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Mirrors > Home > MPE Home > Th. List > vtxdginducedm1fi | Structured version Visualization version GIF version |
Description: The degree of a vertex 𝑣 in the induced subgraph 𝑆 of a pseudograph 𝐺 of finite size obtained by removing one vertex 𝑁 plus the number of edges joining the vertex 𝑣 and the vertex 𝑁 is the degree of the vertex 𝑣 in the pseudograph 𝐺. (Contributed by AV, 18-Dec-2021.) |
Ref | Expression |
---|---|
vtxdginducedm1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
vtxdginducedm1.e | ⊢ 𝐸 = (iEdg‘𝐺) |
vtxdginducedm1.k | ⊢ 𝐾 = (𝑉 ∖ {𝑁}) |
vtxdginducedm1.i | ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} |
vtxdginducedm1.p | ⊢ 𝑃 = (𝐸 ↾ 𝐼) |
vtxdginducedm1.s | ⊢ 𝑆 = 〈𝐾, 𝑃〉 |
vtxdginducedm1.j | ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} |
Ref | Expression |
---|---|
vtxdginducedm1fi | ⊢ (𝐸 ∈ Fin → ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxdginducedm1.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | vtxdginducedm1.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | vtxdginducedm1.k | . . 3 ⊢ 𝐾 = (𝑉 ∖ {𝑁}) | |
4 | vtxdginducedm1.i | . . 3 ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} | |
5 | vtxdginducedm1.p | . . 3 ⊢ 𝑃 = (𝐸 ↾ 𝐼) | |
6 | vtxdginducedm1.s | . . 3 ⊢ 𝑆 = 〈𝐾, 𝑃〉 | |
7 | vtxdginducedm1.j | . . 3 ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} | |
8 | 1, 2, 3, 4, 5, 6, 7 | vtxdginducedm1 27333 | . 2 ⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})) |
9 | 5 | dmeqi 5737 | . . . . . . . . 9 ⊢ dom 𝑃 = dom (𝐸 ↾ 𝐼) |
10 | finresfin 8728 | . . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → (𝐸 ↾ 𝐼) ∈ Fin) | |
11 | dmfi 8786 | . . . . . . . . . 10 ⊢ ((𝐸 ↾ 𝐼) ∈ Fin → dom (𝐸 ↾ 𝐼) ∈ Fin) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝐸 ∈ Fin → dom (𝐸 ↾ 𝐼) ∈ Fin) |
13 | 9, 12 | eqeltrid 2894 | . . . . . . . 8 ⊢ (𝐸 ∈ Fin → dom 𝑃 ∈ Fin) |
14 | 6 | fveq2i 6648 | . . . . . . . . . 10 ⊢ (Vtx‘𝑆) = (Vtx‘〈𝐾, 𝑃〉) |
15 | 1 | fvexi 6659 | . . . . . . . . . . . . 13 ⊢ 𝑉 ∈ V |
16 | 15 | difexi 5196 | . . . . . . . . . . . 12 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
17 | 3, 16 | eqeltri 2886 | . . . . . . . . . . 11 ⊢ 𝐾 ∈ V |
18 | 2 | fvexi 6659 | . . . . . . . . . . . . 13 ⊢ 𝐸 ∈ V |
19 | 18 | resex 5866 | . . . . . . . . . . . 12 ⊢ (𝐸 ↾ 𝐼) ∈ V |
20 | 5, 19 | eqeltri 2886 | . . . . . . . . . . 11 ⊢ 𝑃 ∈ V |
21 | 17, 20 | opvtxfvi 26802 | . . . . . . . . . 10 ⊢ (Vtx‘〈𝐾, 𝑃〉) = 𝐾 |
22 | 14, 21, 3 | 3eqtrri 2826 | . . . . . . . . 9 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
23 | 1, 2, 3, 4, 5, 6 | vtxdginducedm1lem1 27329 | . . . . . . . . . 10 ⊢ (iEdg‘𝑆) = 𝑃 |
24 | 23 | eqcomi 2807 | . . . . . . . . 9 ⊢ 𝑃 = (iEdg‘𝑆) |
25 | eqid 2798 | . . . . . . . . 9 ⊢ dom 𝑃 = dom 𝑃 | |
26 | 22, 24, 25 | vtxdgfisnn0 27265 | . . . . . . . 8 ⊢ ((dom 𝑃 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℕ0) |
27 | 13, 26 | sylan 583 | . . . . . . 7 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℕ0) |
28 | 27 | nn0red 11944 | . . . . . 6 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℝ) |
29 | dmfi 8786 | . . . . . . . . . . 11 ⊢ (𝐸 ∈ Fin → dom 𝐸 ∈ Fin) | |
30 | rabfi 8727 | . . . . . . . . . . 11 ⊢ (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} ∈ Fin) | |
31 | 29, 30 | syl 17 | . . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} ∈ Fin) |
32 | 7, 31 | eqeltrid 2894 | . . . . . . . . 9 ⊢ (𝐸 ∈ Fin → 𝐽 ∈ Fin) |
33 | rabfi 8727 | . . . . . . . . 9 ⊢ (𝐽 ∈ Fin → {𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)} ∈ Fin) | |
34 | hashcl 13713 | . . . . . . . . 9 ⊢ ({𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)} ∈ Fin → (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}) ∈ ℕ0) | |
35 | 32, 33, 34 | 3syl 18 | . . . . . . . 8 ⊢ (𝐸 ∈ Fin → (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}) ∈ ℕ0) |
36 | 35 | adantr 484 | . . . . . . 7 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}) ∈ ℕ0) |
37 | 36 | nn0red 11944 | . . . . . 6 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}) ∈ ℝ) |
38 | 28, 37 | rexaddd 12615 | . . . . 5 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})) = (((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))) |
39 | 38 | eqeq2d 2809 | . . . 4 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})) ↔ ((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))) |
40 | 39 | biimpd 232 | . . 3 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})) → ((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))) |
41 | 40 | ralimdva 3144 | . 2 ⊢ (𝐸 ∈ Fin → (∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))) |
42 | 8, 41 | mpi 20 | 1 ⊢ (𝐸 ∈ Fin → ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 ∀wral 3106 {crab 3110 Vcvv 3441 ∖ cdif 3878 {csn 4525 〈cop 4531 dom cdm 5519 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 + caddc 10529 ℕ0cn0 11885 +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-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: finsumvtxdg2ssteplem4 27338 |
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