Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > finsumvtxdg2ssteplem2 | Structured version Visualization version GIF version |
Description: Lemma for finsumvtxdg2sstep 28205. (Contributed by AV, 12-Dec-2021.) |
Ref | Expression |
---|---|
finsumvtxdg2sstep.v | β’ π = (VtxβπΊ) |
finsumvtxdg2sstep.e | β’ πΈ = (iEdgβπΊ) |
finsumvtxdg2sstep.k | β’ πΎ = (π β {π}) |
finsumvtxdg2sstep.i | β’ πΌ = {π β dom πΈ β£ π β (πΈβπ)} |
finsumvtxdg2sstep.p | β’ π = (πΈ βΎ πΌ) |
finsumvtxdg2sstep.s | β’ π = β¨πΎ, πβ© |
finsumvtxdg2ssteplem.j | β’ π½ = {π β dom πΈ β£ π β (πΈβπ)} |
Ref | Expression |
---|---|
finsumvtxdg2ssteplem2 | β’ (((πΊ β UPGraph β§ π β π) β§ (π β Fin β§ πΈ β Fin)) β ((VtxDegβπΊ)βπ) = ((β―βπ½) + (β―β{π β dom πΈ β£ (πΈβπ) = {π}}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmfi 9195 | . . . 4 β’ (πΈ β Fin β dom πΈ β Fin) | |
2 | 1 | adantl 482 | . . 3 β’ ((π β Fin β§ πΈ β Fin) β dom πΈ β Fin) |
3 | simpr 485 | . . 3 β’ ((πΊ β UPGraph β§ π β π) β π β π) | |
4 | finsumvtxdg2sstep.v | . . . 4 β’ π = (VtxβπΊ) | |
5 | finsumvtxdg2sstep.e | . . . 4 β’ πΈ = (iEdgβπΊ) | |
6 | eqid 2736 | . . . 4 β’ dom πΈ = dom πΈ | |
7 | 4, 5, 6 | vtxdgfival 28125 | . . 3 β’ ((dom πΈ β Fin β§ π β π) β ((VtxDegβπΊ)βπ) = ((β―β{π β dom πΈ β£ π β (πΈβπ)}) + (β―β{π β dom πΈ β£ (πΈβπ) = {π}}))) |
8 | 2, 3, 7 | syl2anr 597 | . 2 β’ (((πΊ β UPGraph β§ π β π) β§ (π β Fin β§ πΈ β Fin)) β ((VtxDegβπΊ)βπ) = ((β―β{π β dom πΈ β£ π β (πΈβπ)}) + (β―β{π β dom πΈ β£ (πΈβπ) = {π}}))) |
9 | finsumvtxdg2ssteplem.j | . . . . . 6 β’ π½ = {π β dom πΈ β£ π β (πΈβπ)} | |
10 | 9 | eqcomi 2745 | . . . . 5 β’ {π β dom πΈ β£ π β (πΈβπ)} = π½ |
11 | 10 | fveq2i 6828 | . . . 4 β’ (β―β{π β dom πΈ β£ π β (πΈβπ)}) = (β―βπ½) |
12 | 11 | a1i 11 | . . 3 β’ (((πΊ β UPGraph β§ π β π) β§ (π β Fin β§ πΈ β Fin)) β (β―β{π β dom πΈ β£ π β (πΈβπ)}) = (β―βπ½)) |
13 | 12 | oveq1d 7352 | . 2 β’ (((πΊ β UPGraph β§ π β π) β§ (π β Fin β§ πΈ β Fin)) β ((β―β{π β dom πΈ β£ π β (πΈβπ)}) + (β―β{π β dom πΈ β£ (πΈβπ) = {π}})) = ((β―βπ½) + (β―β{π β dom πΈ β£ (πΈβπ) = {π}}))) |
14 | 8, 13 | eqtrd 2776 | 1 β’ (((πΊ β UPGraph β§ π β π) β§ (π β Fin β§ πΈ β Fin)) β ((VtxDegβπΊ)βπ) = ((β―βπ½) + (β―β{π β dom πΈ β£ (πΈβπ) = {π}}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1540 β wcel 2105 β wnel 3046 {crab 3403 β cdif 3895 {csn 4573 β¨cop 4579 dom cdm 5620 βΎ cres 5622 βcfv 6479 (class class class)co 7337 Fincfn 8804 + caddc 10975 β―chash 14145 Vtxcvtx 27655 iEdgciedg 27656 UPGraphcupgr 27739 VtxDegcvtxdg 28121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-n0 12335 df-z 12421 df-uz 12684 df-xadd 12950 df-hash 14146 df-vtxdg 28122 |
This theorem is referenced by: finsumvtxdg2sstep 28205 |
Copyright terms: Public domain | W3C validator |