Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 40937 | . . 3 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ 1 < (#‘𝐹)) → ∀𝑖 ∈ (0..^(#‘𝐹))2 ≤ (#‘(𝐼‘(𝐹‘𝑖)))) |
3 | 2 | 3adant1 1072 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(PathS‘𝐺)𝑃 ∧ 1 < (#‘𝐹)) → ∀𝑖 ∈ (0..^(#‘𝐹))2 ≤ (#‘(𝐼‘(𝐹‘𝑖)))) |
4 | pthis1wlk 40933 | . . . . . . 7 ⊢ (𝐹(PathS‘𝐺)𝑃 → 𝐹(1Walks‘𝐺)𝑃) | |
5 | 1 | 1wlkf 40819 | . . . . . . 7 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
6 | simp2 1055 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(#‘𝐹))) → 𝐺 ∈ UPGraph ) | |
7 | wrdsymbcl 13173 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (𝐹‘𝑖) ∈ dom 𝐼) | |
8 | 1 | upgrle2 25771 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹‘𝑖) ∈ dom 𝐼) → (#‘(𝐼‘(𝐹‘𝑖))) ≤ 2) |
9 | 6, 7, 8 | 3imp3i2an 1270 | . . . . . . . . 9 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (#‘(𝐼‘(𝐹‘𝑖))) ≤ 2) |
10 | fvex 6113 | . . . . . . . . . . . . 13 ⊢ (𝐼‘(𝐹‘𝑖)) ∈ V | |
11 | hashxnn0 12989 | . . . . . . . . . . . . 13 ⊢ ((𝐼‘(𝐹‘𝑖)) ∈ V → (#‘(𝐼‘(𝐹‘𝑖))) ∈ ℕ0*) | |
12 | xnn0xr 11245 | . . . . . . . . . . . . 13 ⊢ ((#‘(𝐼‘(𝐹‘𝑖))) ∈ ℕ0* → (#‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ*) | |
13 | 10, 11, 12 | mp2b 10 | . . . . . . . . . . . 12 ⊢ (#‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ* |
14 | 2re 10967 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ | |
15 | 14 | rexri 9976 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ* |
16 | 13, 15 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ ((#‘(𝐼‘(𝐹‘𝑖))) ∈ ℝ* ∧ 2 ∈ ℝ*) |
17 | xrletri3 11861 | . . . . . . . . . . 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 237 | . . . . . . . . 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 707 | . . . . . . . 8 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝐺 ∈ UPGraph ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (2 ≤ (#‘(𝐼‘(𝐹‘𝑖))) → (#‘(𝐼‘(𝐹‘𝑖))) = 2)) |
21 | 20 | 3exp 1256 | . . . . . . 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 1074 | . . . 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 2945 | . 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 2c2 10947 ℕ0*cxnn0 11240 ..^cfzo 12334 #chash 12979 Word cword 13146 iEdgciedg 25674 UPGraph cupgr 25747 1Walksc1wlks 40796 PathScpths 40919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-uhgr 25724 df-upgr 25749 df-1wlks 40800 df-trls 40901 df-pths 40923 |
This theorem is referenced by: (None) |
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