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Theorem elwspths2spth 29996
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.) (Proof shortened by AV, 16-Mar-2022.)
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 elwwlks2.v . . . 4 𝑉 = (Vtx‘𝐺)
21wspthsnwspthsnon 29945 . . 3 (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))
32a1i 11 . 2 (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)))
41elwspths2on 29989 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉𝑐𝑉) → (𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
543expb 1119 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑎𝑉𝑐𝑉)) → (𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
652rexbidva 3217 . 2 (𝐺 ∈ UPGraph → (∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑎𝑉𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
7 rexcom 3287 . . . 4 (∃𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)))
8 wspthnon 29887 . . . . . . 7 (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩))
9 ancom 460 . . . . . . . . 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)))
10 19.41v 1946 . . . . . . . . 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)))
119, 10bitr4i 278 . . . . . . . 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)))
12 simpr 484 . . . . . . . . . . . . . 14 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → 𝑎𝑉)
13 simpr 484 . . . . . . . . . . . . . 14 ((𝑏𝑉𝑐𝑉) → 𝑐𝑉)
1412, 13anim12i 613 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑎𝑉𝑐𝑉))
15 vex 3481 . . . . . . . . . . . . . 14 𝑓 ∈ V
16 s3cli 14916 . . . . . . . . . . . . . 14 ⟨“𝑎𝑏𝑐”⟩ ∈ Word V
1715, 16pm3.2i 470 . . . . . . . . . . . . 13 (𝑓 ∈ V ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
181isspthonpth 29781 . . . . . . . . . . . . 13 (((𝑎𝑉𝑐𝑉) ∧ (𝑓 ∈ V ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)) → (𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)))
1914, 17, 18sylancl 586 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)))
20 wwlknon 29886 . . . . . . . . . . . . 13 (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))
21 2nn0 12540 . . . . . . . . . . . . . . 15 2 ∈ ℕ0
22 iswwlksn 29867 . . . . . . . . . . . . . . 15 (2 ∈ ℕ0 → (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1))))
2321, 22mp1i 13 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1))))
24233anbi1d 1439 . . . . . . . . . . . . 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)))
2520, 24bitrid 283 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
2619, 25anbi12d 632 . . . . . . . . . . 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))))
2726adantr 480 . . . . . . . . . 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))))
2816a1i 11 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
29 simprl1 1217 . . . . . . . . . . . . . . . 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”⟩)
30 spthiswlk 29760 . . . . . . . . . . . . . . . . . . . 20 (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ → 𝑓(Walks‘𝐺)⟨“𝑎𝑏𝑐”⟩)
31 wlklenvm1 29654 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)⟨“𝑎𝑏𝑐”⟩ → (♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1))
32 simpl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1))
33 oveq1 7437 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = ((2 + 1) − 1))
34 2cn 12338 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 ∈ ℂ
35 pncan1 11684 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (2 ∈ ℂ → ((2 + 1) − 1) = 2)
3634, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((2 + 1) − 1) = 2
3733, 36eqtrdi 2790 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
3837adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
39383ad2ant1 1132 . . . . . . . . . . . . . . . . . . . . . . 23 (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
4039adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
4132, 40eqtrd 2774 . . . . . . . . . . . . . . . . . . . . 21 (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (♯‘𝑓) = 2)
4241ex 412 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) → (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → (♯‘𝑓) = 2))
4330, 31, 423syl 18 . . . . . . . . . . . . . . . . . . 19 (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ → (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → (♯‘𝑓) = 2))
44433ad2ant1 1132 . . . . . . . . . . . . . . . . . 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))
4544imp 406 . . . . . . . . . . . . . . . . 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)
4645adantl 481 . . . . . . . . . . . . . . . 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)
47 s3fv0 14926 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
4847elv 3482 . . . . . . . . . . . . . . . . . . 19 (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎
4948eqcomi 2743 . . . . . . . . . . . . . . . . . 18 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0)
50 s3fv1 14927 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏)
5150elv 3482 . . . . . . . . . . . . . . . . . . 19 (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏
5251eqcomi 2743 . . . . . . . . . . . . . . . . . 18 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1)
53 s3fv2 14928 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
5453elv 3482 . . . . . . . . . . . . . . . . . . 19 (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐
5554eqcomi 2743 . . . . . . . . . . . . . . . . . 18 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)
5649, 52, 553pm3.2i 1338 . . . . . . . . . . . . . . . . 17 (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
5756a1i 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)))
5829, 46, 573jca 1127 . . . . . . . . . . . . . . 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))))
59 breq2 5151 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑓(SPaths‘𝐺)𝑝𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
60 fveq1 6905 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘0) = (⟨“𝑎𝑏𝑐”⟩‘0))
6160eqeq2d 2745 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑎 = (𝑝‘0) ↔ 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0)))
62 fveq1 6905 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘1) = (⟨“𝑎𝑏𝑐”⟩‘1))
6362eqeq2d 2745 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑏 = (𝑝‘1) ↔ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1)))
64 fveq1 6905 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘2) = (⟨“𝑎𝑏𝑐”⟩‘2))
6564eqeq2d 2745 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑐 = (𝑝‘2) ↔ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))
6661, 63, 653anbi123d 1435 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) ↔ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))))
6759, 663anbi13d 1437 . . . . . . . . . . . . . . . 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)))))
6867ad2antlr 727 . . . . . . . . . . . . . . 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)))))
6958, 68mpbird 257 . . . . . . . . . . . . . 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))))
7069ex 412 . . . . . . . . . . . . 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)))))
7128, 70spcimedv 3594 . . . . . . . . . . . 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)))))
72 spthiswlk 29760 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(SPaths‘𝐺)𝑝𝑓(Walks‘𝐺)𝑝)
73 wlklenvp1 29650 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
74 oveq1 7437 . . . . . . . . . . . . . . . . . . . . . . . 24 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
75 2p1e3 12405 . . . . . . . . . . . . . . . . . . . . . . . 24 (2 + 1) = 3
7674, 75eqtrdi 2790 . . . . . . . . . . . . . . . . . . . . . . 23 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = 3)
7776eqeq2d 2745 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝑓) = 2 → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
7877biimpcd 249 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑝) = ((♯‘𝑓) + 1) → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3))
7972, 73, 783syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝑓(SPaths‘𝐺)𝑝 → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3))
8079imp 406 . . . . . . . . . . . . . . . . . . 19 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = 3)
81803adant3 1131 . . . . . . . . . . . . . . . . . 18 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (♯‘𝑝) = 3)
8281adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → (♯‘𝑝) = 3)
83 eqcom 2741 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = (𝑝‘0) ↔ (𝑝‘0) = 𝑎)
84 eqcom 2741 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (𝑝‘1) ↔ (𝑝‘1) = 𝑏)
85 eqcom 2741 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = (𝑝‘2) ↔ (𝑝‘2) = 𝑐)
8683, 84, 853anbi123i 1154 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) ↔ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
8786biimpi 216 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
88873ad2ant3 1134 . . . . . . . . . . . . . . . . . 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))
8988adantl 481 . . . . . . . . . . . . . . . . 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))
9082, 89jca 511 . . . . . . . . . . . . . . . 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)))
911wlkpwrd 29649 . . . . . . . . . . . . . . . . . . 19 (𝑓(Walks‘𝐺)𝑝𝑝 ∈ Word 𝑉)
9272, 91syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓(SPaths‘𝐺)𝑝𝑝 ∈ Word 𝑉)
93923ad2ant1 1132 . . . . . . . . . . . . . . . . 17 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → 𝑝 ∈ Word 𝑉)
9412anim1i 615 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑎𝑉 ∧ (𝑏𝑉𝑐𝑉)))
95 3anass 1094 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑉𝑏𝑉𝑐𝑉) ↔ (𝑎𝑉 ∧ (𝑏𝑉𝑐𝑉)))
9694, 95sylibr 234 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑎𝑉𝑏𝑉𝑐𝑉))
97 eqwrds3 14996 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ Word 𝑉 ∧ (𝑎𝑉𝑏𝑉𝑐𝑉)) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))))
9893, 96, 97syl2anr 597 . . . . . . . . . . . . . . . 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))))
9990, 98mpbird 257 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → 𝑝 = ⟨“𝑎𝑏𝑐”⟩)
10059biimpcd 249 . . . . . . . . . . . . . . . . . . . 20 (𝑓(SPaths‘𝐺)𝑝 → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
1011003ad2ant1 1132 . . . . . . . . . . . . . . . . . . 19 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
102101adantl 481 . . . . . . . . . . . . . . . . . 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”⟩))
103102imp 406 . . . . . . . . . . . . . . . . 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”⟩)
10448a1i 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)
105 fveq2 6906 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑓) = 2 → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = (⟨“𝑎𝑏𝑐”⟩‘2))
106105, 54eqtrdi 2790 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑓) = 2 → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)
1071063ad2ant2 1133 . . . . . . . . . . . . . . . . . 18 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)
108107ad2antlr 727 . . . . . . . . . . . . . . . . 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)
109103, 104, 1083jca 1127 . . . . . . . . . . . . . . . 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))
110 wlkiswwlks1 29896 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐺 ∈ UPGraph → (𝑓(Walks‘𝐺)𝑝𝑝 ∈ (WWalks‘𝐺)))
111110adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → (𝑓(Walks‘𝐺)𝑝𝑝 ∈ (WWalks‘𝐺)))
112111adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑓(Walks‘𝐺)𝑝𝑝 ∈ (WWalks‘𝐺)))
11372, 112syl5com 31 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓(SPaths‘𝐺)𝑝 → (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → 𝑝 ∈ (WWalks‘𝐺)))
1141133ad2ant1 1132 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → 𝑝 ∈ (WWalks‘𝐺)))
115114impcom 407 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → 𝑝 ∈ (WWalks‘𝐺))
116115adantr 480 . . . . . . . . . . . . . . . . . . 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))
117 eleq1 2826 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝 ∈ (WWalks‘𝐺) ↔ ⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺)))
118117bicomd 223 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ↔ 𝑝 ∈ (WWalks‘𝐺)))
119118adantl 481 . . . . . . . . . . . . . . . . . . 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)))
120116, 119mpbird 257 . . . . . . . . . . . . . . . . . 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))
121 s3len 14929 . . . . . . . . . . . . . . . . . . 19 (♯‘⟨“𝑎𝑏𝑐”⟩) = 3
122 df-3 12327 . . . . . . . . . . . . . . . . . . 19 3 = (2 + 1)
123121, 122eqtri 2762 . . . . . . . . . . . . . . . . . 18 (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)
124120, 123jctir 520 . . . . . . . . . . . . . . . . 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)))
12554a1i 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)
126124, 104, 1253jca 1127 . . . . . . . . . . . . . . . 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))
127109, 126jca 511 . . . . . . . . . . . . . . 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)))
12899, 127mpdan 687 . . . . . . . . . . . . . 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)))
129128ex 412 . . . . . . . . . . . . 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))))
130129exlimdv 1930 . . . . . . . . . . . 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))))
13171, 130impbid 212 . . . . . . . . . . 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)))))
132131adantr 480 . . . . . . . . . 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)))))
13327, 132bitrd 279 . . . . . . . . 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)))))
134133exbidv 1918 . . . . . . . 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)))))
13511, 134bitrid 283 . . . . . . 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)))))
1368, 135bitrid 283 . . . . . 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)))))
137136pm5.32da 579 . . . . 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))))))
1381372rexbidva 3217 . . . 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))))))
1397, 138bitrid 283 . . 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))))))
140139rexbidva 3174 . 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))))))
1413, 6, 1403bitrd 305 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 206  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  wrex 3067  Vcvv 3477   class class class wbr 5147  cfv 6562  (class class class)co 7430  cc 11150  0cc0 11152  1c1 11153   + caddc 11155  cmin 11489  2c2 12318  3c3 12319  0cn0 12523  chash 14365  Word cword 14548  ⟨“cs3 14877  Vtxcvtx 29027  UPGraphcupgr 29111  Walkscwlks 29628  SPathscspths 29745  SPathsOncspthson 29747  WWalkscwwlks 29854   WWalksN cwwlksn 29855   WWalksNOn cwwlksnon 29856   WSPathsN cwwspthsn 29857   WSPathsNOn cwwspthsnon 29858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-ac2 10500  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-ac 10153  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-concat 14605  df-s1 14630  df-s2 14883  df-s3 14884  df-edg 29079  df-uhgr 29089  df-upgr 29113  df-wlks 29631  df-wlkson 29632  df-trls 29724  df-trlson 29725  df-pths 29748  df-spths 29749  df-spthson 29751  df-wwlks 29859  df-wwlksn 29860  df-wwlksnon 29861  df-wspthsn 29862  df-wspthsnon 29863
This theorem is referenced by: (None)
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