MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eupth2lem3lem3 Structured version   Visualization version   GIF version
Theorem eupth2lem3lem3 30027
Description: Lemma for eupth2lem3 30033, formerly part of proof of eupth2lem3 30033: 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 6891 . . . . . . . 8 (π‘₯ = π‘ˆ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘₯) = ((VtxDegβ€˜π‘‹)β€˜π‘ˆ))
32breq2d 5154 . . . . . . 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 3681 . . . . 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 2814 . . . 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 480 . 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 12616 . . . . . 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 30025 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ∈ β„•0)
2524nn0zd 12606 . . . . . 6 (πœ‘ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ∈ β„€)
2625adantr 480 . . . . 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 30026 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„•0)
2827nn0zd 12606 . . . . . 6 (πœ‘ β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„€)
2928adantr 480 . . . . 5 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„€)
30 z2even 16338 . . . . . . 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 6906 . . . . . . . 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 480 . . . . . . . . . . . . 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 1073 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 4634 . . . . . . . . . . . . 13 ((π‘ƒβ€˜π‘) = π‘ˆ β†’ {(π‘ƒβ€˜π‘)} = {π‘ˆ})
4140eqcoms 2735 . . . . . . . . . . . 12 (π‘ˆ = (π‘ƒβ€˜π‘) β†’ {(π‘ƒβ€˜π‘)} = {π‘ˆ})
4239, 41sylan9eq 2787 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ (πΌβ€˜(πΉβ€˜π‘)) = {π‘ˆ})
4342opeq2d 4876 . . . . . . . . . 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 4636 . . . . . . . . 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 2767 . . . . . . . 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 29304 . . . . . . 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 5180 . . . . . 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 16335 . . . . . . 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 6906 . . . . . . . 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 30017 . . . . . . . . . 10 (πœ‘ β†’ ((π‘ƒβ€˜π‘) ∈ 𝑉 ∧ (π‘ƒβ€˜(𝑁 + 1)) ∈ 𝑉))
5251simpld 494 . . . . . . . . 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 480 . . . . . . . . . 10 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩})
5539opeq2d 4876 . . . . . . . . . . 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 4636 . . . . . . . . . 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 2767 . . . . . . . . 9 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), {(π‘ƒβ€˜π‘)}⟩})
5857adantr 480 . . . . . . . 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 480 . . . . . . . . . 10 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ π‘ˆ ∈ 𝑉)
6059anim1i 614 . . . . . . . . 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 4786 . . . . . . . . 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 29305 . . . . . . 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 5180 . . . . . 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 3024 . . . . 5 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ 2 βˆ₯ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ))
66 dvdsadd2b 16274 . . . . 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 1372 . . . 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 12556 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„‚)
6924nn0cnd 12556 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ∈ β„‚)
7068, 69addcomd 11438 . . . . . 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 5154 . . . . 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 480 . . . 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 484 . . . . 5 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1)))
7675eqeq2d 2738 . . . 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 4740 . . . 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 4550 . . 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 2814 . 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 395  if-wif 1061   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  {crab 3427  Vcvv 3469   βˆ– cdif 3941   βŠ† wss 3944  βˆ…c0 4318  ifcif 4524  {csn 4624  {cpr 4626  βŸ¨cop 4630   class class class wbr 5142   β†Ύ cres 5674   β€œ cima 5675  Fun wfun 6536  β€˜cfv 6542  (class class class)co 7414  0cc0 11130  1c1 11131   + caddc 11133  2c2 12289  β„€cz 12580  ...cfz 13508  ..^cfzo 13651  β™―chash 14313   βˆ₯ cdvds 16222  Vtxcvtx 28796  iEdgciedg 28797  VtxDegcvtxdg 29266  Trailsctrls 29491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-xadd 13117  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-dvds 16223  df-edg 28848  df-uhgr 28858  df-ushgr 28859  df-uspgr 28950  df-vtxdg 29267  df-wlks 29400  df-trls 29493
This theorem is referenced by:  eupth2lem3lem7  30031
  Copyright terms: Public domain W3C validator