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Mirrors > Home > MPE Home > Th. List > upgr2pthnlp | Structured version Visualization version GIF version |
Description: A path of length at least 2 in a pseudograph does not contain a loop. (Contributed by AV, 6-Feb-2021.) |
Ref | Expression |
---|---|
2pthnloop.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
upgr2pthnlp | ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))(♯‘(𝐼‘(𝐹‘𝑖))) = 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2pthnloop.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
2 | 1 | 2pthnloop 27818 | . . 3 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))2 ≤ (♯‘(𝐼‘(𝐹‘𝑖)))) |
3 | 2 | 3adant1 1132 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))2 ≤ (♯‘(𝐼‘(𝐹‘𝑖)))) |
4 | pthiswlk 27814 | . . . . . . 7 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
5 | 1 | wlkf 27702 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
6 | simp2 1139 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → 𝐺 ∈ UPGraph) | |
7 | wrdsymbcl 14082 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑖) ∈ dom 𝐼) | |
8 | 1 | upgrle2 27196 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹‘𝑖) ∈ dom 𝐼) → (♯‘(𝐼‘(𝐹‘𝑖))) ≤ 2) |
9 | 6, 7, 8 | 3imp3i2an 1347 | . . . . . . . . 9 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (♯‘(𝐼‘(𝐹‘𝑖))) ≤ 2) |
10 | fvex 6730 | . . . . . . . . . . . . 13 ⊢ (𝐼‘(𝐹‘𝑖)) ∈ V | |
11 | hashxnn0 13905 | . . . . . . . . . . . . 13 ⊢ ((𝐼‘(𝐹‘𝑖)) ∈ V → (♯‘(𝐼‘(𝐹‘𝑖))) ∈ ℕ0*) | |
12 | xnn0xr 12167 | . . . . . . . . . . . . 13 ⊢ ((♯‘(𝐼‘(𝐹‘𝑖))) ∈ ℕ0* → (♯‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ*) | |
13 | 10, 11, 12 | mp2b 10 | . . . . . . . . . . . 12 ⊢ (♯‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ* |
14 | 2re 11904 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ | |
15 | 14 | rexri 10891 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ* |
16 | 13, 15 | pm3.2i 474 | . . . . . . . . . . 11 ⊢ ((♯‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ* ∧ 2 ∈ ℝ*) |
17 | xrletri3 12744 | . . . . . . . . . . 11 ⊢ (((♯‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ* ∧ 2 ∈ ℝ*) → ((♯‘(𝐼‘(𝐹‘𝑖))) = 2 ↔ ((♯‘(𝐼‘(𝐹‘𝑖))) ≤ 2 ∧ 2 ≤ (♯‘(𝐼‘(𝐹‘𝑖)))))) | |
18 | 16, 17 | mp1i 13 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((♯‘(𝐼‘(𝐹‘𝑖))) = 2 ↔ ((♯‘(𝐼‘(𝐹‘𝑖))) ≤ 2 ∧ 2 ≤ (♯‘(𝐼‘(𝐹‘𝑖)))))) |
19 | 18 | biimprd 251 | . . . . . . . . 9 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (((♯‘(𝐼‘(𝐹‘𝑖))) ≤ 2 ∧ 2 ≤ (♯‘(𝐼‘(𝐹‘𝑖)))) → (♯‘(𝐼‘(𝐹‘𝑖))) = 2)) |
20 | 9, 19 | mpand 695 | . . . . . . . 8 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))) → (♯‘(𝐼‘(𝐹‘𝑖))) = 2)) |
21 | 20 | 3exp 1121 | . . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → (𝐺 ∈ UPGraph → (𝑖 ∈ (0..^(♯‘𝐹)) → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))) → (♯‘(𝐼‘(𝐹‘𝑖))) = 2)))) |
22 | 4, 5, 21 | 3syl 18 | . . . . . 6 ⊢ (𝐹(Paths‘𝐺)𝑃 → (𝐺 ∈ UPGraph → (𝑖 ∈ (0..^(♯‘𝐹)) → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))) → (♯‘(𝐼‘(𝐹‘𝑖))) = 2)))) |
23 | 22 | impcom 411 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑖 ∈ (0..^(♯‘𝐹)) → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))) → (♯‘(𝐼‘(𝐹‘𝑖))) = 2))) |
24 | 23 | 3adant3 1134 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → (𝑖 ∈ (0..^(♯‘𝐹)) → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))) → (♯‘(𝐼‘(𝐹‘𝑖))) = 2))) |
25 | 24 | imp 410 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))) → (♯‘(𝐼‘(𝐹‘𝑖))) = 2)) |
26 | 25 | ralimdva 3100 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → (∀𝑖 ∈ (0..^(♯‘𝐹))2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))) → ∀𝑖 ∈ (0..^(♯‘𝐹))(♯‘(𝐼‘(𝐹‘𝑖))) = 2)) |
27 | 3, 26 | mpd 15 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))(♯‘(𝐼‘(𝐹‘𝑖))) = 2) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 class class class wbr 5053 dom cdm 5551 ‘cfv 6380 (class class class)co 7213 0cc0 10729 1c1 10730 ℝ*cxr 10866 < clt 10867 ≤ cle 10868 2c2 11885 ℕ0*cxnn0 12162 ..^cfzo 13238 ♯chash 13896 Word cword 14069 iEdgciedg 27088 UPGraphcupgr 27171 Walkscwlks 27684 Pathscpths 27799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-oadd 8206 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-n0 12091 df-xnn0 12163 df-z 12177 df-uz 12439 df-fz 13096 df-fzo 13239 df-hash 13897 df-word 14070 df-uhgr 27149 df-upgr 27173 df-wlks 27687 df-trls 27780 df-pths 27803 |
This theorem is referenced by: (None) |
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