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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 4939 | . . 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 13342 | . . . . . 6 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝐼:(0..^(#‘𝐼))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) | |
6 | ffun 6086 | . . . . . 6 ⊢ (𝐼:(0..^(#‘𝐼))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → Fun 𝐼) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → Fun 𝐼) |
8 | 4, 7 | mp1i 13 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → Fun 𝐼) |
9 | vdegp1ai.vf | . . . . 5 ⊢ (Vtx‘𝐹) = 𝑉 | |
10 | 9 | a1i 11 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (Vtx‘𝐹) = 𝑉) |
11 | vdegp1ai.f | . . . . 5 ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑋, 𝑌}”〉) | |
12 | wrdv 13352 | . . . . . . 7 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝐼 ∈ Word V) | |
13 | 4, 12 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
14 | cats1un 13521 | . . . . . 6 ⊢ ((𝐼 ∈ Word V ∧ {𝑋, 𝑌} ∈ V) → (𝐼 ++ 〈“{𝑋, 𝑌}”〉) = (𝐼 ∪ {〈(#‘𝐼), {𝑋, 𝑌}〉})) | |
15 | 13, 14 | mpan 706 | . . . . 5 ⊢ ({𝑋, 𝑌} ∈ V → (𝐼 ++ 〈“{𝑋, 𝑌}”〉) = (𝐼 ∪ {〈(#‘𝐼), {𝑋, 𝑌}〉})) |
16 | 11, 15 | syl5eq 2697 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (iEdg‘𝐹) = (𝐼 ∪ {〈(#‘𝐼), {𝑋, 𝑌}〉})) |
17 | fvexd 6241 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (#‘𝐼) ∈ V) | |
18 | wrdlndm 13353 | . . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → (#‘𝐼) ∉ dom 𝐼) | |
19 | 4, 18 | mp1i 13 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (#‘𝐼) ∉ dom 𝐼) |
20 | vdegp1ai.u | . . . . 5 ⊢ 𝑈 ∈ 𝑉 | |
21 | 20 | a1i 11 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → 𝑈 ∈ 𝑉) |
22 | id 22 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → {𝑋, 𝑌} ∈ V) | |
23 | vdegp1ai.xu | . . . . . . 7 ⊢ 𝑋 ≠ 𝑈 | |
24 | 23 | necomi 2877 | . . . . . 6 ⊢ 𝑈 ≠ 𝑋 |
25 | vdegp1ai.yu | . . . . . . 7 ⊢ 𝑌 ≠ 𝑈 | |
26 | 25 | necomi 2877 | . . . . . 6 ⊢ 𝑈 ≠ 𝑌 |
27 | 24, 26 | prneli 4235 | . . . . 5 ⊢ 𝑈 ∉ {𝑋, 𝑌} |
28 | 27 | a1i 11 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → 𝑈 ∉ {𝑋, 𝑌}) |
29 | 2, 3, 8, 10, 16, 17, 19, 21, 22, 28 | p1evtxdeq 26465 | . . 3 ⊢ ({𝑋, 𝑌} ∈ V → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈)) |
30 | 1, 29 | ax-mp 5 | . 2 ⊢ ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈) |
31 | vdegp1ai.d | . 2 ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃 | |
32 | 30, 31 | eqtri 2673 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∉ wnel 2926 {crab 2945 Vcvv 3231 ∖ cdif 3604 ∪ cun 3605 ∅c0 3948 𝒫 cpw 4191 {csn 4210 {cpr 4212 〈cop 4216 class class class wbr 4685 dom cdm 5143 Fun wfun 5920 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 0cc0 9974 ≤ cle 10113 2c2 11108 ..^cfzo 12504 #chash 13157 Word cword 13323 ++ cconcat 13325 〈“cs1 13326 Vtxcvtx 25919 iEdgciedg 25920 VtxDegcvtxdg 26417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-xnn0 11402 df-z 11416 df-uz 11726 df-xadd 11985 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 df-concat 13333 df-s1 13334 df-vtx 25921 df-iedg 25922 df-vtxdg 26418 |
This theorem is referenced by: konigsberglem1 27230 konigsberglem2 27231 konigsberglem3 27232 |
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