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Theorem eupth2lem3lem3 29472
Description: Lemma for eupth2lem3 29478, formerly part of proof of eupth2lem3 29478: 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 6888 . . . . . . . 8 (π‘₯ = π‘ˆ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘₯) = ((VtxDegβ€˜π‘‹)β€˜π‘ˆ))
32breq2d 5159 . . . . . . 7 (π‘₯ = π‘ˆ β†’ (2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘₯) ↔ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ)))
43notbid 317 . . . . . 6 (π‘₯ = π‘ˆ β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘₯) ↔ Β¬ 2 βˆ₯ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ)))
54elrab3 3683 . . . . 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 2819 . . . 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 280 . . 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 481 . 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 12590 . . . . . 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 29470 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ∈ β„•0)
2524nn0zd 12580 . . . . . 6 (πœ‘ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ∈ β„€)
2625adantr 481 . . . . 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 29471 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„•0)
2827nn0zd 12580 . . . . . 6 (πœ‘ β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„€)
2928adantr 481 . . . . 5 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„€)
30 z2even 16309 . . . . . . 7 2 βˆ₯ 2
3119ad2antrr 724 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ (Vtxβ€˜π‘Œ) = 𝑉)
32 fvexd 6903 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ (πΉβ€˜π‘) ∈ V)
331ad2antrr 724 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ π‘ˆ ∈ 𝑉)
3422ad2antrr 724 . . . . . . . . 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 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‘))))
37 ifptru 1074 . . . . . . . . . . . . . 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 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‘))) ↔ (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 4637 . . . . . . . . . . . . 13 ((π‘ƒβ€˜π‘) = π‘ˆ β†’ {(π‘ƒβ€˜π‘)} = {π‘ˆ})
4140eqcoms 2740 . . . . . . . . . . . 12 (π‘ˆ = (π‘ƒβ€˜π‘) β†’ {(π‘ƒβ€˜π‘)} = {π‘ˆ})
4239, 41sylan9eq 2792 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ = (π‘ƒβ€˜π‘)) β†’ (πΌβ€˜(πΉβ€˜π‘)) = {π‘ˆ})
4342opeq2d 4879 . . . . . . . . . 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 4639 . . . . . . . . 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 2772 . . . . . . . 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 28749 . . . . . . 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 5185 . . . . . 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 16306 . . . . . . 7 2 βˆ₯ 0
4919ad2antrr 724 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)) β†’ (Vtxβ€˜π‘Œ) = 𝑉)
50 fvexd 6903 . . . . . . . 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 29462 . . . . . . . . . 10 (πœ‘ β†’ ((π‘ƒβ€˜π‘) ∈ 𝑉 ∧ (π‘ƒβ€˜(𝑁 + 1)) ∈ 𝑉))
5251simpld 495 . . . . . . . . 9 (πœ‘ β†’ (π‘ƒβ€˜π‘) ∈ 𝑉)
5352ad2antrr 724 . . . . . . . 8 (((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) ∧ π‘ˆ β‰  (π‘ƒβ€˜π‘)) β†’ (π‘ƒβ€˜π‘) ∈ 𝑉)
5422adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩})
5539opeq2d 4879 . . . . . . . . . . 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 4639 . . . . . . . . . 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 2772 . . . . . . . . 9 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), {(π‘ƒβ€˜π‘)}⟩})
5857adantr 481 . . . . . . . 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 481 . . . . . . . . . 10 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ π‘ˆ ∈ 𝑉)
6059anim1i 615 . . . . . . . . 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 4789 . . . . . . . . 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 28750 . . . . . . 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 5185 . . . . . 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 3029 . . . . 5 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ 2 βˆ₯ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ))
66 dvdsadd2b 16245 . . . . 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 1374 . . . 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 12530 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ) ∈ β„‚)
6924nn0cnd 12530 . . . . . . 7 (πœ‘ β†’ ((VtxDegβ€˜π‘‹)β€˜π‘ˆ) ∈ β„‚)
7068, 69addcomd 11412 . . . . . 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 5159 . . . . 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 481 . . . 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 278 . . 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 317 . 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 485 . . . . 5 ((πœ‘ ∧ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1))) β†’ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(𝑁 + 1)))
7675eqeq2d 2743 . . . 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 4743 . . . 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 4553 . . 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 2819 . 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 308 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 396  if-wif 1061   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  {crab 3432  Vcvv 3474   βˆ– cdif 3944   βŠ† wss 3947  βˆ…c0 4321  ifcif 4527  {csn 4627  {cpr 4629  βŸ¨cop 4633   class class class wbr 5147   β†Ύ cres 5677   β€œ cima 5678  Fun wfun 6534  β€˜cfv 6540  (class class class)co 7405  0cc0 11106  1c1 11107   + caddc 11109  2c2 12263  β„€cz 12554  ...cfz 13480  ..^cfzo 13623  β™―chash 14286   βˆ₯ cdvds 16193  Vtxcvtx 28245  iEdgciedg 28246  VtxDegcvtxdg 28711  Trailsctrls 28936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-xadd 13089  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-dvds 16194  df-edg 28297  df-uhgr 28307  df-ushgr 28308  df-uspgr 28399  df-vtxdg 28712  df-wlks 28845  df-trls 28938
This theorem is referenced by:  eupth2lem3lem7  29476
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