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Mirrors > Home > MPE Home > Th. List > vdegp1ai | Structured version Visualization version GIF version |
Description: The induction step for a vertex degree calculation. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑋, 𝑌} to the edge set, where 𝑋 ≠ 𝑈 ≠ 𝑌, yields degree 𝑃 as well. (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 |
vdegp1ai.x | ⊢ 𝑋 ∈ 𝑉 |
vdegp1ai.xu | ⊢ 𝑋 ≠ 𝑈 |
vdegp1ai.y | ⊢ 𝑌 ∈ 𝑉 |
vdegp1ai.yu | ⊢ 𝑌 ≠ 𝑈 |
vdegp1ai.f | ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑋, 𝑌}”〉) |
Ref | Expression |
---|---|
vdegp1ai | ⊢ ((VtxDeg‘𝐹)‘𝑈) = 𝑃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5323 | . . 3 ⊢ {𝑋, 𝑌} ∈ 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 13854 | . . . . . 6 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼:(0..^(♯‘𝐼))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
6 | 5 | ffund 6511 | . . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → Fun 𝐼) |
7 | 4, 6 | mp1i 13 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → Fun 𝐼) |
8 | vdegp1ai.vf | . . . . 5 ⊢ (Vtx‘𝐹) = 𝑉 | |
9 | 8 | a1i 11 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (Vtx‘𝐹) = 𝑉) |
10 | vdegp1ai.f | . . . . 5 ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑋, 𝑌}”〉) | |
11 | wrdv 13865 | . . . . . . 7 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼 ∈ Word V) | |
12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
13 | cats1un 14071 | . . . . . 6 ⊢ ((𝐼 ∈ Word V ∧ {𝑋, 𝑌} ∈ V) → (𝐼 ++ 〈“{𝑋, 𝑌}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑋, 𝑌}〉})) | |
14 | 12, 13 | mpan 686 | . . . . 5 ⊢ ({𝑋, 𝑌} ∈ V → (𝐼 ++ 〈“{𝑋, 𝑌}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑋, 𝑌}〉})) |
15 | 10, 14 | syl5eq 2865 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (iEdg‘𝐹) = (𝐼 ∪ {〈(♯‘𝐼), {𝑋, 𝑌}〉})) |
16 | fvexd 6678 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (♯‘𝐼) ∈ V) | |
17 | wrdlndm 13867 | . . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (♯‘𝐼) ∉ dom 𝐼) | |
18 | 4, 17 | mp1i 13 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (♯‘𝐼) ∉ dom 𝐼) |
19 | vdegp1ai.u | . . . . 5 ⊢ 𝑈 ∈ 𝑉 | |
20 | 19 | a1i 11 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → 𝑈 ∈ 𝑉) |
21 | id 22 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → {𝑋, 𝑌} ∈ V) | |
22 | vdegp1ai.xu | . . . . . . 7 ⊢ 𝑋 ≠ 𝑈 | |
23 | 22 | necomi 3067 | . . . . . 6 ⊢ 𝑈 ≠ 𝑋 |
24 | vdegp1ai.yu | . . . . . . 7 ⊢ 𝑌 ≠ 𝑈 | |
25 | 24 | necomi 3067 | . . . . . 6 ⊢ 𝑈 ≠ 𝑌 |
26 | 23, 25 | prneli 4585 | . . . . 5 ⊢ 𝑈 ∉ {𝑋, 𝑌} |
27 | 26 | a1i 11 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → 𝑈 ∉ {𝑋, 𝑌}) |
28 | 2, 3, 7, 9, 15, 16, 18, 20, 21, 27 | p1evtxdeq 27222 | . . 3 ⊢ ({𝑋, 𝑌} ∈ V → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈)) |
29 | 1, 28 | ax-mp 5 | . 2 ⊢ ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈) |
30 | vdegp1ai.d | . 2 ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃 | |
31 | 29, 30 | eqtri 2841 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∉ wnel 3120 {crab 3139 Vcvv 3492 ∖ cdif 3930 ∪ cun 3931 ∅c0 4288 𝒫 cpw 4535 {csn 4557 {cpr 4559 〈cop 4563 class class class wbr 5057 dom cdm 5548 Fun wfun 6342 ‘cfv 6348 (class class class)co 7145 0cc0 10525 ≤ cle 10664 2c2 11680 ..^cfzo 13021 ♯chash 13678 Word cword 13849 ++ cconcat 13910 〈“cs1 13937 Vtxcvtx 26708 iEdgciedg 26709 VtxDegcvtxdg 27174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-xadd 12496 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-vtx 26710 df-iedg 26711 df-vtxdg 27175 |
This theorem is referenced by: konigsberglem1 27958 konigsberglem2 27959 konigsberglem3 27960 |
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