vdegp1aid - Intuitionistic Logic Explorer

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Theorem vdegp1aid 16721

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 11326	. . . . 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 5537	. . 3 ⊢ (𝜑 → Fun 𝐼)
7		vdegp1aid.vf	. . 3 ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉)
8		vdegp1aid.f	. . . 4 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩))
9		wrdv 11336	. . . . . 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 4349	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ∈ V)
14	11, 12, 13	syl2anc 415	. . . . 5 ⊢ (𝜑 → {𝑋, 𝑌} ∈ V)
15		cats1un 11509	. . . . 5 ⊢ ((𝐼 ∈ Word V ∧ {𝑋, 𝑌} ∈ V) → (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩) = (𝐼 ∪ {⟨(♯‘𝐼), {𝑋, 𝑌}⟩}))
16	10, 14, 15	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩) = (𝐼 ∪ {⟨(♯‘𝐼), {𝑋, 𝑌}⟩}))
17	8, 16	eqtrd 2271	. . 3 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨(♯‘𝐼), {𝑋, 𝑌}⟩}))
18		lencl 11324	. . . 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 11337	. . . 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 16427	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
25	11, 24	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
26	1, 2	wrdupgren 16503	. . . . 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 415	. . . 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 11339	. . . . 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 7251	. . . 4 ⊢ ((𝐼 ∈ Fin ∧ Fun 𝐼) → dom 𝐼 ∈ Fin)
32	30, 6, 31	syl2anc 415	. . 3 ⊢ (𝜑 → dom 𝐼 ∈ Fin)
33		prelpwi 4354	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ∈ 𝒫 𝑉)
34	11, 12, 33	syl2anc 415	. . 3 ⊢ (𝜑 → {𝑋, 𝑌} ∈ 𝒫 𝑉)
35		vdegp1aid.xy	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
36		pr2ne 7539	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ({𝑋, 𝑌} ≈ 2_o ↔ 𝑋 ≠ 𝑌))
37	11, 12, 36	syl2anc 415	. . . 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 2441	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 = 𝑈)
41	40	neqcomd 2243	. . . . . 6 ⊢ (𝜑 → ¬ 𝑈 = 𝑋)
42		vdegp1aid.yu	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑈)
43	42	neneqd 2441	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑌 = 𝑈)
44	43	neqcomd 2243	. . . . . 6 ⊢ (𝜑 → ¬ 𝑈 = 𝑌)
45		ioran 764	. . . . . 6 ⊢ (¬ (𝑈 = 𝑋 ∨ 𝑈 = 𝑌) ↔ (¬ 𝑈 = 𝑋 ∧ ¬ 𝑈 = 𝑌))
46	41, 44, 45	sylanbrc 421	. . . . 5 ⊢ (𝜑 → ¬ (𝑈 = 𝑋 ∨ 𝑈 = 𝑌))
47		elpri 3732	. . . . 5 ⊢ (𝑈 ∈ {𝑋, 𝑌} → (𝑈 = 𝑋 ∨ 𝑈 = 𝑌))
48	46, 47	nsyl 637	. . . 4 ⊢ (𝜑 → ¬ 𝑈 ∈ {𝑋, 𝑌})
49		df-nel 2516	. . . 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 16719	. 2 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈))
52		vdegp1aid.d	. 2 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) = 𝑃)
53	51, 52	eqtrd 2271	1 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = 𝑃)

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 105 ∨ wo 720 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 ∉ wnel 2515 {crab 2532 Vcvv 2821 ∪ cun 3218 𝒫 cpw 3688 {csn 3709 {cpr 3710 ⟨cop 3712 class class class wbr 4130 dom cdm 4774 Fun wfun 5371 ⟶wf 5373 ‘cfv 5377 (class class class)co 6085 1_oc1o 6680 2_oc2o 6681 ≈ cen 7020 Fincfn 7022 0cc0 8180 ℕ₀cn0 9568 ..^cfzo 10560 ♯chash 11230 Word cword 11320 ++ cconcat 11374 ⟨“cs1 11399 Vtxcvtx 16419 iEdgciedg 16420 UPGraphcupgr 16498 VtxDegcvtxdg 16693

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-5 9369 df-6 9370 df-7 9371 df-8 9372 df-9 9373 df-n0 9569 df-z 9650 df-dec 9783 df-uz 9932 df-xadd 10186 df-fz 10423 df-fzo 10561 df-ihash 11231 df-word 11321 df-concat 11375 df-s1 11400 df-ndx 13407 df-slot 13408 df-base 13410 df-edgf 16412 df-vtx 16421 df-iedg 16422 df-upgren 16500 df-vtxdg 16694

This theorem is used by: konigsberglem1 16895 konigsberglem2 16896 konigsberglem3 16897

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