtx 29013 UMGraphcumgr 29098 Walkscwlks 29614 Pathscpths 29730 Cyclesccycls 29805 AcyclicGraphcacycgr 35147 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-upgr 29099 df-umgr 29100 df-wlks 29617 df-trls 29710 df-pths 29734 df-cycls 29807 df-acycgr 35148 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |