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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > usgr2trlspth | Structured version Visualization version GIF version |
Description: In a simple graph, any trail of length 2 is a simple path. (Contributed by AV, 5-Jun-2021.) |
Ref | Expression |
---|---|
usgr2trlspth | ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹(TrailS‘𝐺)𝑃 ↔ 𝐹(SPathS‘𝐺)𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgr2trlncl 40966 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹(TrailS‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2))) | |
2 | 1 | imp 444 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) ∧ 𝐹(TrailS‘𝐺)𝑃) → (𝑃‘0) ≠ (𝑃‘2)) |
3 | trlis1wlk 40905 | . . . . . 6 ⊢ (𝐹(TrailS‘𝐺)𝑃 → 𝐹(1Walks‘𝐺)𝑃) | |
4 | wlkOnwlk 40870 | . . . . . 6 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃) | |
5 | simpll 786 | . . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) ∧ (𝑃‘0) ≠ (𝑃‘2)) → 𝐺 ∈ USGraph ) | |
6 | simplr 788 | . . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) ∧ (𝑃‘0) ≠ (𝑃‘2)) → (#‘𝐹) = 2) | |
7 | fveq2 6103 | . . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 2 → (𝑃‘(#‘𝐹)) = (𝑃‘2)) | |
8 | 7 | eqcomd 2616 | . . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 2 → (𝑃‘2) = (𝑃‘(#‘𝐹))) |
9 | 8 | neeq2d 2842 | . . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 2 → ((𝑃‘0) ≠ (𝑃‘2) ↔ (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))) |
10 | 9 | biimpd 218 | . . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 2 → ((𝑃‘0) ≠ (𝑃‘2) → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))) |
11 | 10 | adantl 481 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → ((𝑃‘0) ≠ (𝑃‘2) → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))) |
12 | 11 | imp 444 | . . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) ∧ (𝑃‘0) ≠ (𝑃‘2)) → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) |
13 | usgr2wlkspth 40965 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) → (𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃 ↔ 𝐹((𝑃‘0)(SPathsOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃)) | |
14 | 5, 6, 12, 13 | syl3anc 1318 | . . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) ∧ (𝑃‘0) ≠ (𝑃‘2)) → (𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃 ↔ 𝐹((𝑃‘0)(SPathsOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃)) |
15 | spthonisspth-av 40956 | . . . . . . . . 9 ⊢ (𝐹((𝑃‘0)(SPathsOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃 → 𝐹(SPathS‘𝐺)𝑃) | |
16 | 14, 15 | syl6bi 242 | . . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) ∧ (𝑃‘0) ≠ (𝑃‘2)) → (𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃 → 𝐹(SPathS‘𝐺)𝑃)) |
17 | 16 | expcom 450 | . . . . . . 7 ⊢ ((𝑃‘0) ≠ (𝑃‘2) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃 → 𝐹(SPathS‘𝐺)𝑃))) |
18 | 17 | com13 86 | . . . . . 6 ⊢ (𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃 → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → ((𝑃‘0) ≠ (𝑃‘2) → 𝐹(SPathS‘𝐺)𝑃))) |
19 | 3, 4, 18 | 3syl 18 | . . . . 5 ⊢ (𝐹(TrailS‘𝐺)𝑃 → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → ((𝑃‘0) ≠ (𝑃‘2) → 𝐹(SPathS‘𝐺)𝑃))) |
20 | 19 | impcom 445 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) ∧ 𝐹(TrailS‘𝐺)𝑃) → ((𝑃‘0) ≠ (𝑃‘2) → 𝐹(SPathS‘𝐺)𝑃)) |
21 | 2, 20 | mpd 15 | . . 3 ⊢ (((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) ∧ 𝐹(TrailS‘𝐺)𝑃) → 𝐹(SPathS‘𝐺)𝑃) |
22 | 21 | ex 449 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹(TrailS‘𝐺)𝑃 → 𝐹(SPathS‘𝐺)𝑃)) |
23 | sPthisPth 40932 | . . 3 ⊢ (𝐹(SPathS‘𝐺)𝑃 → 𝐹(PathS‘𝐺)𝑃) | |
24 | PthisTrl 40931 | . . 3 ⊢ (𝐹(PathS‘𝐺)𝑃 → 𝐹(TrailS‘𝐺)𝑃) | |
25 | 23, 24 | syl 17 | . 2 ⊢ (𝐹(SPathS‘𝐺)𝑃 → 𝐹(TrailS‘𝐺)𝑃) |
26 | 22, 25 | impbid1 214 | 1 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹(TrailS‘𝐺)𝑃 ↔ 𝐹(SPathS‘𝐺)𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 2c2 10947 #chash 12979 USGraph cusgr 40379 1Walksc1wlks 40796 WalksOncwlkson 40798 TrailSctrls 40899 PathScpths 40919 SPathScspths 40920 SPathsOncspthson 40922 |
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-2o 7448 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-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-uhgr 25724 df-upgr 25749 df-umgr 25750 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-1wlks 40800 df-wlks 40801 df-wlkson 40802 df-trls 40901 df-trlson 40902 df-pths 40923 df-spths 40924 df-pthson 40925 df-spthson 40926 |
This theorem is referenced by: usgr2pthspth 40968 |
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