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| Mirrors > Home > MPE Home > Th. List > vdegp1bi | Structured version Visualization version GIF version | ||
| Description: The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑈, 𝑋} to the edge set, where 𝑋 ≠ 𝑈, yields degree 𝑃 + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.) |
| Ref | Expression |
|---|---|
| vdegp1ai.vg | ⊢ 𝑉 = (Vtx‘𝐺) |
| vdegp1ai.u | ⊢ 𝑈 ∈ 𝑉 |
| vdegp1ai.i | ⊢ 𝐼 = (iEdg‘𝐺) |
| vdegp1ai.w | ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} |
| vdegp1ai.d | ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃 |
| vdegp1ai.vf | ⊢ (Vtx‘𝐹) = 𝑉 |
| vdegp1bi.x | ⊢ 𝑋 ∈ 𝑉 |
| vdegp1bi.xu | ⊢ 𝑋 ≠ 𝑈 |
| vdegp1bi.f | ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑈, 𝑋}”〉) |
| Ref | Expression |
|---|---|
| vdegp1bi | ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prex 5437 | . . 3 ⊢ {𝑈, 𝑋} ∈ V | |
| 2 | vdegp1ai.vg | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 3 | vdegp1ai.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 4 | vdegp1ai.w | . . . . 5 ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} | |
| 5 | wrdf 14557 | . . . . . 6 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼:(0..^(♯‘𝐼))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
| 6 | 5 | ffund 6740 | . . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → Fun 𝐼) |
| 7 | 4, 6 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → Fun 𝐼) |
| 8 | vdegp1ai.vf | . . . . 5 ⊢ (Vtx‘𝐹) = 𝑉 | |
| 9 | 8 | a1i 11 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (Vtx‘𝐹) = 𝑉) |
| 10 | vdegp1bi.f | . . . . 5 ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑈, 𝑋}”〉) | |
| 11 | wrdv 14567 | . . . . . . 7 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼 ∈ Word V) | |
| 12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
| 13 | cats1un 14759 | . . . . . 6 ⊢ ((𝐼 ∈ Word V ∧ {𝑈, 𝑋} ∈ V) → (𝐼 ++ 〈“{𝑈, 𝑋}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑈, 𝑋}〉})) | |
| 14 | 12, 13 | mpan 690 | . . . . 5 ⊢ ({𝑈, 𝑋} ∈ V → (𝐼 ++ 〈“{𝑈, 𝑋}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑈, 𝑋}〉})) |
| 15 | 10, 14 | eqtrid 2789 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (iEdg‘𝐹) = (𝐼 ∪ {〈(♯‘𝐼), {𝑈, 𝑋}〉})) |
| 16 | fvexd 6921 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (♯‘𝐼) ∈ V) | |
| 17 | wrdlndm 14568 | . . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (♯‘𝐼) ∉ dom 𝐼) | |
| 18 | 4, 17 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (♯‘𝐼) ∉ dom 𝐼) |
| 19 | vdegp1ai.u | . . . . 5 ⊢ 𝑈 ∈ 𝑉 | |
| 20 | 19 | a1i 11 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → 𝑈 ∈ 𝑉) |
| 21 | vdegp1bi.x | . . . . . 6 ⊢ 𝑋 ∈ 𝑉 | |
| 22 | 19, 21 | pm3.2i 470 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) |
| 23 | prelpwi 5452 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → {𝑈, 𝑋} ∈ 𝒫 𝑉) | |
| 24 | 22, 23 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → {𝑈, 𝑋} ∈ 𝒫 𝑉) |
| 25 | prid1g 4760 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ {𝑈, 𝑋}) | |
| 26 | 19, 25 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → 𝑈 ∈ {𝑈, 𝑋}) |
| 27 | vdegp1bi.xu | . . . . . . . 8 ⊢ 𝑋 ≠ 𝑈 | |
| 28 | 27 | necomi 2995 | . . . . . . 7 ⊢ 𝑈 ≠ 𝑋 |
| 29 | hashprg 14434 | . . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑈 ≠ 𝑋 ↔ (♯‘{𝑈, 𝑋}) = 2)) | |
| 30 | 19, 21, 29 | mp2an 692 | . . . . . . 7 ⊢ (𝑈 ≠ 𝑋 ↔ (♯‘{𝑈, 𝑋}) = 2) |
| 31 | 28, 30 | mpbi 230 | . . . . . 6 ⊢ (♯‘{𝑈, 𝑋}) = 2 |
| 32 | 31 | eqcomi 2746 | . . . . 5 ⊢ 2 = (♯‘{𝑈, 𝑋}) |
| 33 | 2re 12340 | . . . . . 6 ⊢ 2 ∈ ℝ | |
| 34 | 33 | eqlei 11371 | . . . . 5 ⊢ (2 = (♯‘{𝑈, 𝑋}) → 2 ≤ (♯‘{𝑈, 𝑋})) |
| 35 | 32, 34 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → 2 ≤ (♯‘{𝑈, 𝑋})) |
| 36 | 2, 3, 7, 9, 15, 16, 18, 20, 24, 26, 35 | p1evtxdp1 29532 | . . 3 ⊢ ({𝑈, 𝑋} ∈ V → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 1)) |
| 37 | 1, 36 | ax-mp 5 | . 2 ⊢ ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 1) |
| 38 | fzofi 14015 | . . . . 5 ⊢ (0..^(♯‘𝐼)) ∈ Fin | |
| 39 | wrddm 14559 | . . . . . . . 8 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → dom 𝐼 = (0..^(♯‘𝐼))) | |
| 40 | 4, 39 | ax-mp 5 | . . . . . . 7 ⊢ dom 𝐼 = (0..^(♯‘𝐼)) |
| 41 | 40 | eqcomi 2746 | . . . . . 6 ⊢ (0..^(♯‘𝐼)) = dom 𝐼 |
| 42 | 2, 3, 41 | vtxdgfisnn0 29493 | . . . . 5 ⊢ (((0..^(♯‘𝐼)) ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0) |
| 43 | 38, 19, 42 | mp2an 692 | . . . 4 ⊢ ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0 |
| 44 | 43 | nn0rei 12537 | . . 3 ⊢ ((VtxDeg‘𝐺)‘𝑈) ∈ ℝ |
| 45 | 1re 11261 | . . 3 ⊢ 1 ∈ ℝ | |
| 46 | rexadd 13274 | . . 3 ⊢ ((((VtxDeg‘𝐺)‘𝑈) ∈ ℝ ∧ 1 ∈ ℝ) → (((VtxDeg‘𝐺)‘𝑈) +𝑒 1) = (((VtxDeg‘𝐺)‘𝑈) + 1)) | |
| 47 | 44, 45, 46 | mp2an 692 | . 2 ⊢ (((VtxDeg‘𝐺)‘𝑈) +𝑒 1) = (((VtxDeg‘𝐺)‘𝑈) + 1) |
| 48 | vdegp1ai.d | . . 3 ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃 | |
| 49 | 48 | oveq1i 7441 | . 2 ⊢ (((VtxDeg‘𝐺)‘𝑈) + 1) = (𝑃 + 1) |
| 50 | 37, 47, 49 | 3eqtri 2769 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ∅c0 4333 𝒫 cpw 4600 {csn 4626 {cpr 4628 〈cop 4632 class class class wbr 5143 dom cdm 5685 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 ≤ cle 11296 2c2 12321 ℕ0cn0 12526 +𝑒 cxad 13152 ..^cfzo 13694 ♯chash 14369 Word cword 14552 ++ cconcat 14608 〈“cs1 14633 Vtxcvtx 29013 iEdgciedg 29014 VtxDegcvtxdg 29483 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-xadd 13155 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-vtx 29015 df-iedg 29016 df-vtxdg 29484 |
| This theorem is referenced by: vdegp1ci 29556 konigsberglem1 30271 konigsberglem2 30272 |
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