![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5432 | . . 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 14474 | . . . . . 6 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼:(0..^(♯‘𝐼))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
6 | 5 | ffund 6721 | . . . . 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 14484 | . . . . . . 7 ⊢ (𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐼 ∈ Word V) | |
12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
13 | cats1un 14676 | . . . . . 6 ⊢ ((𝐼 ∈ Word V ∧ {𝑋, 𝑌} ∈ V) → (𝐼 ++ 〈“{𝑋, 𝑌}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑋, 𝑌}〉})) | |
14 | 12, 13 | mpan 687 | . . . . 5 ⊢ ({𝑋, 𝑌} ∈ V → (𝐼 ++ 〈“{𝑋, 𝑌}”〉) = (𝐼 ∪ {〈(♯‘𝐼), {𝑋, 𝑌}〉})) |
15 | 10, 14 | eqtrid 2783 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (iEdg‘𝐹) = (𝐼 ∪ {〈(♯‘𝐼), {𝑋, 𝑌}〉})) |
16 | fvexd 6906 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ V → (♯‘𝐼) ∈ V) | |
17 | wrdlndm 14485 | . . . . 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 2994 | . . . . . 6 ⊢ 𝑈 ≠ 𝑋 |
24 | vdegp1ai.yu | . . . . . . 7 ⊢ 𝑌 ≠ 𝑈 | |
25 | 24 | necomi 2994 | . . . . . 6 ⊢ 𝑈 ≠ 𝑌 |
26 | 23, 25 | prneli 4658 | . . . . 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 29038 | . . 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 2759 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∉ wnel 3045 {crab 3431 Vcvv 3473 ∖ cdif 3945 ∪ cun 3946 ∅c0 4322 𝒫 cpw 4602 {csn 4628 {cpr 4630 〈cop 4634 class class class wbr 5148 dom cdm 5676 Fun wfun 6537 ‘cfv 6543 (class class class)co 7412 0cc0 11114 ≤ cle 11254 2c2 12272 ..^cfzo 13632 ♯chash 14295 Word cword 14469 ++ cconcat 14525 〈“cs1 14550 Vtxcvtx 28524 iEdgciedg 28525 VtxDegcvtxdg 28990 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-oadd 8474 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-dju 9900 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-xnn0 12550 df-z 12564 df-uz 12828 df-xadd 13098 df-fz 13490 df-fzo 13633 df-hash 14296 df-word 14470 df-concat 14526 df-s1 14551 df-vtx 28526 df-iedg 28527 df-vtxdg 28991 |
This theorem is referenced by: konigsberglem1 29773 konigsberglem2 29774 konigsberglem3 29775 |
Copyright terms: Public domain | W3C validator |