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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 27319 | . 2 ⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})) |
9 | 5 | dmeqi 5767 | . . . . . . . . 9 ⊢ dom 𝑃 = dom (𝐸 ↾ 𝐼) |
10 | finresfin 8738 | . . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → (𝐸 ↾ 𝐼) ∈ Fin) | |
11 | dmfi 8796 | . . . . . . . . . 10 ⊢ ((𝐸 ↾ 𝐼) ∈ Fin → dom (𝐸 ↾ 𝐼) ∈ Fin) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝐸 ∈ Fin → dom (𝐸 ↾ 𝐼) ∈ Fin) |
13 | 9, 12 | eqeltrid 2917 | . . . . . . . 8 ⊢ (𝐸 ∈ Fin → dom 𝑃 ∈ Fin) |
14 | 6 | fveq2i 6667 | . . . . . . . . . 10 ⊢ (Vtx‘𝑆) = (Vtx‘〈𝐾, 𝑃〉) |
15 | 1 | fvexi 6678 | . . . . . . . . . . . . 13 ⊢ 𝑉 ∈ V |
16 | 15 | difexi 5224 | . . . . . . . . . . . 12 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
17 | 3, 16 | eqeltri 2909 | . . . . . . . . . . 11 ⊢ 𝐾 ∈ V |
18 | 2 | fvexi 6678 | . . . . . . . . . . . . 13 ⊢ 𝐸 ∈ V |
19 | 18 | resex 5893 | . . . . . . . . . . . 12 ⊢ (𝐸 ↾ 𝐼) ∈ V |
20 | 5, 19 | eqeltri 2909 | . . . . . . . . . . 11 ⊢ 𝑃 ∈ V |
21 | 17, 20 | opvtxfvi 26788 | . . . . . . . . . 10 ⊢ (Vtx‘〈𝐾, 𝑃〉) = 𝐾 |
22 | 14, 21, 3 | 3eqtrri 2849 | . . . . . . . . 9 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
23 | 1, 2, 3, 4, 5, 6 | vtxdginducedm1lem1 27315 | . . . . . . . . . 10 ⊢ (iEdg‘𝑆) = 𝑃 |
24 | 23 | eqcomi 2830 | . . . . . . . . 9 ⊢ 𝑃 = (iEdg‘𝑆) |
25 | eqid 2821 | . . . . . . . . 9 ⊢ dom 𝑃 = dom 𝑃 | |
26 | 22, 24, 25 | vtxdgfisnn0 27251 | . . . . . . . 8 ⊢ ((dom 𝑃 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℕ0) |
27 | 13, 26 | sylan 582 | . . . . . . 7 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℕ0) |
28 | 27 | nn0red 11950 | . . . . . 6 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℝ) |
29 | dmfi 8796 | . . . . . . . . . . 11 ⊢ (𝐸 ∈ Fin → dom 𝐸 ∈ Fin) | |
30 | rabfi 8737 | . . . . . . . . . . 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 2917 | . . . . . . . . 9 ⊢ (𝐸 ∈ Fin → 𝐽 ∈ Fin) |
33 | rabfi 8737 | . . . . . . . . 9 ⊢ (𝐽 ∈ Fin → {𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)} ∈ Fin) | |
34 | hashcl 13711 | . . . . . . . . 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 483 | . . . . . . 7 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}) ∈ ℕ0) |
37 | 36 | nn0red 11950 | . . . . . 6 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}) ∈ ℝ) |
38 | 28, 37 | rexaddd 12621 | . . . . 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 2832 | . . . 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 231 | . . 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 3177 | . 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 398 = wceq 1533 ∈ wcel 2110 ∉ wnel 3123 ∀wral 3138 {crab 3142 Vcvv 3494 ∖ cdif 3932 {csn 4560 〈cop 4566 dom cdm 5549 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 + caddc 10534 ℕ0cn0 11891 +𝑒 cxad 12499 ♯chash 13684 Vtxcvtx 26775 iEdgciedg 26776 VtxDegcvtxdg 27241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-xadd 12502 df-fz 12887 df-hash 13685 df-vtx 26777 df-iedg 26778 df-vtxdg 27242 |
This theorem is referenced by: finsumvtxdg2ssteplem4 27324 |
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