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Mirrors > Home > MPE Home > Th. List > p1evtxdp1 | Structured version Visualization version GIF version |
Description: If an edge 𝐸 (not being a loop) which contains vertex 𝑈 is added to a graph 𝐺 (yielding a graph 𝐹), the degree of 𝑈 is increased by 1. (Contributed by AV, 3-Mar-2021.) |
Ref | Expression |
---|---|
p1evtxdeq.v | ⊢ 𝑉 = (Vtx‘𝐺) |
p1evtxdeq.i | ⊢ 𝐼 = (iEdg‘𝐺) |
p1evtxdeq.f | ⊢ (𝜑 → Fun 𝐼) |
p1evtxdeq.fv | ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉) |
p1evtxdeq.fi | ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {〈𝐾, 𝐸〉})) |
p1evtxdeq.k | ⊢ (𝜑 → 𝐾 ∈ 𝑋) |
p1evtxdeq.d | ⊢ (𝜑 → 𝐾 ∉ dom 𝐼) |
p1evtxdeq.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
p1evtxdp1.e | ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) |
p1evtxdp1.n | ⊢ (𝜑 → 𝑈 ∈ 𝐸) |
p1evtxdp1.l | ⊢ (𝜑 → 2 ≤ (♯‘𝐸)) |
Ref | Expression |
---|---|
p1evtxdp1 | ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | p1evtxdeq.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | p1evtxdeq.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | p1evtxdeq.f | . . 3 ⊢ (𝜑 → Fun 𝐼) | |
4 | p1evtxdeq.fv | . . 3 ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉) | |
5 | p1evtxdeq.fi | . . 3 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {〈𝐾, 𝐸〉})) | |
6 | p1evtxdeq.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑋) | |
7 | p1evtxdeq.d | . . 3 ⊢ (𝜑 → 𝐾 ∉ dom 𝐼) | |
8 | p1evtxdeq.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
9 | p1evtxdp1.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | p1evtxdeqlem 27302 | . 2 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈))) |
11 | 1 | fvexi 6659 | . . . . . 6 ⊢ 𝑉 ∈ V |
12 | snex 5297 | . . . . . 6 ⊢ {〈𝐾, 𝐸〉} ∈ V | |
13 | 11, 12 | pm3.2i 474 | . . . . 5 ⊢ (𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) |
14 | opiedgfv 26800 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) → (iEdg‘〈𝑉, {〈𝐾, 𝐸〉}〉) = {〈𝐾, 𝐸〉}) | |
15 | 13, 14 | mp1i 13 | . . . 4 ⊢ (𝜑 → (iEdg‘〈𝑉, {〈𝐾, 𝐸〉}〉) = {〈𝐾, 𝐸〉}) |
16 | opvtxfv 26797 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) → (Vtx‘〈𝑉, {〈𝐾, 𝐸〉}〉) = 𝑉) | |
17 | 13, 16 | mp1i 13 | . . . 4 ⊢ (𝜑 → (Vtx‘〈𝑉, {〈𝐾, 𝐸〉}〉) = 𝑉) |
18 | p1evtxdp1.n | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐸) | |
19 | p1evtxdp1.l | . . . 4 ⊢ (𝜑 → 2 ≤ (♯‘𝐸)) | |
20 | 15, 17, 6, 8, 9, 18, 19 | 1hevtxdg1 27296 | . . 3 ⊢ (𝜑 → ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈) = 1) |
21 | 20 | oveq2d 7151 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈)) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 1)) |
22 | 10, 21 | eqtrd 2833 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 Vcvv 3441 ∪ cun 3879 𝒫 cpw 4497 {csn 4525 〈cop 4531 class class class wbr 5030 dom cdm 5519 Fun wfun 6318 ‘cfv 6324 (class class class)co 7135 1c1 10527 ≤ cle 10665 2c2 11680 +𝑒 cxad 12493 ♯chash 13686 Vtxcvtx 26789 iEdgciedg 26790 VtxDegcvtxdg 27255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-xadd 12496 df-fz 12886 df-hash 13687 df-vtx 26791 df-iedg 26792 df-vtxdg 27256 |
This theorem is referenced by: vdegp1bi 27327 |
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