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Theorem spthonepeq 26511
Description: The endpoints of a simple path between two vertices are equal iff the path is of length 0. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 18-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.)
Assertion
Ref Expression
spthonepeq (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))

Proof of Theorem spthonepeq
StepHypRef Expression
1 eqid 2626 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
21spthonprop 26504 . 2 (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(SPaths‘𝐺)𝑃)))
31istrlson 26466 . . . . . 6 (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)))
433adantl1 1215 . . . . 5 (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)))
5 isspth 26483 . . . . . 6 (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃))
65a1i 11 . . . . 5 (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃)))
74, 6anbi12d 746 . . . 4 (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(SPaths‘𝐺)𝑃) ↔ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃) ∧ (𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃))))
81wlkonprop 26417 . . . . . . . 8 (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
9 wlkcl 26375 . . . . . . . . . . . . 13 (𝐹(Walks‘𝐺)𝑃 → (#‘𝐹) ∈ ℕ0)
101wlkp 26376 . . . . . . . . . . . . 13 (𝐹(Walks‘𝐺)𝑃𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺))
11 df-f1 5855 . . . . . . . . . . . . . . . 16 (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) ↔ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun 𝑃))
12 eqeq2 2637 . . . . . . . . . . . . . . . . . 18 (𝐴 = 𝐵 → ((𝑃‘0) = 𝐴 ↔ (𝑃‘0) = 𝐵))
13 eqtr3 2647 . . . . . . . . . . . . . . . . . . . 20 (((𝑃‘(#‘𝐹)) = 𝐵 ∧ (𝑃‘0) = 𝐵) → (𝑃‘(#‘𝐹)) = (𝑃‘0))
14 elnn0uz 11669 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘𝐹) ∈ ℕ0 ↔ (#‘𝐹) ∈ (ℤ‘0))
15 eluzfz2 12288 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘𝐹) ∈ (ℤ‘0) → (#‘𝐹) ∈ (0...(#‘𝐹)))
1614, 15sylbi 207 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ (0...(#‘𝐹)))
17 0elfz 12374 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝐹) ∈ ℕ0 → 0 ∈ (0...(#‘𝐹)))
1816, 17jca 554 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝐹) ∈ ℕ0 → ((#‘𝐹) ∈ (0...(#‘𝐹)) ∧ 0 ∈ (0...(#‘𝐹))))
19 f1veqaeq 6469 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) ∧ ((#‘𝐹) ∈ (0...(#‘𝐹)) ∧ 0 ∈ (0...(#‘𝐹)))) → ((𝑃‘(#‘𝐹)) = (𝑃‘0) → (#‘𝐹) = 0))
2018, 19sylan2 491 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) ∧ (#‘𝐹) ∈ ℕ0) → ((𝑃‘(#‘𝐹)) = (𝑃‘0) → (#‘𝐹) = 0))
2120ex 450 . . . . . . . . . . . . . . . . . . . . 21 (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → ((#‘𝐹) ∈ ℕ0 → ((𝑃‘(#‘𝐹)) = (𝑃‘0) → (#‘𝐹) = 0)))
2221com13 88 . . . . . . . . . . . . . . . . . . . 20 ((𝑃‘(#‘𝐹)) = (𝑃‘0) → ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → (#‘𝐹) = 0)))
2313, 22syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑃‘(#‘𝐹)) = 𝐵 ∧ (𝑃‘0) = 𝐵) → ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → (#‘𝐹) = 0)))
2423expcom 451 . . . . . . . . . . . . . . . . . 18 ((𝑃‘0) = 𝐵 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → (#‘𝐹) = 0))))
2512, 24syl6bi 243 . . . . . . . . . . . . . . . . 17 (𝐴 = 𝐵 → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → (#‘𝐹) = 0)))))
2625com15 101 . . . . . . . . . . . . . . . 16 (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ0 → (𝐴 = 𝐵 → (#‘𝐹) = 0)))))
2711, 26sylbir 225 . . . . . . . . . . . . . . 15 ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun 𝑃) → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ0 → (𝐴 = 𝐵 → (#‘𝐹) = 0)))))
2827expcom 451 . . . . . . . . . . . . . 14 (Fun 𝑃 → (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ0 → (𝐴 = 𝐵 → (#‘𝐹) = 0))))))
2928com15 101 . . . . . . . . . . . . 13 ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → (Fun 𝑃 → (𝐴 = 𝐵 → (#‘𝐹) = 0))))))
309, 10, 29sylc 65 . . . . . . . . . . . 12 (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → (Fun 𝑃 → (𝐴 = 𝐵 → (#‘𝐹) = 0)))))
31303imp1 1277 . . . . . . . . . . 11 (((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ∧ Fun 𝑃) → (𝐴 = 𝐵 → (#‘𝐹) = 0))
32 fveq2 6150 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) = 0 → (𝑃‘(#‘𝐹)) = (𝑃‘0))
3332eqeq1d 2628 . . . . . . . . . . . . . . . 16 ((#‘𝐹) = 0 → ((𝑃‘(#‘𝐹)) = 𝐵 ↔ (𝑃‘0) = 𝐵))
3433anbi2d 739 . . . . . . . . . . . . . . 15 ((#‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵)))
35 eqtr2 2646 . . . . . . . . . . . . . . 15 (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → 𝐴 = 𝐵)
3634, 35syl6bi 243 . . . . . . . . . . . . . 14 ((#‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → 𝐴 = 𝐵))
3736com12 32 . . . . . . . . . . . . 13 (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((#‘𝐹) = 0 → 𝐴 = 𝐵))
38373adant1 1077 . . . . . . . . . . . 12 ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((#‘𝐹) = 0 → 𝐴 = 𝐵))
3938adantr 481 . . . . . . . . . . 11 (((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ∧ Fun 𝑃) → ((#‘𝐹) = 0 → 𝐴 = 𝐵))
4031, 39impbid 202 . . . . . . . . . 10 (((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ∧ Fun 𝑃) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))
4140ex 450 . . . . . . . . 9 ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (Fun 𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
42413ad2ant3 1082 . . . . . . . 8 (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → (Fun 𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
438, 42syl 17 . . . . . . 7 (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → (Fun 𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
4443adantld 483 . . . . . 6 (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
4544adantr 481 . . . . 5 ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃) → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
4645imp 445 . . . 4 (((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃) ∧ (𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃)) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))
477, 46syl6bi 243 . . 3 (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(SPaths‘𝐺)𝑃) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
48473impia 1258 . 2 (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(SPaths‘𝐺)𝑃)) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))
492, 48syl 17 1 (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  Vcvv 3191   class class class wbr 4618  ccnv 5078  Fun wfun 5844  wf 5846  1-1wf1 5847  cfv 5850  (class class class)co 6605  0cc0 9881  0cn0 11237  cuz 11631  ...cfz 12265  #chash 13054  Vtxcvtx 25769  Walkscwlks 26356  WalksOncwlkson 26357  Trailsctrls 26450  TrailsOnctrlson 26451  SPathscspths 26472  SPathsOncspthson 26474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-hash 13055  df-word 13233  df-wlks 26359  df-wlkson 26360  df-trls 26452  df-trlson 26453  df-pths 26475  df-spths 26476  df-spthson 26478
This theorem is referenced by:  wspthsnonn0vne  26676
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