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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 29561 | . 2 ⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})) | 
| 9 | 5 | dmeqi 5915 | . . . . . . . . 9 ⊢ dom 𝑃 = dom (𝐸 ↾ 𝐼) | 
| 10 | finresfin 9304 | . . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → (𝐸 ↾ 𝐼) ∈ Fin) | |
| 11 | dmfi 9375 | . . . . . . . . . 10 ⊢ ((𝐸 ↾ 𝐼) ∈ Fin → dom (𝐸 ↾ 𝐼) ∈ Fin) | |
| 12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝐸 ∈ Fin → dom (𝐸 ↾ 𝐼) ∈ Fin) | 
| 13 | 9, 12 | eqeltrid 2845 | . . . . . . . 8 ⊢ (𝐸 ∈ Fin → dom 𝑃 ∈ Fin) | 
| 14 | 6 | fveq2i 6909 | . . . . . . . . . 10 ⊢ (Vtx‘𝑆) = (Vtx‘〈𝐾, 𝑃〉) | 
| 15 | 1 | fvexi 6920 | . . . . . . . . . . . . 13 ⊢ 𝑉 ∈ V | 
| 16 | 15 | difexi 5330 | . . . . . . . . . . . 12 ⊢ (𝑉 ∖ {𝑁}) ∈ V | 
| 17 | 3, 16 | eqeltri 2837 | . . . . . . . . . . 11 ⊢ 𝐾 ∈ V | 
| 18 | 2 | fvexi 6920 | . . . . . . . . . . . . 13 ⊢ 𝐸 ∈ V | 
| 19 | 18 | resex 6047 | . . . . . . . . . . . 12 ⊢ (𝐸 ↾ 𝐼) ∈ V | 
| 20 | 5, 19 | eqeltri 2837 | . . . . . . . . . . 11 ⊢ 𝑃 ∈ V | 
| 21 | 17, 20 | opvtxfvi 29026 | . . . . . . . . . 10 ⊢ (Vtx‘〈𝐾, 𝑃〉) = 𝐾 | 
| 22 | 14, 21, 3 | 3eqtrri 2770 | . . . . . . . . 9 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) | 
| 23 | 1, 2, 3, 4, 5, 6 | vtxdginducedm1lem1 29557 | . . . . . . . . . 10 ⊢ (iEdg‘𝑆) = 𝑃 | 
| 24 | 23 | eqcomi 2746 | . . . . . . . . 9 ⊢ 𝑃 = (iEdg‘𝑆) | 
| 25 | eqid 2737 | . . . . . . . . 9 ⊢ dom 𝑃 = dom 𝑃 | |
| 26 | 22, 24, 25 | vtxdgfisnn0 29493 | . . . . . . . 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 12588 | . . . . . 6 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℝ) | 
| 29 | dmfi 9375 | . . . . . . . . . . 11 ⊢ (𝐸 ∈ Fin → dom 𝐸 ∈ Fin) | |
| 30 | rabfi 9303 | . . . . . . . . . . 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 2845 | . . . . . . . . 9 ⊢ (𝐸 ∈ Fin → 𝐽 ∈ Fin) | 
| 33 | rabfi 9303 | . . . . . . . . 9 ⊢ (𝐽 ∈ Fin → {𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)} ∈ Fin) | |
| 34 | hashcl 14395 | . . . . . . . . 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 12588 | . . . . . 6 ⊢ ((𝐸 ∈ Fin ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}) ∈ ℝ) | 
| 38 | 28, 37 | rexaddd 13276 | . . . . 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 2748 | . . . 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 3167 | . 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 1540 ∈ wcel 2108 ∉ wnel 3046 ∀wral 3061 {crab 3436 Vcvv 3480 ∖ cdif 3948 {csn 4626 〈cop 4632 dom cdm 5685 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 + caddc 11158 ℕ0cn0 12526 +𝑒 cxad 13152 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 VtxDegcvtxdg 29483 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-xadd 13155 df-fz 13548 df-hash 14370 df-vtx 29015 df-iedg 29016 df-vtxdg 29484 | 
| This theorem is referenced by: finsumvtxdg2ssteplem4 29566 | 
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