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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 π in the edge set πΈ is π, then adding {π, π} to the edge set, where π β π β π, yields degree π 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 | β’ π = (VtxβπΊ) |
vdegp1ai.u | β’ π β π |
vdegp1ai.i | β’ πΌ = (iEdgβπΊ) |
vdegp1ai.w | β’ πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} |
vdegp1ai.d | β’ ((VtxDegβπΊ)βπ) = π |
vdegp1ai.vf | β’ (VtxβπΉ) = π |
vdegp1ai.x | β’ π β π |
vdegp1ai.xu | β’ π β π |
vdegp1ai.y | β’ π β π |
vdegp1ai.yu | β’ π β π |
vdegp1ai.f | β’ (iEdgβπΉ) = (πΌ ++ β¨β{π, π}ββ©) |
Ref | Expression |
---|---|
vdegp1ai | β’ ((VtxDegβπΉ)βπ) = π |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5438 | . . 3 β’ {π, π} β V | |
2 | vdegp1ai.vg | . . . 4 β’ π = (VtxβπΊ) | |
3 | vdegp1ai.i | . . . 4 β’ πΌ = (iEdgβπΊ) | |
4 | vdegp1ai.w | . . . . 5 β’ πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} | |
5 | wrdf 14511 | . . . . . 6 β’ (πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} β πΌ:(0..^(β―βπΌ))βΆ{π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2}) | |
6 | 5 | ffund 6731 | . . . . 5 β’ (πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} β Fun πΌ) |
7 | 4, 6 | mp1i 13 | . . . 4 β’ ({π, π} β V β Fun πΌ) |
8 | vdegp1ai.vf | . . . . 5 β’ (VtxβπΉ) = π | |
9 | 8 | a1i 11 | . . . 4 β’ ({π, π} β V β (VtxβπΉ) = π) |
10 | vdegp1ai.f | . . . . 5 β’ (iEdgβπΉ) = (πΌ ++ β¨β{π, π}ββ©) | |
11 | wrdv 14521 | . . . . . . 7 β’ (πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} β πΌ β Word V) | |
12 | 4, 11 | ax-mp 5 | . . . . . 6 β’ πΌ β Word V |
13 | cats1un 14713 | . . . . . 6 β’ ((πΌ β Word V β§ {π, π} β V) β (πΌ ++ β¨β{π, π}ββ©) = (πΌ βͺ {β¨(β―βπΌ), {π, π}β©})) | |
14 | 12, 13 | mpan 688 | . . . . 5 β’ ({π, π} β V β (πΌ ++ β¨β{π, π}ββ©) = (πΌ βͺ {β¨(β―βπΌ), {π, π}β©})) |
15 | 10, 14 | eqtrid 2780 | . . . 4 β’ ({π, π} β V β (iEdgβπΉ) = (πΌ βͺ {β¨(β―βπΌ), {π, π}β©})) |
16 | fvexd 6917 | . . . 4 β’ ({π, π} β V β (β―βπΌ) β V) | |
17 | wrdlndm 14522 | . . . . 5 β’ (πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} β (β―βπΌ) β dom πΌ) | |
18 | 4, 17 | mp1i 13 | . . . 4 β’ ({π, π} β V β (β―βπΌ) β dom πΌ) |
19 | vdegp1ai.u | . . . . 5 β’ π β π | |
20 | 19 | a1i 11 | . . . 4 β’ ({π, π} β V β π β π) |
21 | id 22 | . . . 4 β’ ({π, π} β V β {π, π} β V) | |
22 | vdegp1ai.xu | . . . . . . 7 β’ π β π | |
23 | 22 | necomi 2992 | . . . . . 6 β’ π β π |
24 | vdegp1ai.yu | . . . . . . 7 β’ π β π | |
25 | 24 | necomi 2992 | . . . . . 6 β’ π β π |
26 | 23, 25 | prneli 4663 | . . . . 5 β’ π β {π, π} |
27 | 26 | a1i 11 | . . . 4 β’ ({π, π} β V β π β {π, π}) |
28 | 2, 3, 7, 9, 15, 16, 18, 20, 21, 27 | p1evtxdeq 29355 | . . 3 β’ ({π, π} β V β ((VtxDegβπΉ)βπ) = ((VtxDegβπΊ)βπ)) |
29 | 1, 28 | ax-mp 5 | . 2 β’ ((VtxDegβπΉ)βπ) = ((VtxDegβπΊ)βπ) |
30 | vdegp1ai.d | . 2 β’ ((VtxDegβπΊ)βπ) = π | |
31 | 29, 30 | eqtri 2756 | 1 β’ ((VtxDegβπΉ)βπ) = π |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β wcel 2098 β wne 2937 β wnel 3043 {crab 3430 Vcvv 3473 β cdif 3946 βͺ cun 3947 β c0 4326 π« cpw 4606 {csn 4632 {cpr 4634 β¨cop 4638 class class class wbr 5152 dom cdm 5682 Fun wfun 6547 βcfv 6553 (class class class)co 7426 0cc0 11148 β€ cle 11289 2c2 12307 ..^cfzo 13669 β―chash 14331 Word cword 14506 ++ cconcat 14562 β¨βcs1 14587 Vtxcvtx 28837 iEdgciedg 28838 VtxDegcvtxdg 29307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-oadd 8499 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-dju 9934 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-n0 12513 df-xnn0 12585 df-z 12599 df-uz 12863 df-xadd 13135 df-fz 13527 df-fzo 13670 df-hash 14332 df-word 14507 df-concat 14563 df-s1 14588 df-vtx 28839 df-iedg 28840 df-vtxdg 29308 |
This theorem is referenced by: konigsberglem1 30090 konigsberglem2 30091 konigsberglem3 30092 |
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