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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 5325 | . . 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 13860 | . . . . . 6 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼:(0..^(♯‘𝐼))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
6 | 5 | ffund 6513 | . . . . 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 13871 | . . . . . . 7 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼 ∈ Word V) | |
12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
13 | cats1un 14077 | . . . . . 6 ⊢ ((𝐼 ∈ Word V ∧ {𝑈, 𝑋} ∈ V) → (𝐼 ++ 〈“{𝑈, 𝑋}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑈, 𝑋}〉})) | |
14 | 12, 13 | mpan 688 | . . . . 5 ⊢ ({𝑈, 𝑋} ∈ V → (𝐼 ++ 〈“{𝑈, 𝑋}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑈, 𝑋}〉})) |
15 | 10, 14 | syl5eq 2868 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (iEdg‘𝐹) = (𝐼 ∪ {〈(♯‘𝐼), {𝑈, 𝑋}〉})) |
16 | fvexd 6680 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (♯‘𝐼) ∈ V) | |
17 | wrdlndm 13873 | . . . . 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 473 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) |
23 | prelpwi 5332 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → {𝑈, 𝑋} ∈ 𝒫 𝑉) | |
24 | 22, 23 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → {𝑈, 𝑋} ∈ 𝒫 𝑉) |
25 | prid1g 4690 | . . . . 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 3070 | . . . . . . 7 ⊢ 𝑈 ≠ 𝑋 |
29 | hashprg 13750 | . . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑈 ≠ 𝑋 ↔ (♯‘{𝑈, 𝑋}) = 2)) | |
30 | 19, 21, 29 | mp2an 690 | . . . . . . 7 ⊢ (𝑈 ≠ 𝑋 ↔ (♯‘{𝑈, 𝑋}) = 2) |
31 | 28, 30 | mpbi 232 | . . . . . 6 ⊢ (♯‘{𝑈, 𝑋}) = 2 |
32 | 31 | eqcomi 2830 | . . . . 5 ⊢ 2 = (♯‘{𝑈, 𝑋}) |
33 | 2re 11705 | . . . . . 6 ⊢ 2 ∈ ℝ | |
34 | 33 | eqlei 10744 | . . . . 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 27290 | . . 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 13336 | . . . . 5 ⊢ (0..^(♯‘𝐼)) ∈ Fin | |
39 | wrddm 13862 | . . . . . . . 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 2830 | . . . . . 6 ⊢ (0..^(♯‘𝐼)) = dom 𝐼 |
42 | 2, 3, 41 | vtxdgfisnn0 27251 | . . . . 5 ⊢ (((0..^(♯‘𝐼)) ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0) |
43 | 38, 19, 42 | mp2an 690 | . . . 4 ⊢ ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0 |
44 | 43 | nn0rei 11902 | . . 3 ⊢ ((VtxDeg‘𝐺)‘𝑈) ∈ ℝ |
45 | 1re 10635 | . . 3 ⊢ 1 ∈ ℝ | |
46 | rexadd 12619 | . . 3 ⊢ ((((VtxDeg‘𝐺)‘𝑈) ∈ ℝ ∧ 1 ∈ ℝ) → (((VtxDeg‘𝐺)‘𝑈) +𝑒 1) = (((VtxDeg‘𝐺)‘𝑈) + 1)) | |
47 | 44, 45, 46 | mp2an 690 | . 2 ⊢ (((VtxDeg‘𝐺)‘𝑈) +𝑒 1) = (((VtxDeg‘𝐺)‘𝑈) + 1) |
48 | vdegp1ai.d | . . 3 ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃 | |
49 | 48 | oveq1i 7160 | . 2 ⊢ (((VtxDeg‘𝐺)‘𝑈) + 1) = (𝑃 + 1) |
50 | 37, 47, 49 | 3eqtri 2848 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∉ wnel 3123 {crab 3142 Vcvv 3495 ∖ cdif 3933 ∪ cun 3934 ∅c0 4291 𝒫 cpw 4539 {csn 4561 {cpr 4563 〈cop 4567 class class class wbr 5059 dom cdm 5550 Fun wfun 6344 ‘cfv 6350 (class class class)co 7150 Fincfn 8503 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 ≤ cle 10670 2c2 11686 ℕ0cn0 11891 +𝑒 cxad 12499 ..^cfzo 13027 ♯chash 13684 Word cword 13855 ++ cconcat 13916 〈“cs1 13943 Vtxcvtx 26775 iEdgciedg 26776 VtxDegcvtxdg 27241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-xadd 12502 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-vtx 26777 df-iedg 26778 df-vtxdg 27242 |
This theorem is referenced by: vdegp1ci 27314 konigsberglem1 28025 konigsberglem2 28026 |
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