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Theorem elwwlks2 29220
Description: A walk of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 21-Feb-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 14-Mar-2022.)
Hypothesis
Ref Expression
elwwlks2.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
elwwlks2 (𝐺 ∈ UPGraph β†’ (π‘Š ∈ (2 WWalksN 𝐺) ↔ βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))))))
Distinct variable groups:   𝐺,π‘Ž,𝑏,𝑐,𝑓,𝑝   𝑉,π‘Ž,𝑏,𝑐,𝑓,𝑝   π‘Š,π‘Ž,𝑏,𝑐,𝑓,𝑝

Proof of Theorem elwwlks2
StepHypRef Expression
1 elwwlks2.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
21wwlksnwwlksnon 29169 . . 3 (π‘Š ∈ (2 WWalksN 𝐺) ↔ βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 π‘Š ∈ (π‘Ž(2 WWalksNOn 𝐺)𝑐))
32a1i 11 . 2 (𝐺 ∈ UPGraph β†’ (π‘Š ∈ (2 WWalksN 𝐺) ↔ βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 π‘Š ∈ (π‘Ž(2 WWalksNOn 𝐺)𝑐)))
41elwwlks2on 29213 . . . 4 ((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) β†’ (π‘Š ∈ (π‘Ž(2 WWalksNOn 𝐺)𝑐) ↔ βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2))))
543expb 1121 . . 3 ((𝐺 ∈ UPGraph ∧ (π‘Ž ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) β†’ (π‘Š ∈ (π‘Ž(2 WWalksNOn 𝐺)𝑐) ↔ βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2))))
652rexbidva 3218 . 2 (𝐺 ∈ UPGraph β†’ (βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 π‘Š ∈ (π‘Ž(2 WWalksNOn 𝐺)𝑐) ↔ βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2))))
7 rexcom 3288 . . . 4 (βˆƒπ‘ ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)) ↔ βˆƒπ‘ ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)))
8 s3cli 14832 . . . . . . . . . 10 βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∈ Word V
98a1i 11 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∈ Word V)
10 simplr 768 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)
11 simpr 486 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)
1210, 11eqtr4d 2776 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ π‘Š = 𝑝)
1312breq2d 5161 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ (𝑓(Walksβ€˜πΊ)π‘Š ↔ 𝑓(Walksβ€˜πΊ)𝑝))
1413biimpd 228 . . . . . . . . . . . . . 14 (((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ (𝑓(Walksβ€˜πΊ)π‘Š β†’ 𝑓(Walksβ€˜πΊ)𝑝))
1514com12 32 . . . . . . . . . . . . 13 (𝑓(Walksβ€˜πΊ)π‘Š β†’ (((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ 𝑓(Walksβ€˜πΊ)𝑝))
1615adantr 482 . . . . . . . . . . . 12 ((𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2) β†’ (((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ 𝑓(Walksβ€˜πΊ)𝑝))
1716impcom 409 . . . . . . . . . . 11 ((((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ (𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)) β†’ 𝑓(Walksβ€˜πΊ)𝑝)
18 simprr 772 . . . . . . . . . . 11 ((((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ (𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)) β†’ (β™―β€˜π‘“) = 2)
19 vex 3479 . . . . . . . . . . . . . . . 16 π‘Ž ∈ V
20 s3fv0 14842 . . . . . . . . . . . . . . . . 17 (π‘Ž ∈ V β†’ (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜0) = π‘Ž)
2120eqcomd 2739 . . . . . . . . . . . . . . . 16 (π‘Ž ∈ V β†’ π‘Ž = (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜0))
2219, 21mp1i 13 . . . . . . . . . . . . . . 15 (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© β†’ π‘Ž = (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜0))
23 fveq1 6891 . . . . . . . . . . . . . . 15 (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© β†’ (π‘β€˜0) = (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜0))
2422, 23eqtr4d 2776 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© β†’ π‘Ž = (π‘β€˜0))
25 vex 3479 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
26 s3fv1 14843 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ V β†’ (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜1) = 𝑏)
2726eqcomd 2739 . . . . . . . . . . . . . . . 16 (𝑏 ∈ V β†’ 𝑏 = (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜1))
2825, 27mp1i 13 . . . . . . . . . . . . . . 15 (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© β†’ 𝑏 = (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜1))
29 fveq1 6891 . . . . . . . . . . . . . . 15 (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© β†’ (π‘β€˜1) = (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜1))
3028, 29eqtr4d 2776 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© β†’ 𝑏 = (π‘β€˜1))
31 vex 3479 . . . . . . . . . . . . . . . 16 𝑐 ∈ V
32 s3fv2 14844 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ V β†’ (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜2) = 𝑐)
3332eqcomd 2739 . . . . . . . . . . . . . . . 16 (𝑐 ∈ V β†’ 𝑐 = (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜2))
3431, 33mp1i 13 . . . . . . . . . . . . . . 15 (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© β†’ 𝑐 = (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜2))
35 fveq1 6891 . . . . . . . . . . . . . . 15 (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© β†’ (π‘β€˜2) = (βŸ¨β€œπ‘Žπ‘π‘β€βŸ©β€˜2))
3634, 35eqtr4d 2776 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© β†’ 𝑐 = (π‘β€˜2))
3724, 30, 363jca 1129 . . . . . . . . . . . . 13 (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© β†’ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2)))
3837adantl 483 . . . . . . . . . . . 12 (((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2)))
3938adantr 482 . . . . . . . . . . 11 ((((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ (𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)) β†’ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2)))
4017, 18, 393jca 1129 . . . . . . . . . 10 ((((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ (𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)) β†’ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))))
4140ex 414 . . . . . . . . 9 (((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) ∧ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ ((𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2) β†’ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2)))))
429, 41spcimedv 3586 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ ((𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2) β†’ βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2)))))
43 wlklenvp1 28875 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓(Walksβ€˜πΊ)𝑝 β†’ (β™―β€˜π‘) = ((β™―β€˜π‘“) + 1))
44 simpl 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((β™―β€˜π‘) = ((β™―β€˜π‘“) + 1) ∧ (β™―β€˜π‘“) = 2) β†’ (β™―β€˜π‘) = ((β™―β€˜π‘“) + 1))
45 oveq1 7416 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜π‘“) = 2 β†’ ((β™―β€˜π‘“) + 1) = (2 + 1))
4645adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((β™―β€˜π‘) = ((β™―β€˜π‘“) + 1) ∧ (β™―β€˜π‘“) = 2) β†’ ((β™―β€˜π‘“) + 1) = (2 + 1))
4744, 46eqtrd 2773 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((β™―β€˜π‘) = ((β™―β€˜π‘“) + 1) ∧ (β™―β€˜π‘“) = 2) β†’ (β™―β€˜π‘) = (2 + 1))
4847adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘) = ((β™―β€˜π‘“) + 1) ∧ (β™―β€˜π‘“) = 2)) β†’ (β™―β€˜π‘) = (2 + 1))
49 2p1e3 12354 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2 + 1) = 3
5048, 49eqtrdi 2789 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘) = ((β™―β€˜π‘“) + 1) ∧ (β™―β€˜π‘“) = 2)) β†’ (β™―β€˜π‘) = 3)
5150exp32 422 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓(Walksβ€˜πΊ)𝑝 β†’ ((β™―β€˜π‘) = ((β™―β€˜π‘“) + 1) β†’ ((β™―β€˜π‘“) = 2 β†’ (β™―β€˜π‘) = 3)))
5243, 51mpd 15 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓(Walksβ€˜πΊ)𝑝 β†’ ((β™―β€˜π‘“) = 2 β†’ (β™―β€˜π‘) = 3))
5352adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) β†’ ((β™―β€˜π‘“) = 2 β†’ (β™―β€˜π‘) = 3))
5453imp 408 . . . . . . . . . . . . . . . . . . . 20 (((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) ∧ (β™―β€˜π‘“) = 2) β†’ (β™―β€˜π‘) = 3)
55 eqcom 2740 . . . . . . . . . . . . . . . . . . . . . 22 (π‘Ž = (π‘β€˜0) ↔ (π‘β€˜0) = π‘Ž)
5655biimpi 215 . . . . . . . . . . . . . . . . . . . . 21 (π‘Ž = (π‘β€˜0) β†’ (π‘β€˜0) = π‘Ž)
57 eqcom 2740 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = (π‘β€˜1) ↔ (π‘β€˜1) = 𝑏)
5857biimpi 215 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (π‘β€˜1) β†’ (π‘β€˜1) = 𝑏)
59 eqcom 2740 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = (π‘β€˜2) ↔ (π‘β€˜2) = 𝑐)
6059biimpi 215 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = (π‘β€˜2) β†’ (π‘β€˜2) = 𝑐)
6156, 58, 603anim123i 1152 . . . . . . . . . . . . . . . . . . . 20 ((π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2)) β†’ ((π‘β€˜0) = π‘Ž ∧ (π‘β€˜1) = 𝑏 ∧ (π‘β€˜2) = 𝑐))
6254, 61anim12i 614 . . . . . . . . . . . . . . . . . . 19 ((((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) ∧ (β™―β€˜π‘“) = 2) ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ ((β™―β€˜π‘) = 3 ∧ ((π‘β€˜0) = π‘Ž ∧ (π‘β€˜1) = 𝑏 ∧ (π‘β€˜2) = 𝑐)))
631wlkpwrd 28874 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓(Walksβ€˜πΊ)𝑝 β†’ 𝑝 ∈ Word 𝑉)
64 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) β†’ π‘Ž ∈ 𝑉)
6564anim1i 616 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) β†’ (π‘Ž ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
66 3anass 1096 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ↔ (π‘Ž ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
6765, 66sylibr 233 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) β†’ (π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
6867adantr 482 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ (π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
6963, 68anim12i 614 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) β†’ (𝑝 ∈ Word 𝑉 ∧ (π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
7069ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 ((((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) ∧ (β™―β€˜π‘“) = 2) ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ (𝑝 ∈ Word 𝑉 ∧ (π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
71 eqwrds3 14912 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ∈ Word 𝑉 ∧ (π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) β†’ (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ↔ ((β™―β€˜π‘) = 3 ∧ ((π‘β€˜0) = π‘Ž ∧ (π‘β€˜1) = 𝑏 ∧ (π‘β€˜2) = 𝑐))))
7270, 71syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) ∧ (β™―β€˜π‘“) = 2) ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ (𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ↔ ((β™―β€˜π‘) = 3 ∧ ((π‘β€˜0) = π‘Ž ∧ (π‘β€˜1) = 𝑏 ∧ (π‘β€˜2) = 𝑐))))
7362, 72mpbird 257 . . . . . . . . . . . . . . . . . 18 ((((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) ∧ (β™―β€˜π‘“) = 2) ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ 𝑝 = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)
74 simprr 772 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) β†’ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)
7574ad2antrr 725 . . . . . . . . . . . . . . . . . 18 ((((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) ∧ (β™―β€˜π‘“) = 2) ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)
7673, 75eqtr4d 2776 . . . . . . . . . . . . . . . . 17 ((((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) ∧ (β™―β€˜π‘“) = 2) ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ 𝑝 = π‘Š)
7776breq2d 5161 . . . . . . . . . . . . . . . 16 ((((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) ∧ (β™―β€˜π‘“) = 2) ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ (𝑓(Walksβ€˜πΊ)𝑝 ↔ 𝑓(Walksβ€˜πΊ)π‘Š))
7877biimpd 228 . . . . . . . . . . . . . . 15 ((((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) ∧ (β™―β€˜π‘“) = 2) ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ (𝑓(Walksβ€˜πΊ)𝑝 β†’ 𝑓(Walksβ€˜πΊ)π‘Š))
79 simplr 768 . . . . . . . . . . . . . . 15 ((((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) ∧ (β™―β€˜π‘“) = 2) ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ (β™―β€˜π‘“) = 2)
8078, 79jctird 528 . . . . . . . . . . . . . 14 ((((𝑓(Walksβ€˜πΊ)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©)) ∧ (β™―β€˜π‘“) = 2) ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ (𝑓(Walksβ€˜πΊ)𝑝 β†’ (𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)))
8180exp41 436 . . . . . . . . . . . . 13 (𝑓(Walksβ€˜πΊ)𝑝 β†’ ((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ ((β™―β€˜π‘“) = 2 β†’ ((π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2)) β†’ (𝑓(Walksβ€˜πΊ)𝑝 β†’ (𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2))))))
8281com25 99 . . . . . . . . . . . 12 (𝑓(Walksβ€˜πΊ)𝑝 β†’ (𝑓(Walksβ€˜πΊ)𝑝 β†’ ((β™―β€˜π‘“) = 2 β†’ ((π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2)) β†’ ((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ (𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2))))))
8382pm2.43i 52 . . . . . . . . . . 11 (𝑓(Walksβ€˜πΊ)𝑝 β†’ ((β™―β€˜π‘“) = 2 β†’ ((π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2)) β†’ ((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ (𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)))))
84833imp 1112 . . . . . . . . . 10 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ ((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ (𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)))
8584com12 32 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ (𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)))
8685exlimdv 1937 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ (βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))) β†’ (𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)))
8742, 86impbid 211 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ ((𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2) ↔ βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2)))))
8887exbidv 1925 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ©) β†’ (βˆƒπ‘“(𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2) ↔ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2)))))
8988pm5.32da 580 . . . . 5 (((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) β†’ ((π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)) ↔ (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))))))
90892rexbidva 3218 . . . 4 ((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) β†’ (βˆƒπ‘ ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)) ↔ βˆƒπ‘ ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))))))
917, 90bitrid 283 . . 3 ((𝐺 ∈ UPGraph ∧ π‘Ž ∈ 𝑉) β†’ (βˆƒπ‘ ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)) ↔ βˆƒπ‘ ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))))))
9291rexbidva 3177 . 2 (𝐺 ∈ UPGraph β†’ (βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)π‘Š ∧ (β™―β€˜π‘“) = 2)) ↔ βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))))))
933, 6, 923bitrd 305 1 (𝐺 ∈ UPGraph β†’ (π‘Š ∈ (2 WWalksN 𝐺) ↔ βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 (π‘Š = βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∧ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (π‘Ž = (π‘β€˜0) ∧ 𝑏 = (π‘β€˜1) ∧ 𝑐 = (π‘β€˜2))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113  2c2 12267  3c3 12268  β™―chash 14290  Word cword 14464  βŸ¨β€œcs3 14793  Vtxcvtx 28256  UPGraphcupgr 28340  Walkscwlks 28853   WWalksN cwwlksn 29080   WWalksNOn cwwlksnon 29081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-ac2 10458  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-ac 10111  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-concat 14521  df-s1 14546  df-s2 14799  df-s3 14800  df-edg 28308  df-uhgr 28318  df-upgr 28342  df-wlks 28856  df-wwlks 29084  df-wwlksn 29085  df-wwlksnon 29086
This theorem is referenced by: (None)
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