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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 26683 | . . 3 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (#‘𝐹)) → ∀𝑖 ∈ (0..^(#‘𝐹))2 ≤ (#‘(𝐼‘(𝐹‘𝑖)))) |
3 | 2 | 3adant1 1099 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (#‘𝐹)) → ∀𝑖 ∈ (0..^(#‘𝐹))2 ≤ (#‘(𝐼‘(𝐹‘𝑖)))) |
4 | pthiswlk 26679 | . . . . . . 7 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
5 | 1 | wlkf 26566 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
6 | simp2 1082 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(#‘𝐹))) → 𝐺 ∈ UPGraph) | |
7 | wrdsymbcl 13350 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (𝐹‘𝑖) ∈ dom 𝐼) | |
8 | 1 | upgrle2 26045 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹‘𝑖) ∈ dom 𝐼) → (#‘(𝐼‘(𝐹‘𝑖))) ≤ 2) |
9 | 6, 7, 8 | 3imp3i2an 1299 | . . . . . . . . 9 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (#‘(𝐼‘(𝐹‘𝑖))) ≤ 2) |
10 | fvex 6239 | . . . . . . . . . . . . 13 ⊢ (𝐼‘(𝐹‘𝑖)) ∈ V | |
11 | hashxnn0 13167 | . . . . . . . . . . . . 13 ⊢ ((𝐼‘(𝐹‘𝑖)) ∈ V → (#‘(𝐼‘(𝐹‘𝑖))) ∈ ℕ0*) | |
12 | xnn0xr 11406 | . . . . . . . . . . . . 13 ⊢ ((#‘(𝐼‘(𝐹‘𝑖))) ∈ ℕ0* → (#‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ*) | |
13 | 10, 11, 12 | mp2b 10 | . . . . . . . . . . . 12 ⊢ (#‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ* |
14 | 2re 11128 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ | |
15 | 14 | rexri 10135 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ* |
16 | 13, 15 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ ((#‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ* ∧ 2 ∈ ℝ*) |
17 | xrletri3 12023 | . . . . . . . . . . 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 238 | . . . . . . . . 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 711 | . . . . . . . 8 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (2 ≤ (#‘(𝐼‘(𝐹‘𝑖))) → (#‘(𝐼‘(𝐹‘𝑖))) = 2)) |
21 | 20 | 3exp 1283 | . . . . . . 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 445 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑖 ∈ (0..^(#‘𝐹)) → (2 ≤ (#‘(𝐼‘(𝐹‘𝑖))) → (#‘(𝐼‘(𝐹‘𝑖))) = 2))) |
24 | 23 | 3adant3 1101 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (#‘𝐹)) → (𝑖 ∈ (0..^(#‘𝐹)) → (2 ≤ (#‘(𝐼‘(𝐹‘𝑖))) → (#‘(𝐼‘(𝐹‘𝑖))) = 2))) |
25 | 24 | imp 444 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (#‘𝐹)) ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (2 ≤ (#‘(𝐼‘(𝐹‘𝑖))) → (#‘(𝐼‘(𝐹‘𝑖))) = 2)) |
26 | 25 | ralimdva 2991 | . 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 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∀wral 2941 Vcvv 3231 class class class wbr 4685 dom cdm 5143 ‘cfv 5926 (class class class)co 6690 0cc0 9974 1c1 9975 ℝ*cxr 10111 < clt 10112 ≤ cle 10113 2c2 11108 ℕ0*cxnn0 11401 ..^cfzo 12504 #chash 13157 Word cword 13323 iEdgciedg 25920 UPGraphcupgr 26020 Walkscwlks 26548 Pathscpths 26664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ifp 1033 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-n0 11331 df-xnn0 11402 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 df-uhgr 25998 df-upgr 26022 df-wlks 26551 df-trls 26645 df-pths 26668 |
This theorem is referenced by: (None) |
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