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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 29234 | . . 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 481 | . 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 9271 | . . . . . . 7 β’ (π΄ β Fin β {π₯ β π΄ β£ π β (πΌβπ₯)} β Fin) | |
7 | hashcl 14321 | . . . . . . 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 12537 | . . . . 5 β’ (π΄ β Fin β (β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) β β) |
10 | rabfi 9271 | . . . . . . 7 β’ (π΄ β Fin β {π₯ β π΄ β£ (πΌβπ₯) = {π}} β Fin) | |
11 | hashcl 14321 | . . . . . . 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 12537 | . . . . 5 β’ (π΄ β Fin β (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}) β β) |
14 | 9, 13 | jca 511 | . . . 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 480 | . . 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 13217 | . . 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 2766 | 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 395 = wceq 1533 β wcel 2098 {crab 3426 {csn 4623 dom cdm 5669 βcfv 6537 (class class class)co 7405 Fincfn 8941 βcr 11111 + caddc 11115 β0cn0 12476 +π cxad 13096 β―chash 14295 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-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 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-z 12563 df-uz 12827 df-xadd 13099 df-hash 14296 df-vtxdg 29232 |
This theorem is referenced by: vtxdg0e 29240 vtxdgfisnn0 29241 finsumvtxdg2ssteplem2 29312 |
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