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Theorem elwwlks2 28953
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 28902 . . 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 28946 . . . 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 3208 . 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 3272 . . . 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 14776 . . . . . . . . . 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 5118 . . . . . . . . . . . . . . 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 3448 . . . . . . . . . . . . . . . 16 π‘Ž ∈ V
20 s3fv0 14786 . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . 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 3448 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
26 s3fv1 14787 . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . 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 3448 . . . . . . . . . . . . . . . 16 𝑐 ∈ V
32 s3fv2 14788 . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . 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 3553 . . . . . . . 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 28608 . . . . . . . . . . . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12300 . . . . . . . . . . . . . . . . . . . . . . . . 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 28607 . . . . . . . . . . . . . . . . . . . . . 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 14856 . . . . . . . . . . . . . . . . . . . 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 5118 . . . . . . . . . . . . . . . 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 3208 . . . 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 3170 . 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 3070  Vcvv 3444   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  0cc0 11056  1c1 11057   + caddc 11059  2c2 12213  3c3 12214  β™―chash 14236  Word cword 14408  βŸ¨β€œcs3 14737  Vtxcvtx 27989  UPGraphcupgr 28073  Walkscwlks 28586   WWalksN cwwlksn 28813   WWalksNOn cwwlksnon 28814
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-ac2 10404  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-oadd 8417  df-er 8651  df-map 8770  df-pm 8771  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-dju 9842  df-card 9880  df-ac 10057  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-xnn0 12491  df-z 12505  df-uz 12769  df-fz 13431  df-fzo 13574  df-hash 14237  df-word 14409  df-concat 14465  df-s1 14490  df-s2 14743  df-s3 14744  df-edg 28041  df-uhgr 28051  df-upgr 28075  df-wlks 28589  df-wwlks 28817  df-wwlksn 28818  df-wwlksnon 28819
This theorem is referenced by: (None)
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