vdegp1aid - Intuitionistic Logic Explorer

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Theorem vdegp1aid 16232

Description: The induction step for a vertex degree calculation. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑋, 𝑌} to the edge set, where 𝑋 ≠ 𝑈 ≠ 𝑌, yields degree 𝑃 as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
vdegp1ai.vg	⊢ 𝑉 = (Vtx‘𝐺)
vdegp1aid.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
vdegp1ai.i	⊢ 𝐼 = (iEdg‘𝐺)
vdegp1aid.w	⊢ (𝜑 → 𝐼 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)})
vdegp1aid.d	⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) = 𝑃)
vdegp1aid.vf	⊢ (𝜑 → (Vtx‘𝐹) = 𝑉)
vdegp1aid.fi	⊢ (𝜑 → 𝑉 ∈ Fin)
vdegp1aid.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
vdegp1aid.xu	⊢ (𝜑 → 𝑋 ≠ 𝑈)
vdegp1aid.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
vdegp1aid.yu	⊢ (𝜑 → 𝑌 ≠ 𝑈)
vdegp1aid.xy	⊢ (𝜑 → 𝑋 ≠ 𝑌)
vdegp1aid.f	⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩))

**Assertion**
Ref	Expression
vdegp1aid	⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = 𝑃)

Distinct variable groups: 𝑥,𝑈 𝑥,𝑉 𝑥,𝑋 𝑥,𝑌 𝑥,𝐺 Allowed substitution hints: 𝜑(𝑥) 𝑃(𝑥) 𝐹(𝑥) 𝐼(𝑥)

**Proof of Theorem vdegp1aid**
Step	Hyp	Ref	Expression
1		vdegp1ai.vg	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		vdegp1ai.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
3		vdegp1aid.w	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)})
4		wrdf 11166	. . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)} → 𝐼:(0..^(♯‘𝐼))⟶{𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)})
5	3, 4	syl 14	. . . 4 ⊢ (𝜑 → 𝐼:(0..^(♯‘𝐼))⟶{𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)})
6	5	ffund 5493	. . 3 ⊢ (𝜑 → Fun 𝐼)
7		vdegp1aid.vf	. . 3 ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉)
8		vdegp1aid.f	. . . 4 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩))
9		wrdv 11176	. . . . . 6 ⊢ (𝐼 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)} → 𝐼 ∈ Word V)
10	3, 9	syl 14	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ Word V)
11		vdegp1aid.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12		vdegp1aid.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
13		prexg 4307	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ∈ V)
14	11, 12, 13	syl2anc 411	. . . . 5 ⊢ (𝜑 → {𝑋, 𝑌} ∈ V)
15		cats1un 11349	. . . . 5 ⊢ ((𝐼 ∈ Word V ∧ {𝑋, 𝑌} ∈ V) → (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩) = (𝐼 ∪ {⟨(♯‘𝐼), {𝑋, 𝑌}⟩}))
16	10, 14, 15	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩) = (𝐼 ∪ {⟨(♯‘𝐼), {𝑋, 𝑌}⟩}))
17	8, 16	eqtrd 2264	. . 3 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨(♯‘𝐼), {𝑋, 𝑌}⟩}))
18		lencl 11164	. . . 4 ⊢ (𝐼 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)} → (♯‘𝐼) ∈ ℕ₀)
19	3, 18	syl 14	. . 3 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ₀)
20		wrdlndm 11177	. . . 4 ⊢ (𝐼 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)} → (♯‘𝐼) ∉ dom 𝐼)
21	3, 20	syl 14	. . 3 ⊢ (𝜑 → (♯‘𝐼) ∉ dom 𝐼)
22		vdegp1aid.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
23		vdegp1aid.fi	. . 3 ⊢ (𝜑 → 𝑉 ∈ Fin)
24	1	1vgrex 15938	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
25	11, 24	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
26	1, 2	wrdupgren 16014	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝐼 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)}) → (𝐺 ∈ UPGraph ↔ 𝐼 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)}))
27	25, 3, 26	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐺 ∈ UPGraph ↔ 𝐼 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)}))
28	3, 27	mpbird 167	. . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
29		wrdfin 11179	. . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)} → 𝐼 ∈ Fin)
30	3, 29	syl 14	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
31		fundmfi 7179	. . . 4 ⊢ ((𝐼 ∈ Fin ∧ Fun 𝐼) → dom 𝐼 ∈ Fin)
32	30, 6, 31	syl2anc 411	. . 3 ⊢ (𝜑 → dom 𝐼 ∈ Fin)
33		prelpwi 4312	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ∈ 𝒫 𝑉)
34	11, 12, 33	syl2anc 411	. . 3 ⊢ (𝜑 → {𝑋, 𝑌} ∈ 𝒫 𝑉)
35		vdegp1aid.xy	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
36		pr2ne 7440	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ({𝑋, 𝑌} ≈ 2_o ↔ 𝑋 ≠ 𝑌))
37	11, 12, 36	syl2anc 411	. . . 4 ⊢ (𝜑 → ({𝑋, 𝑌} ≈ 2_o ↔ 𝑋 ≠ 𝑌))
38	35, 37	mpbird 167	. . 3 ⊢ (𝜑 → {𝑋, 𝑌} ≈ 2_o)
39		vdegp1aid.xu	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ 𝑈)
40	39	neneqd 2424	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 = 𝑈)
41	40	neqcomd 2236	. . . . . 6 ⊢ (𝜑 → ¬ 𝑈 = 𝑋)
42		vdegp1aid.yu	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑈)
43	42	neneqd 2424	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑌 = 𝑈)
44	43	neqcomd 2236	. . . . . 6 ⊢ (𝜑 → ¬ 𝑈 = 𝑌)
45		ioran 760	. . . . . 6 ⊢ (¬ (𝑈 = 𝑋 ∨ 𝑈 = 𝑌) ↔ (¬ 𝑈 = 𝑋 ∧ ¬ 𝑈 = 𝑌))
46	41, 44, 45	sylanbrc 417	. . . . 5 ⊢ (𝜑 → ¬ (𝑈 = 𝑋 ∨ 𝑈 = 𝑌))
47		elpri 3696	. . . . 5 ⊢ (𝑈 ∈ {𝑋, 𝑌} → (𝑈 = 𝑋 ∨ 𝑈 = 𝑌))
48	46, 47	nsyl 633	. . . 4 ⊢ (𝜑 → ¬ 𝑈 ∈ {𝑋, 𝑌})
49		df-nel 2499	. . . 4 ⊢ (𝑈 ∉ {𝑋, 𝑌} ↔ ¬ 𝑈 ∈ {𝑋, 𝑌})
50	48, 49	sylibr 134	. . 3 ⊢ (𝜑 → 𝑈 ∉ {𝑋, 𝑌})
51	1, 2, 6, 7, 17, 19, 21, 22, 23, 28, 32, 34, 38, 14, 50	p1evtxdeqfi 16230	. 2 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈))
52		vdegp1aid.d	. 2 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) = 𝑃)
53	51, 52	eqtrd 2264	1 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = 𝑃)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∨ wo 716 = wceq 1398 ∈ wcel 2202 ≠ wne 2403 ∉ wnel 2498 {crab 2515 Vcvv 2803 ∪ cun 3199 𝒫 cpw 3656 {csn 3673 {cpr 3674 ⟨cop 3676 class class class wbr 4093 dom cdm 4731 Fun wfun 5327 ⟶wf 5329 ‘cfv 5333 (class class class)co 6028 1_oc1o 6618 2_oc2o 6619 ≈ cen 6950 Fincfn 6952 0cc0 8075 ℕ₀cn0 9445 ..^cfzo 10420 ♯chash 11081 Word cword 11160 ++ cconcat 11214 ⟨“cs1 11239 Vtxcvtx 15930 iEdgciedg 15931 UPGraphcupgr 16009 VtxDegcvtxdg 16204

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-2o 6626 df-oadd 6629 df-er 6745 df-en 6953 df-dom 6954 df-fin 6955 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-5 9248 df-6 9249 df-7 9250 df-8 9251 df-9 9252 df-n0 9446 df-z 9523 df-dec 9655 df-uz 9799 df-xadd 10051 df-fz 10287 df-fzo 10421 df-ihash 11082 df-word 11161 df-concat 11215 df-s1 11240 df-ndx 13146 df-slot 13147 df-base 13149 df-edgf 15923 df-vtx 15932 df-iedg 15933 df-upgren 16011 df-vtxdg 16205

This theorem is referenced by: konigsberglem1 16406 konigsberglem2 16407 konigsberglem3 16408

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