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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 5425 | . . 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 14475 | . . . . . 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 6715 | . . . . 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 14485 | . . . . . . 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 14677 | . . . . . 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 687 | . . . . 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 2778 | . . . 4 β’ ({π, π} β V β (iEdgβπΉ) = (πΌ βͺ {β¨(β―βπΌ), {π, π}β©})) |
16 | fvexd 6900 | . . . 4 β’ ({π, π} β V β (β―βπΌ) β V) | |
17 | wrdlndm 14486 | . . . . 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 2989 | . . . . . 6 β’ π β π |
24 | vdegp1ai.yu | . . . . . . 7 β’ π β π | |
25 | 24 | necomi 2989 | . . . . . 6 β’ π β π |
26 | 23, 25 | prneli 4653 | . . . . 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 29279 | . . 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 2754 | 1 β’ ((VtxDegβπΉ)βπ) = π |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β wcel 2098 β wne 2934 β wnel 3040 {crab 3426 Vcvv 3468 β cdif 3940 βͺ cun 3941 β c0 4317 π« cpw 4597 {csn 4623 {cpr 4625 β¨cop 4629 class class class wbr 5141 dom cdm 5669 Fun wfun 6531 βcfv 6537 (class class class)co 7405 0cc0 11112 β€ cle 11253 2c2 12271 ..^cfzo 13633 β―chash 14295 Word cword 14470 ++ cconcat 14526 β¨βcs1 14551 Vtxcvtx 28764 iEdgciedg 28765 VtxDegcvtxdg 29231 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-xadd 13099 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-concat 14527 df-s1 14552 df-vtx 28766 df-iedg 28767 df-vtxdg 29232 |
This theorem is referenced by: konigsberglem1 30014 konigsberglem2 30015 konigsberglem3 30016 |
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