vdegp1aid - Intuitionistic Logic Explorer

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Theorem vdegp1aid 16125

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 11112	. . . . 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 5483	. . 3 ⊢ (𝜑 → Fun 𝐼)
7		vdegp1aid.vf	. . 3 ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉)
8		vdegp1aid.f	. . . 4 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩))
9		wrdv 11122	. . . . . 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 4299	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ∈ V)
14	11, 12, 13	syl2anc 411	. . . . 5 ⊢ (𝜑 → {𝑋, 𝑌} ∈ V)
15		cats1un 11295	. . . . 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 2262	. . 3 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨(♯‘𝐼), {𝑋, 𝑌}⟩}))
18		lencl 11110	. . . 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 11123	. . . 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 15864	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
25	11, 24	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
26	1, 2	wrdupgren 15940	. . . . 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 11125	. . . . 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 7130	. . . 4 ⊢ ((𝐼 ∈ Fin ∧ Fun 𝐼) → dom 𝐼 ∈ Fin)
32	30, 6, 31	syl2anc 411	. . 3 ⊢ (𝜑 → dom 𝐼 ∈ Fin)
33		prelpwi 4304	. . . 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 7391	. . . . 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 2421	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 = 𝑈)
41	40	neqcomd 2234	. . . . . 6 ⊢ (𝜑 → ¬ 𝑈 = 𝑋)
42		vdegp1aid.yu	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑈)
43	42	neneqd 2421	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑌 = 𝑈)
44	43	neqcomd 2234	. . . . . 6 ⊢ (𝜑 → ¬ 𝑈 = 𝑌)
45		ioran 757	. . . . . 6 ⊢ (¬ (𝑈 = 𝑋 ∨ 𝑈 = 𝑌) ↔ (¬ 𝑈 = 𝑋 ∧ ¬ 𝑈 = 𝑌))
46	41, 44, 45	sylanbrc 417	. . . . 5 ⊢ (𝜑 → ¬ (𝑈 = 𝑋 ∨ 𝑈 = 𝑌))
47		elpri 3690	. . . . 5 ⊢ (𝑈 ∈ {𝑋, 𝑌} → (𝑈 = 𝑋 ∨ 𝑈 = 𝑌))
48	46, 47	nsyl 631	. . . 4 ⊢ (𝜑 → ¬ 𝑈 ∈ {𝑋, 𝑌})
49		df-nel 2496	. . . 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 16123	. 2 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈))
52		vdegp1aid.d	. 2 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) = 𝑃)
53	51, 52	eqtrd 2262	1 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = 𝑃)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∨ wo 713 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 ∉ wnel 2495 {crab 2512 Vcvv 2800 ∪ cun 3196 𝒫 cpw 3650 {csn 3667 {cpr 3668 ⟨cop 3670 class class class wbr 4086 dom cdm 4723 Fun wfun 5318 ⟶wf 5320 ‘cfv 5324 (class class class)co 6013 1_oc1o 6570 2_oc2o 6571 ≈ cen 6902 Fincfn 6904 0cc0 8025 ℕ₀cn0 9395 ..^cfzo 10370 ♯chash 11030 Word cword 11106 ++ cconcat 11160 ⟨“cs1 11185 Vtxcvtx 15856 iEdgciedg 15857 UPGraphcupgr 15935 VtxDegcvtxdg 16097

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-2o 6578 df-oadd 6581 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-5 9198 df-6 9199 df-7 9200 df-8 9201 df-9 9202 df-n0 9396 df-z 9473 df-dec 9605 df-uz 9749 df-xadd 10001 df-fz 10237 df-fzo 10371 df-ihash 11031 df-word 11107 df-concat 11161 df-s1 11186 df-ndx 13078 df-slot 13079 df-base 13081 df-edgf 15849 df-vtx 15858 df-iedg 15859 df-upgren 15937 df-vtxdg 16098

This theorem is referenced by: (None)

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