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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 29576 | . 2 ⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})) |
9 | 5 | dmeqi 5918 | . . . . . . . . 9 ⊢ dom 𝑃 = dom (𝐸 ↾ 𝐼) |
10 | finresfin 9302 | . . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → (𝐸 ↾ 𝐼) ∈ Fin) | |
11 | dmfi 9373 | . . . . . . . . . 10 ⊢ ((𝐸 ↾ 𝐼) ∈ Fin → dom (𝐸 ↾ 𝐼) ∈ Fin) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝐸 ∈ Fin → dom (𝐸 ↾ 𝐼) ∈ Fin) |
13 | 9, 12 | eqeltrid 2843 | . . . . . . . 8 ⊢ (𝐸 ∈ Fin → dom 𝑃 ∈ Fin) |
14 | 6 | fveq2i 6910 | . . . . . . . . . 10 ⊢ (Vtx‘𝑆) = (Vtx‘〈𝐾, 𝑃〉) |
15 | 1 | fvexi 6921 | . . . . . . . . . . . . 13 ⊢ 𝑉 ∈ V |
16 | 15 | difexi 5336 | . . . . . . . . . . . 12 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
17 | 3, 16 | eqeltri 2835 | . . . . . . . . . . 11 ⊢ 𝐾 ∈ V |
18 | 2 | fvexi 6921 | . . . . . . . . . . . . 13 ⊢ 𝐸 ∈ V |
19 | 18 | resex 6049 | . . . . . . . . . . . 12 ⊢ (𝐸 ↾ 𝐼) ∈ V |
20 | 5, 19 | eqeltri 2835 | . . . . . . . . . . 11 ⊢ 𝑃 ∈ V |
21 | 17, 20 | opvtxfvi 29041 | . . . . . . . . . 10 ⊢ (Vtx‘〈𝐾, 𝑃〉) = 𝐾 |
22 | 14, 21, 3 | 3eqtrri 2768 | . . . . . . . . 9 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
23 | 1, 2, 3, 4, 5, 6 | vtxdginducedm1lem1 29572 | . . . . . . . . . 10 ⊢ (iEdg‘𝑆) = 𝑃 |
24 | 23 | eqcomi 2744 | . . . . . . . . 9 ⊢ 𝑃 = (iEdg‘𝑆) |
25 | eqid 2735 | . . . . . . . . 9 ⊢ dom 𝑃 = dom 𝑃 | |
26 | 22, 24, 25 | vtxdgfisnn0 29508 | . . . . . . . 8 ⊢ ((dom 𝑃 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℕ0) |
27 | 13, 26 | sylan 580 | . . . . . . 7 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℕ0) |
28 | 27 | nn0red 12586 | . . . . . 6 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℝ) |
29 | dmfi 9373 | . . . . . . . . . . 11 ⊢ (𝐸 ∈ Fin → dom 𝐸 ∈ Fin) | |
30 | rabfi 9301 | . . . . . . . . . . 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 2843 | . . . . . . . . 9 ⊢ (𝐸 ∈ Fin → 𝐽 ∈ Fin) |
33 | rabfi 9301 | . . . . . . . . 9 ⊢ (𝐽 ∈ Fin → {𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)} ∈ Fin) | |
34 | hashcl 14392 | . . . . . . . . 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 480 | . . . . . . 7 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}) ∈ ℕ0) |
37 | 36 | nn0red 12586 | . . . . . 6 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}) ∈ ℝ) |
38 | 28, 37 | rexaddd 13273 | . . . . 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 2746 | . . . 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 229 | . . 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 3165 | . 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 395 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 ∀wral 3059 {crab 3433 Vcvv 3478 ∖ cdif 3960 {csn 4631 〈cop 4637 dom cdm 5689 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 + caddc 11156 ℕ0cn0 12524 +𝑒 cxad 13150 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 VtxDegcvtxdg 29498 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-xadd 13153 df-fz 13545 df-hash 14367 df-vtx 29030 df-iedg 29031 df-vtxdg 29499 |
This theorem is referenced by: finsumvtxdg2ssteplem4 29581 |
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