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Mirrors > Home > MPE Home > Th. List > vdegp1bi | Structured version Visualization version GIF version |
Description: The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑈, 𝑋} to the edge set, where 𝑋 ≠ 𝑈, yields degree 𝑃 + 1. (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 |
vdegp1bi.x | ⊢ 𝑋 ∈ 𝑉 |
vdegp1bi.xu | ⊢ 𝑋 ≠ 𝑈 |
vdegp1bi.f | ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑈, 𝑋}”〉) |
Ref | Expression |
---|---|
vdegp1bi | ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5443 | . . 3 ⊢ {𝑈, 𝑋} ∈ 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 14554 | . . . . . 6 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼:(0..^(♯‘𝐼))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
6 | 5 | ffund 6741 | . . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → Fun 𝐼) |
7 | 4, 6 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → Fun 𝐼) |
8 | vdegp1ai.vf | . . . . 5 ⊢ (Vtx‘𝐹) = 𝑉 | |
9 | 8 | a1i 11 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (Vtx‘𝐹) = 𝑉) |
10 | vdegp1bi.f | . . . . 5 ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑈, 𝑋}”〉) | |
11 | wrdv 14564 | . . . . . . 7 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼 ∈ Word V) | |
12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
13 | cats1un 14756 | . . . . . 6 ⊢ ((𝐼 ∈ Word V ∧ {𝑈, 𝑋} ∈ V) → (𝐼 ++ 〈“{𝑈, 𝑋}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑈, 𝑋}〉})) | |
14 | 12, 13 | mpan 690 | . . . . 5 ⊢ ({𝑈, 𝑋} ∈ V → (𝐼 ++ 〈“{𝑈, 𝑋}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑈, 𝑋}〉})) |
15 | 10, 14 | eqtrid 2787 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (iEdg‘𝐹) = (𝐼 ∪ {〈(♯‘𝐼), {𝑈, 𝑋}〉})) |
16 | fvexd 6922 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (♯‘𝐼) ∈ V) | |
17 | wrdlndm 14565 | . . . . 5 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (♯‘𝐼) ∉ dom 𝐼) | |
18 | 4, 17 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → (♯‘𝐼) ∉ dom 𝐼) |
19 | vdegp1ai.u | . . . . 5 ⊢ 𝑈 ∈ 𝑉 | |
20 | 19 | a1i 11 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → 𝑈 ∈ 𝑉) |
21 | vdegp1bi.x | . . . . . 6 ⊢ 𝑋 ∈ 𝑉 | |
22 | 19, 21 | pm3.2i 470 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) |
23 | prelpwi 5458 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → {𝑈, 𝑋} ∈ 𝒫 𝑉) | |
24 | 22, 23 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → {𝑈, 𝑋} ∈ 𝒫 𝑉) |
25 | prid1g 4765 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ {𝑈, 𝑋}) | |
26 | 19, 25 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → 𝑈 ∈ {𝑈, 𝑋}) |
27 | vdegp1bi.xu | . . . . . . . 8 ⊢ 𝑋 ≠ 𝑈 | |
28 | 27 | necomi 2993 | . . . . . . 7 ⊢ 𝑈 ≠ 𝑋 |
29 | hashprg 14431 | . . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑈 ≠ 𝑋 ↔ (♯‘{𝑈, 𝑋}) = 2)) | |
30 | 19, 21, 29 | mp2an 692 | . . . . . . 7 ⊢ (𝑈 ≠ 𝑋 ↔ (♯‘{𝑈, 𝑋}) = 2) |
31 | 28, 30 | mpbi 230 | . . . . . 6 ⊢ (♯‘{𝑈, 𝑋}) = 2 |
32 | 31 | eqcomi 2744 | . . . . 5 ⊢ 2 = (♯‘{𝑈, 𝑋}) |
33 | 2re 12338 | . . . . . 6 ⊢ 2 ∈ ℝ | |
34 | 33 | eqlei 11369 | . . . . 5 ⊢ (2 = (♯‘{𝑈, 𝑋}) → 2 ≤ (♯‘{𝑈, 𝑋})) |
35 | 32, 34 | mp1i 13 | . . . 4 ⊢ ({𝑈, 𝑋} ∈ V → 2 ≤ (♯‘{𝑈, 𝑋})) |
36 | 2, 3, 7, 9, 15, 16, 18, 20, 24, 26, 35 | p1evtxdp1 29547 | . . 3 ⊢ ({𝑈, 𝑋} ∈ V → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 1)) |
37 | 1, 36 | ax-mp 5 | . 2 ⊢ ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 1) |
38 | fzofi 14012 | . . . . 5 ⊢ (0..^(♯‘𝐼)) ∈ Fin | |
39 | wrddm 14556 | . . . . . . . 8 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → dom 𝐼 = (0..^(♯‘𝐼))) | |
40 | 4, 39 | ax-mp 5 | . . . . . . 7 ⊢ dom 𝐼 = (0..^(♯‘𝐼)) |
41 | 40 | eqcomi 2744 | . . . . . 6 ⊢ (0..^(♯‘𝐼)) = dom 𝐼 |
42 | 2, 3, 41 | vtxdgfisnn0 29508 | . . . . 5 ⊢ (((0..^(♯‘𝐼)) ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0) |
43 | 38, 19, 42 | mp2an 692 | . . . 4 ⊢ ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0 |
44 | 43 | nn0rei 12535 | . . 3 ⊢ ((VtxDeg‘𝐺)‘𝑈) ∈ ℝ |
45 | 1re 11259 | . . 3 ⊢ 1 ∈ ℝ | |
46 | rexadd 13271 | . . 3 ⊢ ((((VtxDeg‘𝐺)‘𝑈) ∈ ℝ ∧ 1 ∈ ℝ) → (((VtxDeg‘𝐺)‘𝑈) +𝑒 1) = (((VtxDeg‘𝐺)‘𝑈) + 1)) | |
47 | 44, 45, 46 | mp2an 692 | . 2 ⊢ (((VtxDeg‘𝐺)‘𝑈) +𝑒 1) = (((VtxDeg‘𝐺)‘𝑈) + 1) |
48 | vdegp1ai.d | . . 3 ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃 | |
49 | 48 | oveq1i 7441 | . 2 ⊢ (((VtxDeg‘𝐺)‘𝑈) + 1) = (𝑃 + 1) |
50 | 37, 47, 49 | 3eqtri 2767 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∉ wnel 3044 {crab 3433 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ∅c0 4339 𝒫 cpw 4605 {csn 4631 {cpr 4633 〈cop 4637 class class class wbr 5148 dom cdm 5689 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ≤ cle 11294 2c2 12319 ℕ0cn0 12524 +𝑒 cxad 13150 ..^cfzo 13691 ♯chash 14366 Word cword 14549 ++ cconcat 14605 〈“cs1 14630 Vtxcvtx 29028 iEdgciedg 29029 VtxDegcvtxdg 29498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-xadd 13153 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-vtx 29030 df-iedg 29031 df-vtxdg 29499 |
This theorem is referenced by: vdegp1ci 29571 konigsberglem1 30281 konigsberglem2 30282 |
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