vdegp1aid - Intuitionistic Logic Explorer

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Theorem vdegp1aid 16538

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 11293	. . . . 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 5535	. . 3 ⊢ (𝜑 → Fun 𝐼)
7		vdegp1aid.vf	. . 3 ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉)
8		vdegp1aid.f	. . . 4 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩))
9		wrdv 11303	. . . . . 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 4347	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ∈ V)
14	11, 12, 13	syl2anc 415	. . . . 5 ⊢ (𝜑 → {𝑋, 𝑌} ∈ V)
15		cats1un 11476	. . . . 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 11291	. . . 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 11304	. . . 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 16244	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
25	11, 24	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
26	1, 2	wrdupgren 16320	. . . . 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 11306	. . . . 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 7245	. . . 4 ⊢ ((𝐼 ∈ Fin ∧ Fun 𝐼) → dom 𝐼 ∈ Fin)
32	30, 6, 31	syl2anc 415	. . 3 ⊢ (𝜑 → dom 𝐼 ∈ Fin)
33		prelpwi 4352	. . . 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 7532	. . . . 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 3731	. . . . 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 16536	. 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

Syntax hints: ¬ 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 3708 {cpr 3709 ⟨cop 3711 class class class wbr 4128 dom cdm 4772 Fun wfun 5369 ⟶wf 5371 ‘cfv 5375 (class class class)co 6079 1_oc1o 6674 2_oc2o 6675 ≈ cen 7014 Fincfn 7016 0cc0 8173 ℕ₀cn0 9546 ..^cfzo 10532 ♯chash 11197 Word cword 11287 ++ cconcat 11341 ⟨“cs1 11366 Vtxcvtx 16236 iEdgciedg 16237 UPGraphcupgr 16315 VtxDegcvtxdg 16510

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 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 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289

This theorem 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 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-irdg 6635 df-frec 6656 df-1o 6681 df-2o 6682 df-oadd 6685 df-er 6801 df-en 7017 df-dom 7018 df-fin 7019 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-5 9349 df-6 9350 df-7 9351 df-8 9352 df-9 9353 df-n0 9547 df-z 9628 df-dec 9761 df-uz 9905 df-xadd 10158 df-fz 10395 df-fzo 10533 df-ihash 11198 df-word 11288 df-concat 11342 df-s1 11367 df-ndx 13338 df-slot 13339 df-base 13341 df-edgf 16229 df-vtx 16238 df-iedg 16239 df-upgren 16317 df-vtxdg 16511

This theorem is referenced by: konigsberglem1 16712 konigsberglem2 16713 konigsberglem3 16714

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