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Theorem eupth2lem3lem3 29177
Description: Lemma for eupth2lem3 29183, formerly part of proof of eupth2lem3 29183: If a loop {(π‘ƒβ€˜π‘), (π‘ƒβ€˜(𝑁 + 1))} is added to a trail, the degree of the vertices with odd degree remains odd (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 21-Feb-2021.)
Hypotheses
Ref Expression
trlsegvdeg.v 𝑉 = (Vtxβ€˜πΊ)
trlsegvdeg.i 𝐼 = (iEdgβ€˜πΊ)
trlsegvdeg.f (πœ‘ β†’ Fun 𝐼)
trlsegvdeg.n (πœ‘ β†’ 𝑁 ∈ (0..^(β™―β€˜πΉ)))
trlsegvdeg.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
trlsegvdeg.w (πœ‘ β†’ 𝐹(Trailsβ€˜πΊ)𝑃)
trlsegvdeg.vx (πœ‘ β†’ (Vtxβ€˜π‘‹) = 𝑉)
trlsegvdeg.vy (πœ‘ β†’ (Vtxβ€˜π‘Œ) = 𝑉)
trlsegvdeg.vz (πœ‘ β†’ (Vtxβ€˜π‘) = 𝑉)
trlsegvdeg.ix (πœ‘ β†’ (iEdgβ€˜π‘‹) = (𝐼 β†Ύ (𝐹 β€œ (0..^𝑁))))
trlsegvdeg.iy (πœ‘ β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩})
trlsegvdeg.iz (πœ‘ β†’ (iEdgβ€˜π‘) = (𝐼 β†Ύ (𝐹 β€œ (0...𝑁))))
eupth2lem3.o (πœ‘ β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘)}))
eupth2lem3lem3.e (πœ‘ β†’ if-((π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1)), (πΌβ€˜(πΉβ€˜π‘)) = {(π‘ƒβ€˜π‘)}, {(π‘ƒβ€˜π‘), (π‘ƒβ€˜(𝑁 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘))))
Assertion
Ref Expression
eupth2lem3lem3 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (Β¬ 2 βˆ₯ (((VtxDegβ€˜π‘‹)β€˜π‘ˆ) + ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ)) ↔ π‘ˆ ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑁 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑁 + 1))})))
Distinct variable groups:   π‘₯,π‘ˆ   π‘₯,𝑉   π‘₯,𝑋
Allowed substitution hints:   πœ‘(π‘₯)   𝑃(π‘₯)   𝐹(π‘₯)   𝐺(π‘₯)   𝐼(π‘₯)   𝑁(π‘₯)   π‘Œ(π‘₯)   𝑍(π‘₯)

Proof of Theorem eupth2lem3lem3
StepHypRef Expression
1 trlsegvdeg.u . . . . 5 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
2 fveq2 6843 . . . . . . . 8 (π‘₯ = π‘ˆ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘₯) = ((VtxDegβ€˜π‘‹)β€˜π‘ˆ))
32breq2d 5118 . . . . . . 7 (π‘₯ = π‘ˆ β†’ (2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘₯) ↔ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ)))
43notbid 318 . . . . . 6 (π‘₯ = π‘ˆ β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘₯) ↔ Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ)))
54elrab3 3647 . . . . 5 (π‘ˆ ∈ 𝑉 β†’ (π‘ˆ ∈ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘₯)} ↔ Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ)))
61, 5syl 17 . . . 4 (πœ‘ β†’ (π‘ˆ ∈ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘₯)} ↔ Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ)))
7 eupth2lem3.o . . . . 5 (πœ‘ β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘)}))
87eleq2d 2824 . . . 4 (πœ‘ β†’ (π‘ˆ ∈ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘₯)} ↔ π‘ˆ ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘)})))
96, 8bitr3d 281 . . 3 (πœ‘ β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ↔ π‘ˆ ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘)})))
109adantr 482 . 2 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ↔ π‘ˆ ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘)})))
11 2z 12536 . . . . . 6 2 ∈ β„€
1211a1i 11 . . . . 5 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ 2 ∈ β„€)
13 trlsegvdeg.v . . . . . . . 8 𝑉 = (Vtxβ€˜πΊ)
14 trlsegvdeg.i . . . . . . . 8 𝐼 = (iEdgβ€˜πΊ)
15 trlsegvdeg.f . . . . . . . 8 (πœ‘ β†’ Fun 𝐼)
16 trlsegvdeg.n . . . . . . . 8 (πœ‘ β†’ 𝑁 ∈ (0..^(β™―β€˜πΉ)))
17 trlsegvdeg.w . . . . . . . 8 (πœ‘ β†’ 𝐹(Trailsβ€˜πΊ)𝑃)
18 trlsegvdeg.vx . . . . . . . 8 (πœ‘ β†’ (Vtxβ€˜π‘‹) = 𝑉)
19 trlsegvdeg.vy . . . . . . . 8 (πœ‘ β†’ (Vtxβ€˜π‘Œ) = 𝑉)
20 trlsegvdeg.vz . . . . . . . 8 (πœ‘ β†’ (Vtxβ€˜π‘) = 𝑉)
21 trlsegvdeg.ix . . . . . . . 8 (πœ‘ β†’ (iEdgβ€˜π‘‹) = (𝐼 β†Ύ (𝐹 β€œ (0..^𝑁))))
22 trlsegvdeg.iy . . . . . . . 8 (πœ‘ β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩})
23 trlsegvdeg.iz . . . . . . . 8 (πœ‘ β†’ (iEdgβ€˜π‘) = (𝐼 β†Ύ (𝐹 β€œ (0...𝑁))))
2413, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23eupth2lem3lem1 29175 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ∈ β„•0)
2524nn0zd 12526 . . . . . 6 (πœ‘ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ∈ β„€)
2625adantr 482 . . . . 5 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ∈ β„€)
2713, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23eupth2lem3lem2 29176 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„•0)
2827nn0zd 12526 . . . . . 6 (πœ‘ β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„€)
2928adantr 482 . . . . 5 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„€)
30 z2even 16253 . . . . . . 7 2 βˆ₯ 2
3119ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ (Vtxβ€˜π‘Œ) = 𝑉)
32 fvexd 6858 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ (πΉβ€˜π‘) ∈ V)
331ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ π‘ˆ ∈ 𝑉)
3422ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩})
35 eupth2lem3lem3.e . . . . . . . . . . . . . 14 (πœ‘ β†’ if-((π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1)), (πΌβ€˜(πΉβ€˜π‘)) = {(π‘ƒβ€˜π‘)}, {(π‘ƒβ€˜π‘), (π‘ƒβ€˜(𝑁 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘))))
3635adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ if-((π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1)), (πΌβ€˜(πΉβ€˜π‘)) = {(π‘ƒβ€˜π‘)}, {(π‘ƒβ€˜π‘), (π‘ƒβ€˜(𝑁 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘))))
37 ifptru 1075 . . . . . . . . . . . . . 14 ((π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1)) β†’ (if-((π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1)), (πΌβ€˜(πΉβ€˜π‘)) = {(π‘ƒβ€˜π‘)}, {(π‘ƒβ€˜π‘), (π‘ƒβ€˜(𝑁 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘))) ↔ (πΌβ€˜(πΉβ€˜π‘)) = {(π‘ƒβ€˜π‘)}))
3837adantl 483 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (if-((π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1)), (πΌβ€˜(πΉβ€˜π‘)) = {(π‘ƒβ€˜π‘)}, {(π‘ƒβ€˜π‘), (π‘ƒβ€˜(𝑁 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘))) ↔ (πΌβ€˜(πΉβ€˜π‘)) = {(π‘ƒβ€˜π‘)}))
3936, 38mpbid 231 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (πΌβ€˜(πΉβ€˜π‘)) = {(π‘ƒβ€˜π‘)})
40 sneq 4597 . . . . . . . . . . . . 13 ((π‘ƒβ€˜π‘) = π‘ˆ β†’ {(π‘ƒβ€˜π‘)} = {π‘ˆ})
4140eqcoms 2745 . . . . . . . . . . . 12 (π‘ˆ = (π‘ƒβ€˜π‘) β†’ {(π‘ƒβ€˜π‘)} = {π‘ˆ})
4239, 41sylan9eq 2797 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ (πΌβ€˜(πΉβ€˜π‘)) = {π‘ˆ})
4342opeq2d 4838 . . . . . . . . . 10 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ ⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩ = ⟨(πΉβ€˜π‘), {π‘ˆ}⟩)
4443sneqd 4599 . . . . . . . . 9 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩} = {⟨(πΉβ€˜π‘), {π‘ˆ}⟩})
4534, 44eqtrd 2777 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), {π‘ˆ}⟩})
4631, 32, 33, 451loopgrvd2 28454 . . . . . . 7 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) = 2)
4730, 46breqtrrid 5144 . . . . . 6 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ 2 βˆ₯ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ))
48 z0even 16250 . . . . . . 7 2 βˆ₯ 0
4919ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)) β†’ (Vtxβ€˜π‘Œ) = 𝑉)
50 fvexd 6858 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)) β†’ (πΉβ€˜π‘) ∈ V)
5113, 14, 15, 16, 1, 17trlsegvdeglem1 29167 . . . . . . . . . 10 (πœ‘ β†’ ((π‘ƒβ€˜π‘) ∈ 𝑉 ∧ (π‘ƒβ€˜(𝑁 + 1)) ∈ 𝑉))
5251simpld 496 . . . . . . . . 9 (πœ‘ β†’ (π‘ƒβ€˜π‘) ∈ 𝑉)
5352ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)) β†’ (π‘ƒβ€˜π‘) ∈ 𝑉)
5422adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩})
5539opeq2d 4838 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ ⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩ = ⟨(πΉβ€˜π‘), {(π‘ƒβ€˜π‘)}⟩)
5655sneqd 4599 . . . . . . . . . 10 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩} = {⟨(πΉβ€˜π‘), {(π‘ƒβ€˜π‘)}⟩})
5754, 56eqtrd 2777 . . . . . . . . 9 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), {(π‘ƒβ€˜π‘)}⟩})
5857adantr 482 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)) β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), {(π‘ƒβ€˜π‘)}⟩})
591adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ π‘ˆ ∈ 𝑉)
6059anim1i 616 . . . . . . . . 9 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)) β†’ (π‘ˆ ∈ 𝑉 ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)))
61 eldifsn 4748 . . . . . . . . 9 (π‘ˆ ∈ (𝑉 βˆ– {(π‘ƒβ€˜π‘)}) ↔ (π‘ˆ ∈ 𝑉 ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)))
6260, 61sylibr 233 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)) β†’ π‘ˆ ∈ (𝑉 βˆ– {(π‘ƒβ€˜π‘)}))
6349, 50, 53, 58, 621loopgrvd0 28455 . . . . . . 7 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)) β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) = 0)
6448, 63breqtrrid 5144 . . . . . 6 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)) β†’ 2 βˆ₯ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ))
6547, 64pm2.61dane 3033 . . . . 5 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ 2 βˆ₯ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ))
66 dvdsadd2b 16189 . . . . 5 ((2 ∈ β„€ ∧ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ∈ β„€ ∧ (((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„€ ∧ 2 βˆ₯ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ))) β†’ (2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ↔ 2 βˆ₯ (((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) + ((VtxDegβ€˜π‘‹)β€˜π‘ˆ))))
6712, 26, 29, 65, 66syl112anc 1375 . . . 4 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ↔ 2 βˆ₯ (((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) + ((VtxDegβ€˜π‘‹)β€˜π‘ˆ))))
6827nn0cnd 12476 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„‚)
6924nn0cnd 12476 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ∈ β„‚)
7068, 69addcomd 11358 . . . . . 6 (πœ‘ β†’ (((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) + ((VtxDegβ€˜π‘‹)β€˜π‘ˆ)) = (((VtxDegβ€˜π‘‹)β€˜π‘ˆ) + ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ)))
7170breq2d 5118 . . . . 5 (πœ‘ β†’ (2 βˆ₯ (((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) + ((VtxDegβ€˜π‘‹)β€˜π‘ˆ)) ↔ 2 βˆ₯ (((VtxDegβ€˜π‘‹)β€˜π‘ˆ) + ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ))))
7271adantr 482 . . . 4 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (2 βˆ₯ (((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) + ((VtxDegβ€˜π‘‹)β€˜π‘ˆ)) ↔ 2 βˆ₯ (((VtxDegβ€˜π‘‹)β€˜π‘ˆ) + ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ))))
7367, 72bitrd 279 . . 3 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ↔ 2 βˆ₯ (((VtxDegβ€˜π‘‹)β€˜π‘ˆ) + ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ))))
7473notbid 318 . 2 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ↔ Β¬ 2 βˆ₯ (((VtxDegβ€˜π‘‹)β€˜π‘ˆ) + ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ))))
75 simpr 486 . . . . 5 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1)))
7675eqeq2d 2748 . . . 4 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘) ↔ (π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑁 + 1))))
7775preq2d 4702 . . . 4 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑁 + 1))})
7876, 77ifbieq2d 4513 . . 3 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘)}) = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑁 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑁 + 1))}))
7978eleq2d 2824 . 2 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (π‘ˆ ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘)}) ↔ π‘ˆ ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑁 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑁 + 1))})))
8010, 74, 793bitr3d 309 1 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (Β¬ 2 βˆ₯ (((VtxDegβ€˜π‘‹)β€˜π‘ˆ) + ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ)) ↔ π‘ˆ ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑁 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑁 + 1))})))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397  if-wif 1062   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  {crab 3408  Vcvv 3446   βˆ– cdif 3908   βŠ† wss 3911  βˆ…c0 4283  ifcif 4487  {csn 4587  {cpr 4589  βŸ¨cop 4593   class class class wbr 5106   β†Ύ cres 5636   β€œ cima 5637  Fun wfun 6491  β€˜cfv 6497  (class class class)co 7358  0cc0 11052  1c1 11053   + caddc 11055  2c2 12209  β„€cz 12500  ...cfz 13425  ..^cfzo 13568  β™―chash 14231   βˆ₯ cdvds 16137  Vtxcvtx 27950  iEdgciedg 27951  VtxDegcvtxdg 28416  Trailsctrls 28641
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  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-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-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-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-oadd 8417  df-er 8649  df-map 8768  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-dju 9838  df-card 9876  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-nn 12155  df-2 12217  df-n0 12415  df-xnn0 12487  df-z 12501  df-uz 12765  df-xadd 13035  df-fz 13426  df-fzo 13569  df-hash 14232  df-word 14404  df-dvds 16138  df-edg 28002  df-uhgr 28012  df-ushgr 28013  df-uspgr 28104  df-vtxdg 28417  df-wlks 28550  df-trls 28643
This theorem is referenced by:  eupth2lem3lem7  29181
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