Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  spthonepeq Structured version   Visualization version   GIF version

Theorem spthonepeq 26117
 Description: The endpoints of a simple path between two vertices are equal if and only if the path is of length 0 (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
Assertion
Ref Expression
spthonepeq (𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))

Proof of Theorem spthonepeq
StepHypRef Expression
1 spthonprp 26115 . 2 (𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃)))
2 iswlkon 26062 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
3 isspth 26099 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 SPaths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃)))
433adant3 1074 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝑉 SPaths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃)))
52, 4anbi12d 743 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → ((𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃) ↔ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ∧ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃))))
6 2mwlk 26049 . . . . . . . 8 (𝐹(𝑉 Walks 𝐸)𝑃 → (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉))
7 lencl 13179 . . . . . . . . 9 (𝐹 ∈ Word dom 𝐸 → (#‘𝐹) ∈ ℕ0)
87anim1i 590 . . . . . . . 8 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉) → ((#‘𝐹) ∈ ℕ0𝑃:(0...(#‘𝐹))⟶𝑉))
9 df-f1 5809 . . . . . . . . . . . 12 (𝑃:(0...(#‘𝐹))–1-1𝑉 ↔ (𝑃:(0...(#‘𝐹))⟶𝑉 ∧ Fun 𝑃))
10 eqeq2 2621 . . . . . . . . . . . . . 14 (𝐴 = 𝐵 → ((𝑃‘0) = 𝐴 ↔ (𝑃‘0) = 𝐵))
11 eqtr3 2631 . . . . . . . . . . . . . . . 16 (((𝑃‘(#‘𝐹)) = 𝐵 ∧ (𝑃‘0) = 𝐵) → (𝑃‘(#‘𝐹)) = (𝑃‘0))
12 elnn0uz 11601 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝐹) ∈ ℕ0 ↔ (#‘𝐹) ∈ (ℤ‘0))
13 eluzfz2 12220 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝐹) ∈ (ℤ‘0) → (#‘𝐹) ∈ (0...(#‘𝐹)))
1412, 13sylbi 206 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ (0...(#‘𝐹)))
15 0elfz 12305 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝐹) ∈ ℕ0 → 0 ∈ (0...(#‘𝐹)))
1614, 15jca 553 . . . . . . . . . . . . . . . . . . 19 ((#‘𝐹) ∈ ℕ0 → ((#‘𝐹) ∈ (0...(#‘𝐹)) ∧ 0 ∈ (0...(#‘𝐹))))
17 f1veqaeq 6418 . . . . . . . . . . . . . . . . . . 19 ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ((#‘𝐹) ∈ (0...(#‘𝐹)) ∧ 0 ∈ (0...(#‘𝐹)))) → ((𝑃‘(#‘𝐹)) = (𝑃‘0) → (#‘𝐹) = 0))
1816, 17sylan2 490 . . . . . . . . . . . . . . . . . 18 ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ (#‘𝐹) ∈ ℕ0) → ((𝑃‘(#‘𝐹)) = (𝑃‘0) → (#‘𝐹) = 0))
1918ex 449 . . . . . . . . . . . . . . . . 17 (𝑃:(0...(#‘𝐹))–1-1𝑉 → ((#‘𝐹) ∈ ℕ0 → ((𝑃‘(#‘𝐹)) = (𝑃‘0) → (#‘𝐹) = 0)))
2019com13 86 . . . . . . . . . . . . . . . 16 ((𝑃‘(#‘𝐹)) = (𝑃‘0) → ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → (#‘𝐹) = 0)))
2111, 20syl 17 . . . . . . . . . . . . . . 15 (((𝑃‘(#‘𝐹)) = 𝐵 ∧ (𝑃‘0) = 𝐵) → ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → (#‘𝐹) = 0)))
2221expcom 450 . . . . . . . . . . . . . 14 ((𝑃‘0) = 𝐵 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → (#‘𝐹) = 0))))
2310, 22syl6bi 242 . . . . . . . . . . . . 13 (𝐴 = 𝐵 → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → (#‘𝐹) = 0)))))
2423com15 99 . . . . . . . . . . . 12 (𝑃:(0...(#‘𝐹))–1-1𝑉 → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ0 → (𝐴 = 𝐵 → (#‘𝐹) = 0)))))
259, 24sylbir 224 . . . . . . . . . . 11 ((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ Fun 𝑃) → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ0 → (𝐴 = 𝐵 → (#‘𝐹) = 0)))))
2625expcom 450 . . . . . . . . . 10 (Fun 𝑃 → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ0 → (𝐴 = 𝐵 → (#‘𝐹) = 0))))))
2726com15 99 . . . . . . . . 9 ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → (Fun 𝑃 → (𝐴 = 𝐵 → (#‘𝐹) = 0))))))
2827imp 444 . . . . . . . 8 (((#‘𝐹) ∈ ℕ0𝑃:(0...(#‘𝐹))⟶𝑉) → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → (Fun 𝑃 → (𝐴 = 𝐵 → (#‘𝐹) = 0)))))
296, 8, 283syl 18 . . . . . . 7 (𝐹(𝑉 Walks 𝐸)𝑃 → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → (Fun 𝑃 → (𝐴 = 𝐵 → (#‘𝐹) = 0)))))
30293imp1 1272 . . . . . 6 (((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ∧ Fun 𝑃) → (𝐴 = 𝐵 → (#‘𝐹) = 0))
31 fveq2 6103 . . . . . . . . . . . 12 ((#‘𝐹) = 0 → (𝑃‘(#‘𝐹)) = (𝑃‘0))
3231eqeq1d 2612 . . . . . . . . . . 11 ((#‘𝐹) = 0 → ((𝑃‘(#‘𝐹)) = 𝐵 ↔ (𝑃‘0) = 𝐵))
3332anbi2d 736 . . . . . . . . . 10 ((#‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵)))
34 eqtr2 2630 . . . . . . . . . 10 (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → 𝐴 = 𝐵)
3533, 34syl6bi 242 . . . . . . . . 9 ((#‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → 𝐴 = 𝐵))
3635com12 32 . . . . . . . 8 (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((#‘𝐹) = 0 → 𝐴 = 𝐵))
37363adant1 1072 . . . . . . 7 ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((#‘𝐹) = 0 → 𝐴 = 𝐵))
3837adantr 480 . . . . . 6 (((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ∧ Fun 𝑃) → ((#‘𝐹) = 0 → 𝐴 = 𝐵))
3930, 38impbid 201 . . . . 5 (((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ∧ Fun 𝑃) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))
4039adantrl 748 . . . 4 (((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ∧ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃)) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))
415, 40syl6bi 242 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → ((𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
4241imp 444 . 2 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃)) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))
431, 42syl 17 1 (𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕ0cn0 11169  ℤ≥cuz 11563  ...cfz 12197  #chash 12979  Word cword 13146   Walks cwalk 26026   Trails ctrail 26027   SPaths cspath 26029   WalkOn cwlkon 26030   SPathOn cspthon 26033 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-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-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-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator