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Mirrors > Home > MPE Home > Th. List > p1evtxdeq | Structured version Visualization version GIF version |
Description: If an edge πΈ which does not contain vertex π is added to a graph πΊ (yielding a graph πΉ), the degree of π is the same in both graphs. (Contributed by AV, 2-Mar-2021.) |
Ref | Expression |
---|---|
p1evtxdeq.v | β’ π = (VtxβπΊ) |
p1evtxdeq.i | β’ πΌ = (iEdgβπΊ) |
p1evtxdeq.f | β’ (π β Fun πΌ) |
p1evtxdeq.fv | β’ (π β (VtxβπΉ) = π) |
p1evtxdeq.fi | β’ (π β (iEdgβπΉ) = (πΌ βͺ {β¨πΎ, πΈβ©})) |
p1evtxdeq.k | β’ (π β πΎ β π) |
p1evtxdeq.d | β’ (π β πΎ β dom πΌ) |
p1evtxdeq.u | β’ (π β π β π) |
p1evtxdeq.e | β’ (π β πΈ β π) |
p1evtxdeq.n | β’ (π β π β πΈ) |
Ref | Expression |
---|---|
p1evtxdeq | β’ (π β ((VtxDegβπΉ)βπ) = ((VtxDegβπΊ)βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | p1evtxdeq.v | . . 3 β’ π = (VtxβπΊ) | |
2 | p1evtxdeq.i | . . 3 β’ πΌ = (iEdgβπΊ) | |
3 | p1evtxdeq.f | . . 3 β’ (π β Fun πΌ) | |
4 | p1evtxdeq.fv | . . 3 β’ (π β (VtxβπΉ) = π) | |
5 | p1evtxdeq.fi | . . 3 β’ (π β (iEdgβπΉ) = (πΌ βͺ {β¨πΎ, πΈβ©})) | |
6 | p1evtxdeq.k | . . 3 β’ (π β πΎ β π) | |
7 | p1evtxdeq.d | . . 3 β’ (π β πΎ β dom πΌ) | |
8 | p1evtxdeq.u | . . 3 β’ (π β π β π) | |
9 | p1evtxdeq.e | . . 3 β’ (π β πΈ β π) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | p1evtxdeqlem 27924 | . 2 β’ (π β ((VtxDegβπΉ)βπ) = (((VtxDegβπΊ)βπ) +π ((VtxDegββ¨π, {β¨πΎ, πΈβ©}β©)βπ))) |
11 | 1 | fvexi 6818 | . . . . . 6 β’ π β V |
12 | snex 5363 | . . . . . 6 β’ {β¨πΎ, πΈβ©} β V | |
13 | 11, 12 | pm3.2i 472 | . . . . 5 β’ (π β V β§ {β¨πΎ, πΈβ©} β V) |
14 | opiedgfv 27422 | . . . . 5 β’ ((π β V β§ {β¨πΎ, πΈβ©} β V) β (iEdgββ¨π, {β¨πΎ, πΈβ©}β©) = {β¨πΎ, πΈβ©}) | |
15 | 13, 14 | mp1i 13 | . . . 4 β’ (π β (iEdgββ¨π, {β¨πΎ, πΈβ©}β©) = {β¨πΎ, πΈβ©}) |
16 | opvtxfv 27419 | . . . . 5 β’ ((π β V β§ {β¨πΎ, πΈβ©} β V) β (Vtxββ¨π, {β¨πΎ, πΈβ©}β©) = π) | |
17 | 13, 16 | mp1i 13 | . . . 4 β’ (π β (Vtxββ¨π, {β¨πΎ, πΈβ©}β©) = π) |
18 | p1evtxdeq.n | . . . 4 β’ (π β π β πΈ) | |
19 | 15, 17, 6, 8, 9, 18 | 1hevtxdg0 27917 | . . 3 β’ (π β ((VtxDegββ¨π, {β¨πΎ, πΈβ©}β©)βπ) = 0) |
20 | 19 | oveq2d 7323 | . 2 β’ (π β (((VtxDegβπΊ)βπ) +π ((VtxDegββ¨π, {β¨πΎ, πΈβ©}β©)βπ)) = (((VtxDegβπΊ)βπ) +π 0)) |
21 | 1 | vtxdgelxnn0 27884 | . . . 4 β’ (π β π β ((VtxDegβπΊ)βπ) β β0*) |
22 | xnn0xr 12356 | . . . 4 β’ (((VtxDegβπΊ)βπ) β β0* β ((VtxDegβπΊ)βπ) β β*) | |
23 | 8, 21, 22 | 3syl 18 | . . 3 β’ (π β ((VtxDegβπΊ)βπ) β β*) |
24 | 23 | xaddid1d 13023 | . 2 β’ (π β (((VtxDegβπΊ)βπ) +π 0) = ((VtxDegβπΊ)βπ)) |
25 | 10, 20, 24 | 3eqtrd 2780 | 1 β’ (π β ((VtxDegβπΉ)βπ) = ((VtxDegβπΊ)βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1539 β wcel 2104 β wnel 3047 Vcvv 3437 βͺ cun 3890 {csn 4565 β¨cop 4571 dom cdm 5600 Fun wfun 6452 βcfv 6458 (class class class)co 7307 0cc0 10917 β*cxr 11054 β0*cxnn0 12351 +π cxad 12892 Vtxcvtx 27411 iEdgciedg 27412 VtxDegcvtxdg 27877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-oadd 8332 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-dju 9703 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-n0 12280 df-xnn0 12352 df-z 12366 df-uz 12629 df-xadd 12895 df-fz 13286 df-hash 14091 df-vtx 27413 df-iedg 27414 df-vtxdg 27878 |
This theorem is referenced by: vdegp1ai 27948 |
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