Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > vdegp1bi | 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 {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.) |
Ref | Expression |
---|---|
vdeg0i.v | ⊢ 𝑉 ∈ V |
vdegp1ai.1 | ⊢ (⊤ → 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
vdegp1ai.u | ⊢ 𝑈 ∈ 𝑉 |
vdegp1ai.2 | ⊢ ((𝑉 VDeg 𝐸)‘𝑈) = 𝑃 |
vdegp1bi.3 | ⊢ 𝑄 = (𝑃 + 1) |
vdegp1bi.4 | ⊢ 𝑋 ∈ 𝑉 |
vdegp1bi.5 | ⊢ 𝑋 ≠ 𝑈 |
vdegp1bi.f | ⊢ 𝐹 = (𝐸 ++ 〈“{𝑈, 𝑋}”〉) |
Ref | Expression |
---|---|
vdegp1bi | ⊢ ((𝑉 VDeg 𝐹)‘𝑈) = 𝑄 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vdegp1ai.1 | . . . . . 6 ⊢ (⊤ → 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) | |
2 | wrdf 13165 | . . . . . 6 ⊢ (𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝐸:(0..^(#‘𝐸))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) | |
3 | ffn 5958 | . . . . . 6 ⊢ (𝐸:(0..^(#‘𝐸))⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝐸 Fn (0..^(#‘𝐸))) | |
4 | 1, 2, 3 | 3syl 18 | . . . . 5 ⊢ (⊤ → 𝐸 Fn (0..^(#‘𝐸))) |
5 | fvex 6113 | . . . . . . 7 ⊢ (#‘𝐸) ∈ V | |
6 | prex 4836 | . . . . . . 7 ⊢ {𝑈, 𝑋} ∈ V | |
7 | 5, 6 | f1osn 6088 | . . . . . 6 ⊢ {〈(#‘𝐸), {𝑈, 𝑋}〉}:{(#‘𝐸)}–1-1-onto→{{𝑈, 𝑋}} |
8 | f1ofn 6051 | . . . . . 6 ⊢ ({〈(#‘𝐸), {𝑈, 𝑋}〉}:{(#‘𝐸)}–1-1-onto→{{𝑈, 𝑋}} → {〈(#‘𝐸), {𝑈, 𝑋}〉} Fn {(#‘𝐸)}) | |
9 | 7, 8 | mp1i 13 | . . . . 5 ⊢ (⊤ → {〈(#‘𝐸), {𝑈, 𝑋}〉} Fn {(#‘𝐸)}) |
10 | fzofi 12635 | . . . . . 6 ⊢ (0..^(#‘𝐸)) ∈ Fin | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (⊤ → (0..^(#‘𝐸)) ∈ Fin) |
12 | snfi 7923 | . . . . . 6 ⊢ {(#‘𝐸)} ∈ Fin | |
13 | 12 | a1i 11 | . . . . 5 ⊢ (⊤ → {(#‘𝐸)} ∈ Fin) |
14 | fzonel 12352 | . . . . . . 7 ⊢ ¬ (#‘𝐸) ∈ (0..^(#‘𝐸)) | |
15 | disjsn 4192 | . . . . . . 7 ⊢ (((0..^(#‘𝐸)) ∩ {(#‘𝐸)}) = ∅ ↔ ¬ (#‘𝐸) ∈ (0..^(#‘𝐸))) | |
16 | 14, 15 | mpbir 220 | . . . . . 6 ⊢ ((0..^(#‘𝐸)) ∩ {(#‘𝐸)}) = ∅ |
17 | 16 | a1i 11 | . . . . 5 ⊢ (⊤ → ((0..^(#‘𝐸)) ∩ {(#‘𝐸)}) = ∅) |
18 | vdeg0i.v | . . . . . . 7 ⊢ 𝑉 ∈ V | |
19 | 1 | trud 1484 | . . . . . . 7 ⊢ 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} |
20 | wrdumgra 25845 | . . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) → (𝑉 UMGrph 𝐸 ↔ 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})) | |
21 | 18, 19, 20 | mp2an 704 | . . . . . 6 ⊢ (𝑉 UMGrph 𝐸 ↔ 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
22 | 1, 21 | sylibr 223 | . . . . 5 ⊢ (⊤ → 𝑉 UMGrph 𝐸) |
23 | vdegp1ai.u | . . . . . . 7 ⊢ 𝑈 ∈ 𝑉 | |
24 | vdegp1bi.4 | . . . . . . 7 ⊢ 𝑋 ∈ 𝑉 | |
25 | umgra1 25855 | . . . . . . 7 ⊢ (((𝑉 ∈ V ∧ (#‘𝐸) ∈ V) ∧ (𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → 𝑉 UMGrph {〈(#‘𝐸), {𝑈, 𝑋}〉}) | |
26 | 18, 5, 23, 24, 25 | mp4an 705 | . . . . . 6 ⊢ 𝑉 UMGrph {〈(#‘𝐸), {𝑈, 𝑋}〉} |
27 | 26 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝑉 UMGrph {〈(#‘𝐸), {𝑈, 𝑋}〉}) |
28 | 23 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝑈 ∈ 𝑉) |
29 | 4, 9, 11, 13, 17, 22, 27, 28 | vdgrfiun 26429 | . . . 4 ⊢ (⊤ → ((𝑉 VDeg (𝐸 ∪ {〈(#‘𝐸), {𝑈, 𝑋}〉}))‘𝑈) = (((𝑉 VDeg 𝐸)‘𝑈) + ((𝑉 VDeg {〈(#‘𝐸), {𝑈, 𝑋}〉})‘𝑈))) |
30 | 29 | trud 1484 | . . 3 ⊢ ((𝑉 VDeg (𝐸 ∪ {〈(#‘𝐸), {𝑈, 𝑋}〉}))‘𝑈) = (((𝑉 VDeg 𝐸)‘𝑈) + ((𝑉 VDeg {〈(#‘𝐸), {𝑈, 𝑋}〉})‘𝑈)) |
31 | vdegp1ai.2 | . . . 4 ⊢ ((𝑉 VDeg 𝐸)‘𝑈) = 𝑃 | |
32 | 18 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝑉 ∈ V) |
33 | 5 | a1i 11 | . . . . . 6 ⊢ (⊤ → (#‘𝐸) ∈ V) |
34 | 24 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝑋 ∈ 𝑉) |
35 | vdegp1bi.5 | . . . . . . 7 ⊢ 𝑋 ≠ 𝑈 | |
36 | 35 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝑋 ≠ 𝑈) |
37 | 32, 33, 28, 34, 36 | vdgr1b 26431 | . . . . 5 ⊢ (⊤ → ((𝑉 VDeg {〈(#‘𝐸), {𝑈, 𝑋}〉})‘𝑈) = 1) |
38 | 37 | trud 1484 | . . . 4 ⊢ ((𝑉 VDeg {〈(#‘𝐸), {𝑈, 𝑋}〉})‘𝑈) = 1 |
39 | 31, 38 | oveq12i 6561 | . . 3 ⊢ (((𝑉 VDeg 𝐸)‘𝑈) + ((𝑉 VDeg {〈(#‘𝐸), {𝑈, 𝑋}〉})‘𝑈)) = (𝑃 + 1) |
40 | 30, 39 | eqtri 2632 | . 2 ⊢ ((𝑉 VDeg (𝐸 ∪ {〈(#‘𝐸), {𝑈, 𝑋}〉}))‘𝑈) = (𝑃 + 1) |
41 | vdegp1bi.f | . . . . 5 ⊢ 𝐹 = (𝐸 ++ 〈“{𝑈, 𝑋}”〉) | |
42 | 18, 23, 24 | umgrabi 26510 | . . . . . . 7 ⊢ (⊤ → {𝑈, 𝑋} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
43 | 42 | trud 1484 | . . . . . 6 ⊢ {𝑈, 𝑋} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} |
44 | cats1un 13327 | . . . . . 6 ⊢ ((𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ∧ {𝑈, 𝑋} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) → (𝐸 ++ 〈“{𝑈, 𝑋}”〉) = (𝐸 ∪ {〈(#‘𝐸), {𝑈, 𝑋}〉})) | |
45 | 19, 43, 44 | mp2an 704 | . . . . 5 ⊢ (𝐸 ++ 〈“{𝑈, 𝑋}”〉) = (𝐸 ∪ {〈(#‘𝐸), {𝑈, 𝑋}〉}) |
46 | 41, 45 | eqtri 2632 | . . . 4 ⊢ 𝐹 = (𝐸 ∪ {〈(#‘𝐸), {𝑈, 𝑋}〉}) |
47 | 46 | oveq2i 6560 | . . 3 ⊢ (𝑉 VDeg 𝐹) = (𝑉 VDeg (𝐸 ∪ {〈(#‘𝐸), {𝑈, 𝑋}〉})) |
48 | 47 | fveq1i 6104 | . 2 ⊢ ((𝑉 VDeg 𝐹)‘𝑈) = ((𝑉 VDeg (𝐸 ∪ {〈(#‘𝐸), {𝑈, 𝑋}〉}))‘𝑈) |
49 | vdegp1bi.3 | . 2 ⊢ 𝑄 = (𝑃 + 1) | |
50 | 40, 48, 49 | 3eqtr4i 2642 | 1 ⊢ ((𝑉 VDeg 𝐹)‘𝑈) = 𝑄 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {cpr 4127 〈cop 4131 class class class wbr 4583 Fn wfn 5799 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 2c2 10947 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 〈“cs1 13149 UMGrph cumg 25841 VDeg cvdg 26420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-umgra 25842 df-vdgr 26421 |
This theorem is referenced by: vdegp1ci 26513 konigsberg 26514 |
Copyright terms: Public domain | W3C validator |