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Mirrors > Home > MPE Home > Th. List > vdegp1ai | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
vdegp1ai.vg | ⊢ 𝑉 = (Vtx‘𝐺) |
vdegp1ai.u | ⊢ 𝑈 ∈ 𝑉 |
vdegp1ai.i | ⊢ 𝐼 = (iEdg‘𝐺) |
vdegp1ai.w | ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} |
vdegp1ai.d | ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃 |
vdegp1ai.vf | ⊢ (Vtx‘𝐹) = 𝑉 |
vdegp1ai.x | ⊢ 𝑋 ∈ 𝑉 |
vdegp1ai.xu | ⊢ 𝑋 ≠ 𝑈 |
vdegp1ai.y | ⊢ 𝑌 ∈ 𝑉 |
vdegp1ai.yu | ⊢ 𝑌 ≠ 𝑈 |
vdegp1ai.f | ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑋, 𝑌}”〉) |
Ref | Expression |
---|---|
vdegp1ai | ⊢ ((VtxDeg‘𝐹)‘𝑈) = 𝑃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5333 | . . 3 ⊢ {𝑋, 𝑌} ∈ V | |
2 | vdegp1ai.vg | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | vdegp1ai.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | vdegp1ai.w | . . . . 5 ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} | |
5 | wrdf 13867 | . . . . . 6 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼:(0..^(♯‘𝐼))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
6 | 5 | ffund 6518 | . . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → Fun 𝐼) |
7 | 4, 6 | mp1i 13 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → Fun 𝐼) |
8 | vdegp1ai.vf | . . . . 5 ⊢ (Vtx‘𝐹) = 𝑉 | |
9 | 8 | a1i 11 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (Vtx‘𝐹) = 𝑉) |
10 | vdegp1ai.f | . . . . 5 ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑋, 𝑌}”〉) | |
11 | wrdv 13878 | . . . . . . 7 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼 ∈ Word V) | |
12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
13 | cats1un 14083 | . . . . . 6 ⊢ ((𝐼 ∈ Word V ∧ {𝑋, 𝑌} ∈ V) → (𝐼 ++ 〈“{𝑋, 𝑌}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑋, 𝑌}〉})) | |
14 | 12, 13 | mpan 688 | . . . . 5 ⊢ ({𝑋, 𝑌} ∈ V → (𝐼 ++ 〈“{𝑋, 𝑌}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑋, 𝑌}〉})) |
15 | 10, 14 | syl5eq 2868 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (iEdg‘𝐹) = (𝐼 ∪ {〈(♯‘𝐼), {𝑋, 𝑌}〉})) |
16 | fvexd 6685 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (♯‘𝐼) ∈ V) | |
17 | wrdlndm 13879 | . . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (♯‘𝐼) ∉ dom 𝐼) | |
18 | 4, 17 | mp1i 13 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (♯‘𝐼) ∉ dom 𝐼) |
19 | vdegp1ai.u | . . . . 5 ⊢ 𝑈 ∈ 𝑉 | |
20 | 19 | a1i 11 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → 𝑈 ∈ 𝑉) |
21 | id 22 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → {𝑋, 𝑌} ∈ V) | |
22 | vdegp1ai.xu | . . . . . . 7 ⊢ 𝑋 ≠ 𝑈 | |
23 | 22 | necomi 3070 | . . . . . 6 ⊢ 𝑈 ≠ 𝑋 |
24 | vdegp1ai.yu | . . . . . . 7 ⊢ 𝑌 ≠ 𝑈 | |
25 | 24 | necomi 3070 | . . . . . 6 ⊢ 𝑈 ≠ 𝑌 |
26 | 23, 25 | prneli 4595 | . . . . 5 ⊢ 𝑈 ∉ {𝑋, 𝑌} |
27 | 26 | a1i 11 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → 𝑈 ∉ {𝑋, 𝑌}) |
28 | 2, 3, 7, 9, 15, 16, 18, 20, 21, 27 | p1evtxdeq 27295 | . . 3 ⊢ ({𝑋, 𝑌} ∈ V → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈)) |
29 | 1, 28 | ax-mp 5 | . 2 ⊢ ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈) |
30 | vdegp1ai.d | . 2 ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃 | |
31 | 29, 30 | eqtri 2844 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∉ wnel 3123 {crab 3142 Vcvv 3494 ∖ cdif 3933 ∪ cun 3934 ∅c0 4291 𝒫 cpw 4539 {csn 4567 {cpr 4569 〈cop 4573 class class class wbr 5066 dom cdm 5555 Fun wfun 6349 ‘cfv 6355 (class class class)co 7156 0cc0 10537 ≤ cle 10676 2c2 11693 ..^cfzo 13034 ♯chash 13691 Word cword 13862 ++ cconcat 13922 〈“cs1 13949 Vtxcvtx 26781 iEdgciedg 26782 VtxDegcvtxdg 27247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-xadd 12509 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-vtx 26783 df-iedg 26784 df-vtxdg 27248 |
This theorem is referenced by: konigsberglem1 28031 konigsberglem2 28032 konigsberglem3 28033 |
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