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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 5310 | . . 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 14039 | . . . . . 6 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼:(0..^(♯‘𝐼))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
6 | 5 | ffund 6527 | . . . . 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 14049 | . . . . . . 7 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼 ∈ Word V) | |
12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
13 | cats1un 14251 | . . . . . 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 | syl5eq 2783 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (iEdg‘𝐹) = (𝐼 ∪ {〈(♯‘𝐼), {𝑈, 𝑋}〉})) |
16 | fvexd 6710 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (♯‘𝐼) ∈ V) | |
17 | wrdlndm 14050 | . . . . 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 474 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) |
23 | prelpwi 5317 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → {𝑈, 𝑋} ∈ 𝒫 𝑉) | |
24 | 22, 23 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → {𝑈, 𝑋} ∈ 𝒫 𝑉) |
25 | prid1g 4662 | . . . . 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 2986 | . . . . . . 7 ⊢ 𝑈 ≠ 𝑋 |
29 | hashprg 13927 | . . . . . . . 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 233 | . . . . . 6 ⊢ (♯‘{𝑈, 𝑋}) = 2 |
32 | 31 | eqcomi 2745 | . . . . 5 ⊢ 2 = (♯‘{𝑈, 𝑋}) |
33 | 2re 11869 | . . . . . 6 ⊢ 2 ∈ ℝ | |
34 | 33 | eqlei 10907 | . . . . 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 27556 | . . 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 13512 | . . . . 5 ⊢ (0..^(♯‘𝐼)) ∈ Fin | |
39 | wrddm 14041 | . . . . . . . 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 2745 | . . . . . 6 ⊢ (0..^(♯‘𝐼)) = dom 𝐼 |
42 | 2, 3, 41 | vtxdgfisnn0 27517 | . . . . 5 ⊢ (((0..^(♯‘𝐼)) ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0) |
43 | 38, 19, 42 | mp2an 692 | . . . 4 ⊢ ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0 |
44 | 43 | nn0rei 12066 | . . 3 ⊢ ((VtxDeg‘𝐺)‘𝑈) ∈ ℝ |
45 | 1re 10798 | . . 3 ⊢ 1 ∈ ℝ | |
46 | rexadd 12787 | . . 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 7201 | . 2 ⊢ (((VtxDeg‘𝐺)‘𝑈) + 1) = (𝑃 + 1) |
50 | 37, 47, 49 | 3eqtri 2763 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∉ wnel 3036 {crab 3055 Vcvv 3398 ∖ cdif 3850 ∪ cun 3851 ∅c0 4223 𝒫 cpw 4499 {csn 4527 {cpr 4529 〈cop 4533 class class class wbr 5039 dom cdm 5536 Fun wfun 6352 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 ℝcr 10693 0cc0 10694 1c1 10695 + caddc 10697 ≤ cle 10833 2c2 11850 ℕ0cn0 12055 +𝑒 cxad 12667 ..^cfzo 13203 ♯chash 13861 Word cword 14034 ++ cconcat 14090 〈“cs1 14117 Vtxcvtx 27041 iEdgciedg 27042 VtxDegcvtxdg 27507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-n0 12056 df-xnn0 12128 df-z 12142 df-uz 12404 df-xadd 12670 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-concat 14091 df-s1 14118 df-vtx 27043 df-iedg 27044 df-vtxdg 27508 |
This theorem is referenced by: vdegp1ci 27580 konigsberglem1 28289 konigsberglem2 28290 |
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