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Theorem usg2spthonot0 26416
 Description: A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
Assertion
Ref Expression
usg2spthonot0 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))

Proof of Theorem usg2spthonot0
StepHypRef Expression
1 ne0i 3880 . . . . 5 (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ≠ ∅)
2 2spontn0vne 26414 . . . . 5 ((𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ≠ ∅ → 𝐴𝐶)
31, 2syl 17 . . . 4 (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → 𝐴𝐶)
4 simpl 472 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝑉 USGrph 𝐸)
54adantl 481 . . . . . . . . . 10 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → 𝑉 USGrph 𝐸)
6 3simpb 1052 . . . . . . . . . . . 12 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐴𝑉𝐶𝑉))
76adantl 481 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴𝑉𝐶𝑉))
87adantl 481 . . . . . . . . . 10 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (𝐴𝑉𝐶𝑉))
9 simpl 472 . . . . . . . . . 10 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → 𝐴𝐶)
10 2pthwlkonot 26412 . . . . . . . . . 10 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) = (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))
115, 8, 9, 10syl3anc 1318 . . . . . . . . 9 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) = (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))
1211eleq2d 2673 . . . . . . . 8 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)))
13 usgrav 25867 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
1413, 6anim12i 588 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐶𝑉)))
1514adantl 481 . . . . . . . . . 10 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐶𝑉)))
16 el2wlkonotot1 26401 . . . . . . . . . 10 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐶𝑉)) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ (𝑆 = 𝐴𝑇 = 𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇))))
1715, 16syl 17 . . . . . . . . 9 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ (𝑆 = 𝐴𝑇 = 𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇))))
18 df-3an 1033 . . . . . . . . 9 ((𝑆 = 𝐴𝑇 = 𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) ↔ ((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)))
1917, 18syl6bb 275 . . . . . . . 8 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇))))
2012, 19bitrd 267 . . . . . . 7 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇))))
21 simpll 786 . . . . . . . . . . . . 13 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ 𝐴𝐶) → 𝑆 = 𝐴)
22 simpr 476 . . . . . . . . . . . . . 14 ((𝑆 = 𝐴𝑇 = 𝐶) → 𝑇 = 𝐶)
2322adantr 480 . . . . . . . . . . . . 13 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ 𝐴𝐶) → 𝑇 = 𝐶)
24 simpr 476 . . . . . . . . . . . . 13 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ 𝐴𝐶) → 𝐴𝐶)
2521, 23, 243jca 1235 . . . . . . . . . . . 12 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ 𝐴𝐶) → (𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶))
2625ex 449 . . . . . . . . . . 11 ((𝑆 = 𝐴𝑇 = 𝐶) → (𝐴𝐶 → (𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶)))
2726adantr 480 . . . . . . . . . 10 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) → (𝐴𝐶 → (𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶)))
2827com12 32 . . . . . . . . 9 (𝐴𝐶 → (((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) → (𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶)))
2928adantr 480 . . . . . . . 8 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) → (𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶)))
305adantl 481 . . . . . . . . . . . 12 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ (𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))) → 𝑉 USGrph 𝐸)
31 simprrr 801 . . . . . . . . . . . . 13 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ (𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))) → (𝐴𝑉𝐵𝑉𝐶𝑉))
32 eleq1 2676 . . . . . . . . . . . . . . . 16 (𝑆 = 𝐴 → (𝑆𝑉𝐴𝑉))
3332adantr 480 . . . . . . . . . . . . . . 15 ((𝑆 = 𝐴𝑇 = 𝐶) → (𝑆𝑉𝐴𝑉))
34 eleq1 2676 . . . . . . . . . . . . . . . 16 (𝑇 = 𝐶 → (𝑇𝑉𝐶𝑉))
3534adantl 481 . . . . . . . . . . . . . . 15 ((𝑆 = 𝐴𝑇 = 𝐶) → (𝑇𝑉𝐶𝑉))
3633, 353anbi13d 1393 . . . . . . . . . . . . . 14 ((𝑆 = 𝐴𝑇 = 𝐶) → ((𝑆𝑉𝐵𝑉𝑇𝑉) ↔ (𝐴𝑉𝐵𝑉𝐶𝑉)))
3736adantr 480 . . . . . . . . . . . . 13 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ (𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))) → ((𝑆𝑉𝐵𝑉𝑇𝑉) ↔ (𝐴𝑉𝐵𝑉𝐶𝑉)))
3831, 37mpbird 246 . . . . . . . . . . . 12 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ (𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))) → (𝑆𝑉𝐵𝑉𝑇𝑉))
39 usg2wlkonot 26410 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸 ∧ (𝑆𝑉𝐵𝑉𝑇𝑉)) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇) ↔ ({𝑆, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝑇} ∈ ran 𝐸)))
4030, 38, 39syl2anc 691 . . . . . . . . . . 11 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ (𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇) ↔ ({𝑆, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝑇} ∈ ran 𝐸)))
41 preq1 4212 . . . . . . . . . . . . . . . 16 (𝑆 = 𝐴 → {𝑆, 𝐵} = {𝐴, 𝐵})
4241adantr 480 . . . . . . . . . . . . . . 15 ((𝑆 = 𝐴𝑇 = 𝐶) → {𝑆, 𝐵} = {𝐴, 𝐵})
4342eleq1d 2672 . . . . . . . . . . . . . 14 ((𝑆 = 𝐴𝑇 = 𝐶) → ({𝑆, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
44 preq2 4213 . . . . . . . . . . . . . . . 16 (𝑇 = 𝐶 → {𝐵, 𝑇} = {𝐵, 𝐶})
4544adantl 481 . . . . . . . . . . . . . . 15 ((𝑆 = 𝐴𝑇 = 𝐶) → {𝐵, 𝑇} = {𝐵, 𝐶})
4645eleq1d 2672 . . . . . . . . . . . . . 14 ((𝑆 = 𝐴𝑇 = 𝐶) → ({𝐵, 𝑇} ∈ ran 𝐸 ↔ {𝐵, 𝐶} ∈ ran 𝐸))
4743, 46anbi12d 743 . . . . . . . . . . . . 13 ((𝑆 = 𝐴𝑇 = 𝐶) → (({𝑆, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝑇} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
4847biimpd 218 . . . . . . . . . . . 12 ((𝑆 = 𝐴𝑇 = 𝐶) → (({𝑆, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝑇} ∈ ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
4948adantr 480 . . . . . . . . . . 11 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ (𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))) → (({𝑆, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝑇} ∈ ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
5040, 49sylbid 229 . . . . . . . . . 10 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ (𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
5150impancom 455 . . . . . . . . 9 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) → ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
5251com12 32 . . . . . . . 8 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
5329, 52jcad 554 . . . . . . 7 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) → ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
5420, 53sylbid 229 . . . . . 6 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
5554ex 449 . . . . 5 (𝐴𝐶 → ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
5655com23 84 . . . 4 (𝐴𝐶 → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
573, 56mpcom 37 . . 3 (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
5857com12 32 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
59 simpll 786 . . . . . . . 8 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶)) → 𝑉 USGrph 𝐸)
60 eleq1 2676 . . . . . . . . . . . . . . 15 (𝐴 = 𝑆 → (𝐴𝑉𝑆𝑉))
6160eqcoms 2618 . . . . . . . . . . . . . 14 (𝑆 = 𝐴 → (𝐴𝑉𝑆𝑉))
6261adantr 480 . . . . . . . . . . . . 13 ((𝑆 = 𝐴𝑇 = 𝐶) → (𝐴𝑉𝑆𝑉))
63 eleq1 2676 . . . . . . . . . . . . . . 15 (𝐶 = 𝑇 → (𝐶𝑉𝑇𝑉))
6463eqcoms 2618 . . . . . . . . . . . . . 14 (𝑇 = 𝐶 → (𝐶𝑉𝑇𝑉))
6564adantl 481 . . . . . . . . . . . . 13 ((𝑆 = 𝐴𝑇 = 𝐶) → (𝐶𝑉𝑇𝑉))
6662, 653anbi13d 1393 . . . . . . . . . . . 12 ((𝑆 = 𝐴𝑇 = 𝐶) → ((𝐴𝑉𝐵𝑉𝐶𝑉) ↔ (𝑆𝑉𝐵𝑉𝑇𝑉)))
6766biimpd 218 . . . . . . . . . . 11 ((𝑆 = 𝐴𝑇 = 𝐶) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝑆𝑉𝐵𝑉𝑇𝑉)))
6867adantld 482 . . . . . . . . . 10 ((𝑆 = 𝐴𝑇 = 𝐶) → ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝑆𝑉𝐵𝑉𝑇𝑉)))
69683adant3 1074 . . . . . . . . 9 ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) → ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝑆𝑉𝐵𝑉𝑇𝑉)))
7069impcom 445 . . . . . . . 8 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶)) → (𝑆𝑉𝐵𝑉𝑇𝑉))
7159, 70, 39syl2anc 691 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶)) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇) ↔ ({𝑆, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝑇} ∈ ran 𝐸)))
72473adant3 1074 . . . . . . . 8 ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) → (({𝑆, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝑇} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
7372adantl 481 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶)) → (({𝑆, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝑇} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
7471, 73bitr2d 268 . . . . . 6 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)))
7574pm5.32da 671 . . . . 5 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇))))
76 df-3an 1033 . . . . . . . 8 ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ↔ ((𝑆 = 𝐴𝑇 = 𝐶) ∧ 𝐴𝐶))
77 ancom 465 . . . . . . . 8 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ 𝐴𝐶) ↔ (𝐴𝐶 ∧ (𝑆 = 𝐴𝑇 = 𝐶)))
7876, 77bitri 263 . . . . . . 7 ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ↔ (𝐴𝐶 ∧ (𝑆 = 𝐴𝑇 = 𝐶)))
7978anbi1i 727 . . . . . 6 (((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) ↔ ((𝐴𝐶 ∧ (𝑆 = 𝐴𝑇 = 𝐶)) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)))
80 anass 679 . . . . . 6 (((𝐴𝐶 ∧ (𝑆 = 𝐴𝑇 = 𝐶)) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) ↔ (𝐴𝐶 ∧ ((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇))))
8118bicomi 213 . . . . . . 7 (((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) ↔ (𝑆 = 𝐴𝑇 = 𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)))
8281anbi2i 726 . . . . . 6 ((𝐴𝐶 ∧ ((𝑆 = 𝐴𝑇 = 𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇))) ↔ (𝐴𝐶 ∧ (𝑆 = 𝐴𝑇 = 𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇))))
8379, 80, 823bitri 285 . . . . 5 (((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) ↔ (𝐴𝐶 ∧ (𝑆 = 𝐴𝑇 = 𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇))))
8475, 83syl6bb 275 . . . 4 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (𝐴𝐶 ∧ (𝑆 = 𝐴𝑇 = 𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)))))
8514, 16syl 17 . . . . . 6 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ (𝑆 = 𝐴𝑇 = 𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇))))
8685bicomd 212 . . . . 5 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝑆 = 𝐴𝑇 = 𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇)) ↔ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)))
8786anbi2d 736 . . . 4 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴𝐶 ∧ (𝑆 = 𝐴𝑇 = 𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝑆(𝑉 2WalksOnOt 𝐸)𝑇))) ↔ (𝐴𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))))
8884, 87bitrd 267 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (𝐴𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))))
89 simpll 786 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → 𝑉 USGrph 𝐸)
907adantr 480 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → (𝐴𝑉𝐶𝑉))
91 simpr 476 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → 𝐴𝐶)
9210eqcomd 2616 . . . . . . 7 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) = (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))
9389, 90, 91, 92syl3anc 1318 . . . . . 6 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) = (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))
9493eleq2d 2673 . . . . 5 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)))
9594biimpd 218 . . . 4 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) → ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)))
9695expimpd 627 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴𝐶 ∧ ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)) → ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)))
9788, 96sylbid 229 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)))
9858, 97impbid 201 1 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  {cpr 4127  ⟨cotp 4133   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   2WalksOnOt c2wlkonot 26382   2SPathOnOt c2pthonot 26384 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-rmo 2904  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-ot 4134  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-cda 8873  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-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-2wlkonot 26385  df-2spthonot 26387 This theorem is referenced by:  usg2spthonot1  26417
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