vtxdgfival - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > vtxdgfival		Structured version Visualization version GIF version

Theorem vtxdgfival 27836

Description: The degree of a vertex for graphs of finite size. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 8-Dec-2020.) (Revised by AV, 22-Mar-2021.)

**Hypotheses**
Ref	Expression
vtxdgval.v	⊢ 𝑉 = (Vtx‘𝐺)
vtxdgval.i	⊢ 𝐼 = (iEdg‘𝐺)
vtxdgval.a	⊢ 𝐴 = dom 𝐼

**Assertion**
Ref	Expression
vtxdgfival	⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐺 𝑥,𝑈 Allowed substitution hints: 𝐼(𝑥) 𝑉(𝑥)

**Proof of Theorem vtxdgfival**
Step	Hyp	Ref	Expression
1		vtxdgval.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		vtxdgval.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
3		vtxdgval.a	. . . 4 ⊢ 𝐴 = dom 𝐼
4	1, 2, 3	vtxdgval 27835	. . 3 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +_𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
5	4	adantl 482	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +_𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
6		rabfi 9044	. . . . . . 7 ⊢ (𝐴 ∈ Fin → {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)} ∈ Fin)
7		hashcl 14071	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)} ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) ∈ ℕ₀)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) ∈ ℕ₀)
9	8	nn0red 12294	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) ∈ ℝ)
10		rabfi 9044	. . . . . . 7 ⊢ (𝐴 ∈ Fin → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} ∈ Fin)
11		hashcl 14071	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) ∈ ℕ₀)
12	10, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) ∈ ℕ₀)
13	12	nn0red 12294	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) ∈ ℝ)
14	9, 13	jca 512	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) ∈ ℝ ∧ (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) ∈ ℝ))
15	14	adantr 481	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) ∈ ℝ ∧ (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) ∈ ℝ))
16		rexadd 12966	. . 3 ⊢ (((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) ∈ ℝ ∧ (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) ∈ ℝ) → ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +_𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
17	15, 16	syl 17	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +_𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
18	5, 17	eqtrd 2778	1 ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 {csn 4561 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 ℝcr 10870 + caddc 10874 ℕ₀cn0 12233 +_𝑒 cxad 12846 ♯chash 14044 Vtxcvtx 27366 iEdgciedg 27367 VtxDegcvtxdg 27832

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-xadd 12849 df-hash 14045 df-vtxdg 27833

This theorem is referenced by: vtxdg0e 27841 vtxdgfisnn0 27842 finsumvtxdg2ssteplem2 27913

W3C validator