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Mirrors > Home > MPE Home > Th. List > vtxdgfival | Structured version Visualization version GIF version |
Description: The degree of a vertex for graphs of finite size. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 8-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
Ref | Expression |
---|---|
vtxdgval.v | β’ π = (VtxβπΊ) |
vtxdgval.i | β’ πΌ = (iEdgβπΊ) |
vtxdgval.a | β’ π΄ = dom πΌ |
Ref | Expression |
---|---|
vtxdgfival | β’ ((π΄ β Fin β§ π β π) β ((VtxDegβπΊ)βπ) = ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) + (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxdgval.v | . . . 4 β’ π = (VtxβπΊ) | |
2 | vtxdgval.i | . . . 4 β’ πΌ = (iEdgβπΊ) | |
3 | vtxdgval.a | . . . 4 β’ π΄ = dom πΌ | |
4 | 1, 2, 3 | vtxdgval 29326 | . . 3 β’ (π β π β ((VtxDegβπΊ)βπ) = ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) +π (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}))) |
5 | 4 | adantl 480 | . 2 β’ ((π΄ β Fin β§ π β π) β ((VtxDegβπΊ)βπ) = ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) +π (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}))) |
6 | rabfi 9292 | . . . . . . 7 β’ (π΄ β Fin β {π₯ β π΄ β£ π β (πΌβπ₯)} β Fin) | |
7 | hashcl 14347 | . . . . . . 7 β’ ({π₯ β π΄ β£ π β (πΌβπ₯)} β Fin β (β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) β β0) | |
8 | 6, 7 | syl 17 | . . . . . 6 β’ (π΄ β Fin β (β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) β β0) |
9 | 8 | nn0red 12563 | . . . . 5 β’ (π΄ β Fin β (β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) β β) |
10 | rabfi 9292 | . . . . . . 7 β’ (π΄ β Fin β {π₯ β π΄ β£ (πΌβπ₯) = {π}} β Fin) | |
11 | hashcl 14347 | . . . . . . 7 β’ ({π₯ β π΄ β£ (πΌβπ₯) = {π}} β Fin β (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}) β β0) | |
12 | 10, 11 | syl 17 | . . . . . 6 β’ (π΄ β Fin β (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}) β β0) |
13 | 12 | nn0red 12563 | . . . . 5 β’ (π΄ β Fin β (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}) β β) |
14 | 9, 13 | jca 510 | . . . 4 β’ (π΄ β Fin β ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) β β β§ (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}) β β)) |
15 | 14 | adantr 479 | . . 3 β’ ((π΄ β Fin β§ π β π) β ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) β β β§ (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}) β β)) |
16 | rexadd 13243 | . . 3 β’ (((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) β β β§ (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}) β β) β ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) +π (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}})) = ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) + (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}))) | |
17 | 15, 16 | syl 17 | . 2 β’ ((π΄ β Fin β§ π β π) β ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) +π (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}})) = ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) + (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}))) |
18 | 5, 17 | eqtrd 2765 | 1 β’ ((π΄ β Fin β§ π β π) β ((VtxDegβπΊ)βπ) = ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) + (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {crab 3419 {csn 4624 dom cdm 5672 βcfv 6543 (class class class)co 7416 Fincfn 8962 βcr 11137 + caddc 11141 β0cn0 12502 +π cxad 13122 β―chash 14321 Vtxcvtx 28853 iEdgciedg 28854 VtxDegcvtxdg 29323 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-xadd 13125 df-hash 14322 df-vtxdg 29324 |
This theorem is referenced by: vtxdg0e 29332 vtxdgfisnn0 29333 finsumvtxdg2ssteplem2 29404 |
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