vdegp1aid - Intuitionistic Logic Explorer

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Theorem vdegp1aid 16296

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 11223	. . . . 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 5511	. . 3 ⊢ (𝜑 → Fun 𝐼)
7		vdegp1aid.vf	. . 3 ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉)
8		vdegp1aid.f	. . . 4 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩))
9		wrdv 11233	. . . . . 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 4324	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ∈ V)
14	11, 12, 13	syl2anc 411	. . . . 5 ⊢ (𝜑 → {𝑋, 𝑌} ∈ V)
15		cats1un 11406	. . . . 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 2265	. . 3 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨(♯‘𝐼), {𝑋, 𝑌}⟩}))
18		lencl 11221	. . . 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 11234	. . . 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 16002	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
25	11, 24	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
26	1, 2	wrdupgren 16078	. . . . 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 11236	. . . . 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 7203	. . . 4 ⊢ ((𝐼 ∈ Fin ∧ Fun 𝐼) → dom 𝐼 ∈ Fin)
32	30, 6, 31	syl2anc 411	. . 3 ⊢ (𝜑 → dom 𝐼 ∈ Fin)
33		prelpwi 4329	. . . 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 7488	. . . . 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 2433	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 = 𝑈)
41	40	neqcomd 2237	. . . . . 6 ⊢ (𝜑 → ¬ 𝑈 = 𝑋)
42		vdegp1aid.yu	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑈)
43	42	neneqd 2433	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑌 = 𝑈)
44	43	neqcomd 2237	. . . . . 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 3711	. . . . 5 ⊢ (𝑈 ∈ {𝑋, 𝑌} → (𝑈 = 𝑋 ∨ 𝑈 = 𝑌))
48	46, 47	nsyl 633	. . . 4 ⊢ (𝜑 → ¬ 𝑈 ∈ {𝑋, 𝑌})
49		df-nel 2508	. . . 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 16294	. 2 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈))
52		vdegp1aid.d	. 2 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) = 𝑃)
53	51, 52	eqtrd 2265	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 2203 ≠ wne 2412 ∉ wnel 2507 {crab 2524 Vcvv 2812 ∪ cun 3208 𝒫 cpw 3668 {csn 3688 {cpr 3689 ⟨cop 3691 class class class wbr 4108 dom cdm 4748 Fun wfun 5345 ⟶wf 5347 ‘cfv 5351 (class class class)co 6049 1_oc1o 6639 2_oc2o 6640 ≈ cen 6972 Fincfn 6974 0cc0 8123 ℕ₀cn0 9492 ..^cfzo 10472 ♯chash 11133 Word cword 11217 ++ cconcat 11271 ⟨“cs1 11296 Vtxcvtx 15994 iEdgciedg 15995 UPGraphcupgr 16073 VtxDegcvtxdg 16268

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 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-addcom 8223 ax-mulcom 8224 ax-addass 8225 ax-mulass 8226 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-1rid 8230 ax-0id 8231 ax-rnegex 8232 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239

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 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-irdg 6600 df-frec 6621 df-1o 6646 df-2o 6647 df-oadd 6650 df-er 6766 df-en 6975 df-dom 6976 df-fin 6977 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-inn 9234 df-2 9292 df-3 9293 df-4 9294 df-5 9295 df-6 9296 df-7 9297 df-8 9298 df-9 9299 df-n0 9493 df-z 9574 df-dec 9706 df-uz 9850 df-xadd 10102 df-fz 10339 df-fzo 10473 df-ihash 11134 df-word 11218 df-concat 11272 df-s1 11297 df-ndx 13204 df-slot 13205 df-base 13207 df-edgf 15987 df-vtx 15996 df-iedg 15997 df-upgren 16075 df-vtxdg 16269

This theorem is referenced by: konigsberglem1 16470 konigsberglem2 16471 konigsberglem3 16472

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