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Theorem vtxdgfival 29929

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 29928	. . 3 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +_𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
5	4	adantl 487	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +_𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
6		rabfi 9241	. . . . . . 7 ⊢ (𝐴 ∈ Fin → {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)} ∈ Fin)
7		hashcl 14420	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)} ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) ∈ ℕ₀)
8	6, 7	syl 18	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) ∈ ℕ₀)
9	8	nn0red 12590	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) ∈ ℝ)
10		rabfi 9241	. . . . . . 7 ⊢ (𝐴 ∈ Fin → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} ∈ Fin)
11		hashcl 14420	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) ∈ ℕ₀)
12	10, 11	syl 18	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) ∈ ℕ₀)
13	12	nn0red 12590	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) ∈ ℝ)
14	9, 13	jca 521	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) ∈ ℝ ∧ (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) ∈ ℝ))
15	14	adantr 486	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) ∈ ℝ ∧ (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) ∈ ℝ))
16		rexadd 13284	. . 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 18	. 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 2795	1 ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {crab 3412 {csn 4584 dom cdm 5655 ‘cfv 6533 (class class class)co 7413 Fincfn 8952 ℝcr 11123 + caddc 11127 ℕ₀cn0 12528 +_𝑒 cxad 13161 ♯chash 14394 Vtxcvtx 29453 iEdgciedg 29454 VtxDegcvtxdg 29925

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-n0 12529 df-z 12616 df-uz 12888 df-xadd 13164 df-hash 14395 df-vtxdg 29926

This theorem is used by: vtxdg0e 29934 vtxdgfisnn0 29935 finsumvtxdg2ssteplem2 30006

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