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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 27514 | . . 3 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))2 ≤ (♯‘(𝐼‘(𝐹‘𝑖)))) |
3 | 2 | 3adant1 1126 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))2 ≤ (♯‘(𝐼‘(𝐹‘𝑖)))) |
4 | pthiswlk 27510 | . . . . . . 7 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
5 | 1 | wlkf 27398 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
6 | simp2 1133 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → 𝐺 ∈ UPGraph) | |
7 | wrdsymbcl 13878 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑖) ∈ dom 𝐼) | |
8 | 1 | upgrle2 26892 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹‘𝑖) ∈ dom 𝐼) → (♯‘(𝐼‘(𝐹‘𝑖))) ≤ 2) |
9 | 6, 7, 8 | 3imp3i2an 1341 | . . . . . . . . 9 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (♯‘(𝐼‘(𝐹‘𝑖))) ≤ 2) |
10 | fvex 6685 | . . . . . . . . . . . . 13 ⊢ (𝐼‘(𝐹‘𝑖)) ∈ V | |
11 | hashxnn0 13702 | . . . . . . . . . . . . 13 ⊢ ((𝐼‘(𝐹‘𝑖)) ∈ V → (♯‘(𝐼‘(𝐹‘𝑖))) ∈ ℕ0*) | |
12 | xnn0xr 11975 | . . . . . . . . . . . . 13 ⊢ ((♯‘(𝐼‘(𝐹‘𝑖))) ∈ ℕ0* → (♯‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ*) | |
13 | 10, 11, 12 | mp2b 10 | . . . . . . . . . . . 12 ⊢ (♯‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ* |
14 | 2re 11714 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ | |
15 | 14 | rexri 10701 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ* |
16 | 13, 15 | pm3.2i 473 | . . . . . . . . . . 11 ⊢ ((♯‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ* ∧ 2 ∈ ℝ*) |
17 | xrletri3 12550 | . . . . . . . . . . 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 250 | . . . . . . . . 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 693 | . . . . . . . 8 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))) → (♯‘(𝐼‘(𝐹‘𝑖))) = 2)) |
21 | 20 | 3exp 1115 | . . . . . . 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 410 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑖 ∈ (0..^(♯‘𝐹)) → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))) → (♯‘(𝐼‘(𝐹‘𝑖))) = 2))) |
24 | 23 | 3adant3 1128 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → (𝑖 ∈ (0..^(♯‘𝐹)) → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))) → (♯‘(𝐼‘(𝐹‘𝑖))) = 2))) |
25 | 24 | imp 409 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))) → (♯‘(𝐼‘(𝐹‘𝑖))) = 2)) |
26 | 25 | ralimdva 3179 | . 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 class class class wbr 5068 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 2c2 11695 ℕ0*cxnn0 11970 ..^cfzo 13036 ♯chash 13693 Word cword 13864 iEdgciedg 26784 UPGraphcupgr 26867 Walkscwlks 27380 Pathscpths 27495 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-uhgr 26845 df-upgr 26869 df-wlks 27383 df-trls 27476 df-pths 27499 |
This theorem is referenced by: (None) |
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