Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elwspths2spth Structured version   Visualization version   GIF version

Theorem elwspths2spth 26746
 Description: A simple path of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 28-Feb-2018.) (Revised by AV, 18-May-2021.)
Hypothesis
Ref Expression
elwwlks2.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
elwspths2spth (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
Distinct variable groups:   𝐺,𝑎,𝑏,𝑐,𝑓,𝑝   𝑉,𝑎,𝑏,𝑐,𝑓,𝑝   𝑊,𝑎,𝑏,𝑐,𝑓,𝑝

Proof of Theorem elwspths2spth
StepHypRef Expression
1 2nn0 11261 . . 3 2 ∈ ℕ0
2 elwwlks2.v . . . 4 𝑉 = (Vtx‘𝐺)
32wspthsnwspthsnon 26697 . . 3 ((2 ∈ ℕ0𝐺 ∈ UPGraph ) → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)))
41, 3mpan 705 . 2 (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)))
52elwspths2on 26738 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉𝑐𝑉) → (𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
653expb 1263 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑎𝑉𝑐𝑉)) → (𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
762rexbidva 3050 . 2 (𝐺 ∈ UPGraph → (∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑎𝑉𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
8 rexcom 3092 . . . 4 (∃𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)))
9 simpr 477 . . . . . . . . . 10 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → 𝑎𝑉)
10 simpr 477 . . . . . . . . . 10 ((𝑏𝑉𝑐𝑉) → 𝑐𝑉)
119, 10anim12i 589 . . . . . . . . 9 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑎𝑉𝑐𝑉))
122wspthnon 26629 . . . . . . . . 9 ((𝑎𝑉𝑐𝑉) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩)))
1311, 12syl 17 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩)))
1413adantr 481 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩)))
15 ancom 466 . . . . . . . . 9 ((⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩) ↔ (∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
16 19.41v 1911 . . . . . . . . 9 (∃𝑓(𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ (∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
1715, 16bitr4i 267 . . . . . . . 8 ((⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩) ↔ ∃𝑓(𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
18 vex 3192 . . . . . . . . . . . . . 14 𝑓 ∈ V
19 s3cli 13570 . . . . . . . . . . . . . 14 ⟨“𝑎𝑏𝑐”⟩ ∈ Word V
2018, 19pm3.2i 471 . . . . . . . . . . . . 13 (𝑓 ∈ V ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
212isspthonpth 26531 . . . . . . . . . . . . 13 (((𝑎𝑉𝑐𝑉) ∧ (𝑓 ∈ V ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)) → (𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐)))
2211, 20, 21sylancl 693 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐)))
232wwlknon 26628 . . . . . . . . . . . . . 14 ((𝑎𝑉𝑐𝑉) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
2411, 23syl 17 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
25 iswwlksn 26616 . . . . . . . . . . . . . . . 16 (2 ∈ ℕ0 → (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1))))
261, 25ax-mp 5 . . . . . . . . . . . . . . 15 (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)))
2726a1i 11 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1))))
28273anbi1d 1400 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ((⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) ↔ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
2924, 28bitrd 268 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
3022, 29anbi12d 746 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ((𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
3130adantr 481 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
3219a1i 11 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
33 simprl1 1104 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩)
34 spthiswlk 26510 . . . . . . . . . . . . . . . . . . . 20 (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ → 𝑓(Walks‘𝐺)⟨“𝑎𝑏𝑐”⟩)
35 wlklenvm1 26404 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)⟨“𝑎𝑏𝑐”⟩ → (#‘𝑓) = ((#‘⟨“𝑎𝑏𝑐”⟩) − 1))
36 simpl 473 . . . . . . . . . . . . . . . . . . . . . 22 (((#‘𝑓) = ((#‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (#‘𝑓) = ((#‘⟨“𝑎𝑏𝑐”⟩) − 1))
37 oveq1 6617 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1) → ((#‘⟨“𝑎𝑏𝑐”⟩) − 1) = ((2 + 1) − 1))
38 2cn 11043 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 ∈ ℂ
39 pncan1 10406 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (2 ∈ ℂ → ((2 + 1) − 1) = 2)
4038, 39ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((2 + 1) − 1) = 2
4137, 40syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1) → ((#‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
4241adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) → ((#‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
43423ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . . . 23 (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → ((#‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
4443adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((#‘𝑓) = ((#‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → ((#‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
4536, 44eqtrd 2655 . . . . . . . . . . . . . . . . . . . . 21 (((#‘𝑓) = ((#‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (#‘𝑓) = 2)
4645ex 450 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑓) = ((#‘⟨“𝑎𝑏𝑐”⟩) − 1) → (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → (#‘𝑓) = 2))
4734, 35, 463syl 18 . . . . . . . . . . . . . . . . . . 19 (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ → (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → (#‘𝑓) = 2))
48473ad2ant1 1080 . . . . . . . . . . . . . . . . . 18 ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) → (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → (#‘𝑓) = 2))
4948imp 445 . . . . . . . . . . . . . . . . 17 (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (#‘𝑓) = 2)
5049adantl 482 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (#‘𝑓) = 2)
51 vex 3192 . . . . . . . . . . . . . . . . . . . 20 𝑎 ∈ V
52 s3fv0 13580 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
5351, 52ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎
5453eqcomi 2630 . . . . . . . . . . . . . . . . . 18 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0)
55 vex 3192 . . . . . . . . . . . . . . . . . . . 20 𝑏 ∈ V
56 s3fv1 13581 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏)
5755, 56ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏
5857eqcomi 2630 . . . . . . . . . . . . . . . . . 18 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1)
59 vex 3192 . . . . . . . . . . . . . . . . . . . 20 𝑐 ∈ V
60 s3fv2 13582 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
6159, 60ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐
6261eqcomi 2630 . . . . . . . . . . . . . . . . . 18 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)
6354, 58, 623pm3.2i 1237 . . . . . . . . . . . . . . . . 17 (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
6463a1i 11 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))
6533, 50, 643jca 1240 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (#‘𝑓) = 2 ∧ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))))
66 breq2 4622 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑓(SPaths‘𝐺)𝑝𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
67 fveq1 6152 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘0) = (⟨“𝑎𝑏𝑐”⟩‘0))
6867eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑎 = (𝑝‘0) ↔ 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0)))
69 fveq1 6152 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘1) = (⟨“𝑎𝑏𝑐”⟩‘1))
7069eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑏 = (𝑝‘1) ↔ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1)))
71 fveq1 6152 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘2) = (⟨“𝑎𝑏𝑐”⟩‘2))
7271eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑐 = (𝑝‘2) ↔ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))
7368, 70, 723anbi123d 1396 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) ↔ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))))
7466, 733anbi13d 1398 . . . . . . . . . . . . . . . 16 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → ((𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (#‘𝑓) = 2 ∧ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))))
7574ad2antlr 762 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → ((𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (#‘𝑓) = 2 ∧ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))))
7665, 75mpbird 247 . . . . . . . . . . . . . 14 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))
7776ex 450 . . . . . . . . . . . . 13 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
7832, 77spcimedv 3281 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
79 spthiswlk 26510 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(SPaths‘𝐺)𝑝𝑓(Walks‘𝐺)𝑝)
80 wlklenvp1 26401 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(Walks‘𝐺)𝑝 → (#‘𝑝) = ((#‘𝑓) + 1))
81 oveq1 6617 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝑓) = 2 → ((#‘𝑓) + 1) = (2 + 1))
82 2p1e3 11103 . . . . . . . . . . . . . . . . . . . . . . . 24 (2 + 1) = 3
8381, 82syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑓) = 2 → ((#‘𝑓) + 1) = 3)
8483eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑓) = 2 → ((#‘𝑝) = ((#‘𝑓) + 1) ↔ (#‘𝑝) = 3))
8584biimpcd 239 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑝) = ((#‘𝑓) + 1) → ((#‘𝑓) = 2 → (#‘𝑝) = 3))
8679, 80, 853syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝑓(SPaths‘𝐺)𝑝 → ((#‘𝑓) = 2 → (#‘𝑝) = 3))
8786imp 445 . . . . . . . . . . . . . . . . . . 19 ((𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2) → (#‘𝑝) = 3)
88873adant3 1079 . . . . . . . . . . . . . . . . . 18 ((𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (#‘𝑝) = 3)
8988adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → (#‘𝑝) = 3)
90 eqcom 2628 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = (𝑝‘0) ↔ (𝑝‘0) = 𝑎)
91 eqcom 2628 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (𝑝‘1) ↔ (𝑝‘1) = 𝑏)
92 eqcom 2628 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = (𝑝‘2) ↔ (𝑝‘2) = 𝑐)
9390, 91, 923anbi123i 1249 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) ↔ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
9493biimpi 206 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
95943ad2ant3 1082 . . . . . . . . . . . . . . . . . 18 ((𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
9695adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
9789, 96jca 554 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → ((#‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐)))
982wlkpwrd 26400 . . . . . . . . . . . . . . . . . . 19 (𝑓(Walks‘𝐺)𝑝𝑝 ∈ Word 𝑉)
9979, 98syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓(SPaths‘𝐺)𝑝𝑝 ∈ Word 𝑉)
100993ad2ant1 1080 . . . . . . . . . . . . . . . . 17 ((𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → 𝑝 ∈ Word 𝑉)
1019anim1i 591 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑎𝑉 ∧ (𝑏𝑉𝑐𝑉)))
102 3anass 1040 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑉𝑏𝑉𝑐𝑉) ↔ (𝑎𝑉 ∧ (𝑏𝑉𝑐𝑉)))
103101, 102sylibr 224 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑎𝑉𝑏𝑉𝑐𝑉))
104 eqwrds3 13646 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ Word 𝑉 ∧ (𝑎𝑉𝑏𝑉𝑐𝑉)) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((#‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))))
105100, 103, 104syl2anr 495 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((#‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))))
10697, 105mpbird 247 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → 𝑝 = ⟨“𝑎𝑏𝑐”⟩)
10766biimpcd 239 . . . . . . . . . . . . . . . . . . . 20 (𝑓(SPaths‘𝐺)𝑝 → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
1081073ad2ant1 1080 . . . . . . . . . . . . . . . . . . 19 ((𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
109108adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
110109imp 445 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩)
11153a1i 11 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
112 fveq2 6153 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑓) = 2 → (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = (⟨“𝑎𝑏𝑐”⟩‘2))
113112, 61syl6eq 2671 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑓) = 2 → (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐)
1141133ad2ant2 1081 . . . . . . . . . . . . . . . . . 18 ((𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐)
115114ad2antlr 762 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐)
116110, 111, 1153jca 1240 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐))
117 wlkiswwlks1 26639 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐺 ∈ UPGraph → (𝑓(Walks‘𝐺)𝑝𝑝 ∈ (WWalks‘𝐺)))
118117adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → (𝑓(Walks‘𝐺)𝑝𝑝 ∈ (WWalks‘𝐺)))
119118adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑓(Walks‘𝐺)𝑝𝑝 ∈ (WWalks‘𝐺)))
12079, 119syl5com 31 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓(SPaths‘𝐺)𝑝 → (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → 𝑝 ∈ (WWalks‘𝐺)))
1211203ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → 𝑝 ∈ (WWalks‘𝐺)))
122121impcom 446 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → 𝑝 ∈ (WWalks‘𝐺))
123122adantr 481 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑝 ∈ (WWalks‘𝐺))
124 eleq1 2686 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝 ∈ (WWalks‘𝐺) ↔ ⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺)))
125124bicomd 213 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ↔ 𝑝 ∈ (WWalks‘𝐺)))
126125adantl 482 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ↔ 𝑝 ∈ (WWalks‘𝐺)))
127123, 126mpbird 247 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → ⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺))
128 s3len 13583 . . . . . . . . . . . . . . . . . . 19 (#‘⟨“𝑎𝑏𝑐”⟩) = 3
129 df-3 11032 . . . . . . . . . . . . . . . . . . 19 3 = (2 + 1)
130128, 129eqtri 2643 . . . . . . . . . . . . . . . . . 18 (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)
131127, 130jctir 560 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)))
13261a1i 11 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
133131, 111, 1323jca 1240 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))
134116, 133jca 554 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
135106, 134mpdan 701 . . . . . . . . . . . . . 14 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
136135ex 450 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ((𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
137136exlimdv 1858 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
13878, 137impbid 202 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) ↔ ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
139138adantr 481 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(#‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) ↔ ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
14031, 139bitrd 268 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
141140exbidv 1847 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (∃𝑓(𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
14217, 141syl5bb 272 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩) ↔ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
14314, 142bitrd 268 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
144143pm5.32da 672 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ((𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
1451442rexbidva 3050 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → (∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
1468, 145syl5bb 272 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → (∃𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
147146rexbidva 3043 . 2 (𝐺 ∈ UPGraph → (∃𝑎𝑉𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
1484, 7, 1473bitrd 294 1 (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∃wrex 2908  Vcvv 3189   class class class wbr 4618  ‘cfv 5852  (class class class)co 6610  ℂcc 9886  0cc0 9888  1c1 9889   + caddc 9891   − cmin 10218  2c2 11022  3c3 11023  ℕ0cn0 11244  #chash 13065  Word cword 13238  ⟨“cs3 13532  Vtxcvtx 25791   UPGraph cupgr 25888  Walkscwlks 26379  SPathscspths 26495  SPathsOncspthson 26497  WWalkscwwlks 26603   WWalksN cwwlksn 26604   WWalksNOn cwwlksnon 26605   WSPathsN cwwspthsn 26606   WSPathsNOn cwwspthsnon 26607 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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-ac2 9237  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965 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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  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-se 5039  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 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-ac 8891  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-xnn0 11316  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-hash 13066  df-word 13246  df-concat 13248  df-s1 13249  df-s2 13538  df-s3 13539  df-edg 25857  df-uhgr 25866  df-upgr 25890  df-wlks 26382  df-wlkson 26383  df-trls 26475  df-trlson 26476  df-pths 26498  df-spths 26499  df-spthson 26501  df-wwlks 26608  df-wwlksn 26609  df-wwlksnon 26610  df-wspthsn 26611  df-wspthsnon 26612 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator