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Mirrors > Home > MPE Home > Th. List > pthd | Structured version Visualization version GIF version |
Description: Two words representing a trail which also represent a path in a graph. (Contributed by AV, 10-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
pthd.p | ⊢ (𝜑 → 𝑃 ∈ Word V) |
pthd.r | ⊢ 𝑅 = ((#‘𝑃) − 1) |
pthd.s | ⊢ (𝜑 → ∀𝑖 ∈ (0..^(#‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))) |
pthd.f | ⊢ (#‘𝐹) = 𝑅 |
pthd.t | ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
Ref | Expression |
---|---|
pthd | ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pthd.t | . 2 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | |
2 | pthd.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ Word V) | |
3 | pthd.f | . . . 4 ⊢ (#‘𝐹) = 𝑅 | |
4 | pthd.r | . . . 4 ⊢ 𝑅 = ((#‘𝑃) − 1) | |
5 | 3, 4 | eqtri 2644 | . . 3 ⊢ (#‘𝐹) = ((#‘𝑃) − 1) |
6 | pthd.s | . . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(#‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))) | |
7 | 3 | oveq2i 6661 | . . . . . 6 ⊢ (1..^(#‘𝐹)) = (1..^𝑅) |
8 | 7 | raleqi 3142 | . . . . 5 ⊢ (∀𝑗 ∈ (1..^(#‘𝐹))(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ↔ ∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))) |
9 | 8 | ralbii 2980 | . . . 4 ⊢ (∀𝑖 ∈ (0..^(#‘𝑃))∀𝑗 ∈ (1..^(#‘𝐹))(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ↔ ∀𝑖 ∈ (0..^(#‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))) |
10 | 6, 9 | sylibr 224 | . . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(#‘𝑃))∀𝑗 ∈ (1..^(#‘𝐹))(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))) |
11 | 2, 5, 10 | pthdlem1 26662 | . 2 ⊢ (𝜑 → Fun ◡(𝑃 ↾ (1..^(#‘𝐹)))) |
12 | 2, 5, 10 | pthdlem2 26664 | . 2 ⊢ (𝜑 → ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) |
13 | ispth 26619 | . 2 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)) | |
14 | 1, 11, 12, 13 | syl3anbrc 1246 | 1 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∩ cin 3573 ∅c0 3915 {cpr 4179 class class class wbr 4653 ◡ccnv 5113 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 − cmin 10266 ..^cfzo 12465 #chash 13117 Word cword 13291 Trailsctrls 26587 Pathscpths 26608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-wlks 26495 df-trls 26589 df-pths 26612 |
This theorem is referenced by: 2pthd 26836 3pthd 27034 |
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