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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 5335 | . . 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 13869 | . . . . . 6 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼:(0..^(♯‘𝐼))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
6 | 5 | ffund 6520 | . . . . 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 13880 | . . . . . . 7 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼 ∈ Word V) | |
12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
13 | cats1un 14085 | . . . . . 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 2870 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (iEdg‘𝐹) = (𝐼 ∪ {〈(♯‘𝐼), {𝑈, 𝑋}〉})) |
16 | fvexd 6687 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (♯‘𝐼) ∈ V) | |
17 | wrdlndm 13881 | . . . . 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 5342 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → {𝑈, 𝑋} ∈ 𝒫 𝑉) | |
24 | 22, 23 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → {𝑈, 𝑋} ∈ 𝒫 𝑉) |
25 | prid1g 4698 | . . . . 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 3072 | . . . . . . 7 ⊢ 𝑈 ≠ 𝑋 |
29 | hashprg 13759 | . . . . . . . 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 2832 | . . . . 5 ⊢ 2 = (♯‘{𝑈, 𝑋}) |
33 | 2re 11714 | . . . . . 6 ⊢ 2 ∈ ℝ | |
34 | 33 | eqlei 10752 | . . . . 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 27298 | . . 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 13345 | . . . . 5 ⊢ (0..^(♯‘𝐼)) ∈ Fin | |
39 | wrddm 13871 | . . . . . . . 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 2832 | . . . . . 6 ⊢ (0..^(♯‘𝐼)) = dom 𝐼 |
42 | 2, 3, 41 | vtxdgfisnn0 27259 | . . . . 5 ⊢ (((0..^(♯‘𝐼)) ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0) |
43 | 38, 19, 42 | mp2an 690 | . . . 4 ⊢ ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0 |
44 | 43 | nn0rei 11911 | . . 3 ⊢ ((VtxDeg‘𝐺)‘𝑈) ∈ ℝ |
45 | 1re 10643 | . . 3 ⊢ 1 ∈ ℝ | |
46 | rexadd 12628 | . . 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 7168 | . 2 ⊢ (((VtxDeg‘𝐺)‘𝑈) + 1) = (𝑃 + 1) |
50 | 37, 47, 49 | 3eqtri 2850 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∉ wnel 3125 {crab 3144 Vcvv 3496 ∖ cdif 3935 ∪ cun 3936 ∅c0 4293 𝒫 cpw 4541 {csn 4569 {cpr 4571 〈cop 4575 class class class wbr 5068 dom cdm 5557 Fun wfun 6351 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 ≤ cle 10678 2c2 11695 ℕ0cn0 11900 +𝑒 cxad 12508 ..^cfzo 13036 ♯chash 13693 Word cword 13864 ++ cconcat 13924 〈“cs1 13951 Vtxcvtx 26783 iEdgciedg 26784 VtxDegcvtxdg 27249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-xadd 12511 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-vtx 26785 df-iedg 26786 df-vtxdg 27250 |
This theorem is referenced by: vdegp1ci 27322 konigsberglem1 28033 konigsberglem2 28034 |
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