Proof of Theorem eupth2lem3lem3
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | trlsegvdeg.u | . . . . 5
⊢ (𝜑 → 𝑈 ∈ 𝑉) | 
| 2 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑥 = 𝑈 → ((VtxDeg‘𝑋)‘𝑥) = ((VtxDeg‘𝑋)‘𝑈)) | 
| 3 | 2 | breq2d 5155 | . . . . . . 7
⊢ (𝑥 = 𝑈 → (2 ∥ ((VtxDeg‘𝑋)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝑋)‘𝑈))) | 
| 4 | 3 | notbid 318 | . . . . . 6
⊢ (𝑥 = 𝑈 → (¬ 2 ∥
((VtxDeg‘𝑋)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈))) | 
| 5 | 4 | elrab3 3693 | . . . . 5
⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥
((VtxDeg‘𝑋)‘𝑥)} ↔ ¬ 2 ∥
((VtxDeg‘𝑋)‘𝑈))) | 
| 6 | 1, 5 | syl 17 | . . . 4
⊢ (𝜑 → (𝑈 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥
((VtxDeg‘𝑋)‘𝑥)} ↔ ¬ 2 ∥
((VtxDeg‘𝑋)‘𝑈))) | 
| 7 |  | eupth2lem3.o | . . . . 5
⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥
((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) | 
| 8 | 7 | eleq2d 2827 | . . . 4
⊢ (𝜑 → (𝑈 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥
((VtxDeg‘𝑋)‘𝑥)} ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))) | 
| 9 | 6, 8 | bitr3d 281 | . . 3
⊢ (𝜑 → (¬ 2 ∥
((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))) | 
| 10 | 9 | adantr 480 | . 2
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥
((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))) | 
| 11 |  | 2z 12649 | . . . . . 6
⊢ 2 ∈
ℤ | 
| 12 | 11 | a1i 11 | . . . . 5
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → 2 ∈
ℤ) | 
| 13 |  | trlsegvdeg.v | . . . . . . . 8
⊢ 𝑉 = (Vtx‘𝐺) | 
| 14 |  | trlsegvdeg.i | . . . . . . . 8
⊢ 𝐼 = (iEdg‘𝐺) | 
| 15 |  | trlsegvdeg.f | . . . . . . . 8
⊢ (𝜑 → Fun 𝐼) | 
| 16 |  | trlsegvdeg.n | . . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) | 
| 17 |  | trlsegvdeg.w | . . . . . . . 8
⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | 
| 18 |  | trlsegvdeg.vx | . . . . . . . 8
⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) | 
| 19 |  | trlsegvdeg.vy | . . . . . . . 8
⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) | 
| 20 |  | trlsegvdeg.vz | . . . . . . . 8
⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) | 
| 21 |  | trlsegvdeg.ix | . . . . . . . 8
⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | 
| 22 |  | trlsegvdeg.iy | . . . . . . . 8
⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | 
| 23 |  | trlsegvdeg.iz | . . . . . . . 8
⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) | 
| 24 | 13, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23 | eupth2lem3lem1 30247 | . . . . . . 7
⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈
ℕ0) | 
| 25 | 24 | nn0zd 12639 | . . . . . 6
⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℤ) | 
| 26 | 25 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → ((VtxDeg‘𝑋)‘𝑈) ∈ ℤ) | 
| 27 | 13, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23 | eupth2lem3lem2 30248 | . . . . . . 7
⊢ (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈
ℕ0) | 
| 28 | 27 | nn0zd 12639 | . . . . . 6
⊢ (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℤ) | 
| 29 | 28 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → ((VtxDeg‘𝑌)‘𝑈) ∈ ℤ) | 
| 30 |  | z2even 16407 | . . . . . . 7
⊢ 2 ∥
2 | 
| 31 | 19 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃‘𝑁)) → (Vtx‘𝑌) = 𝑉) | 
| 32 |  | fvexd 6921 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃‘𝑁)) → (𝐹‘𝑁) ∈ V) | 
| 33 | 1 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃‘𝑁)) → 𝑈 ∈ 𝑉) | 
| 34 | 22 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃‘𝑁)) → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | 
| 35 |  | eupth2lem3lem3.e | . . . . . . . . . . . . . 14
⊢ (𝜑 → if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁)))) | 
| 36 | 35 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁)))) | 
| 37 |  | ifptru 1075 | . . . . . . . . . . . . . 14
⊢ ((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)) → (if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁))) ↔ (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)})) | 
| 38 | 37 | adantl 481 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁))) ↔ (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)})) | 
| 39 | 36, 38 | mpbid 232 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}) | 
| 40 |  | sneq 4636 | . . . . . . . . . . . . 13
⊢ ((𝑃‘𝑁) = 𝑈 → {(𝑃‘𝑁)} = {𝑈}) | 
| 41 | 40 | eqcoms 2745 | . . . . . . . . . . . 12
⊢ (𝑈 = (𝑃‘𝑁) → {(𝑃‘𝑁)} = {𝑈}) | 
| 42 | 39, 41 | sylan9eq 2797 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃‘𝑁)) → (𝐼‘(𝐹‘𝑁)) = {𝑈}) | 
| 43 | 42 | opeq2d 4880 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃‘𝑁)) → 〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉 = 〈(𝐹‘𝑁), {𝑈}〉) | 
| 44 | 43 | sneqd 4638 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃‘𝑁)) → {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉} = {〈(𝐹‘𝑁), {𝑈}〉}) | 
| 45 | 34, 44 | eqtrd 2777 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃‘𝑁)) → (iEdg‘𝑌) = {〈(𝐹‘𝑁), {𝑈}〉}) | 
| 46 | 31, 32, 33, 45 | 1loopgrvd2 29521 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃‘𝑁)) → ((VtxDeg‘𝑌)‘𝑈) = 2) | 
| 47 | 30, 46 | breqtrrid 5181 | . . . . . 6
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃‘𝑁)) → 2 ∥ ((VtxDeg‘𝑌)‘𝑈)) | 
| 48 |  | z0even 16404 | . . . . . . 7
⊢ 2 ∥
0 | 
| 49 | 19 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃‘𝑁)) → (Vtx‘𝑌) = 𝑉) | 
| 50 |  | fvexd 6921 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃‘𝑁)) → (𝐹‘𝑁) ∈ V) | 
| 51 | 13, 14, 15, 16, 1, 17 | trlsegvdeglem1 30239 | . . . . . . . . . 10
⊢ (𝜑 → ((𝑃‘𝑁) ∈ 𝑉 ∧ (𝑃‘(𝑁 + 1)) ∈ 𝑉)) | 
| 52 | 51 | simpld 494 | . . . . . . . . 9
⊢ (𝜑 → (𝑃‘𝑁) ∈ 𝑉) | 
| 53 | 52 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃‘𝑁)) → (𝑃‘𝑁) ∈ 𝑉) | 
| 54 | 22 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | 
| 55 | 39 | opeq2d 4880 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → 〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉 = 〈(𝐹‘𝑁), {(𝑃‘𝑁)}〉) | 
| 56 | 55 | sneqd 4638 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉} = {〈(𝐹‘𝑁), {(𝑃‘𝑁)}〉}) | 
| 57 | 54, 56 | eqtrd 2777 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (iEdg‘𝑌) = {〈(𝐹‘𝑁), {(𝑃‘𝑁)}〉}) | 
| 58 | 57 | adantr 480 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃‘𝑁)) → (iEdg‘𝑌) = {〈(𝐹‘𝑁), {(𝑃‘𝑁)}〉}) | 
| 59 | 1 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → 𝑈 ∈ 𝑉) | 
| 60 | 59 | anim1i 615 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃‘𝑁)) → (𝑈 ∈ 𝑉 ∧ 𝑈 ≠ (𝑃‘𝑁))) | 
| 61 |  | eldifsn 4786 | . . . . . . . . 9
⊢ (𝑈 ∈ (𝑉 ∖ {(𝑃‘𝑁)}) ↔ (𝑈 ∈ 𝑉 ∧ 𝑈 ≠ (𝑃‘𝑁))) | 
| 62 | 60, 61 | sylibr 234 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃‘𝑁)) → 𝑈 ∈ (𝑉 ∖ {(𝑃‘𝑁)})) | 
| 63 | 49, 50, 53, 58, 62 | 1loopgrvd0 29522 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃‘𝑁)) → ((VtxDeg‘𝑌)‘𝑈) = 0) | 
| 64 | 48, 63 | breqtrrid 5181 | . . . . . 6
⊢ (((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃‘𝑁)) → 2 ∥ ((VtxDeg‘𝑌)‘𝑈)) | 
| 65 | 47, 64 | pm2.61dane 3029 | . . . . 5
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → 2 ∥
((VtxDeg‘𝑌)‘𝑈)) | 
| 66 |  | dvdsadd2b 16343 | . . . . 5
⊢ ((2
∈ ℤ ∧ ((VtxDeg‘𝑋)‘𝑈) ∈ ℤ ∧ (((VtxDeg‘𝑌)‘𝑈) ∈ ℤ ∧ 2 ∥
((VtxDeg‘𝑌)‘𝑈))) → (2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)))) | 
| 67 | 12, 26, 29, 65, 66 | syl112anc 1376 | . . . 4
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥
((VtxDeg‘𝑋)‘𝑈) ↔ 2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)))) | 
| 68 | 27 | nn0cnd 12589 | . . . . . . 7
⊢ (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℂ) | 
| 69 | 24 | nn0cnd 12589 | . . . . . . 7
⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℂ) | 
| 70 | 68, 69 | addcomd 11463 | . . . . . 6
⊢ (𝜑 → (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))) | 
| 71 | 70 | breq2d 5155 | . . . . 5
⊢ (𝜑 → (2 ∥
(((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))) | 
| 72 | 71 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥
(((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))) | 
| 73 | 67, 72 | bitrd 279 | . . 3
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥
((VtxDeg‘𝑋)‘𝑈) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))) | 
| 74 | 73 | notbid 318 | . 2
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥
((VtxDeg‘𝑋)‘𝑈) ↔ ¬ 2 ∥
(((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))) | 
| 75 |  | simpr 484 | . . . . 5
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) | 
| 76 | 75 | eqeq2d 2748 | . . . 4
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → ((𝑃‘0) = (𝑃‘𝑁) ↔ (𝑃‘0) = (𝑃‘(𝑁 + 1)))) | 
| 77 | 75 | preq2d 4740 | . . . 4
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → {(𝑃‘0), (𝑃‘𝑁)} = {(𝑃‘0), (𝑃‘(𝑁 + 1))}) | 
| 78 | 76, 77 | ifbieq2d 4552 | . . 3
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}) = if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})) | 
| 79 | 78 | eleq2d 2827 | . 2
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (𝑈 ∈ if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) | 
| 80 | 10, 74, 79 | 3bitr3d 309 | 1
⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥
(((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) |