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Mirrors > Home > MPE Home > Th. List > Mathboxes > p1evtxdeq | Structured version Visualization version GIF version |
Description: If an edge 𝐸 which does not contain vertex 𝑈 is added to a graph 𝐺 (yielding a graph 𝐹), the degree of 𝑈 is the same in both graphs. (Contributed by AV, 2-Mar-2021.) |
Ref | Expression |
---|---|
p1evtxdeq.v | ⊢ 𝑉 = (Vtx‘𝐺) |
p1evtxdeq.i | ⊢ 𝐼 = (iEdg‘𝐺) |
p1evtxdeq.f | ⊢ (𝜑 → Fun 𝐼) |
p1evtxdeq.fv | ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉) |
p1evtxdeq.fi | ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {〈𝐾, 𝐸〉})) |
p1evtxdeq.k | ⊢ (𝜑 → 𝐾 ∈ 𝑋) |
p1evtxdeq.d | ⊢ (𝜑 → 𝐾 ∉ dom 𝐼) |
p1evtxdeq.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
p1evtxdeq.e | ⊢ (𝜑 → 𝐸 ∈ 𝑌) |
p1evtxdeq.n | ⊢ (𝜑 → 𝑈 ∉ 𝐸) |
Ref | Expression |
---|---|
p1evtxdeq | ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | p1evtxdeq.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | p1evtxdeq.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | p1evtxdeq.f | . . 3 ⊢ (𝜑 → Fun 𝐼) | |
4 | p1evtxdeq.fv | . . 3 ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉) | |
5 | p1evtxdeq.fi | . . 3 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {〈𝐾, 𝐸〉})) | |
6 | p1evtxdeq.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑋) | |
7 | p1evtxdeq.d | . . 3 ⊢ (𝜑 → 𝐾 ∉ dom 𝐼) | |
8 | p1evtxdeq.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
9 | p1evtxdeq.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑌) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | p1evtxdeqlem 40728 | . 2 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈))) |
11 | fvex 6113 | . . . . . . 7 ⊢ (Vtx‘𝐺) ∈ V | |
12 | 1, 11 | eqeltri 2684 | . . . . . 6 ⊢ 𝑉 ∈ V |
13 | snex 4835 | . . . . . 6 ⊢ {〈𝐾, 𝐸〉} ∈ V | |
14 | 12, 13 | pm3.2i 470 | . . . . 5 ⊢ (𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) |
15 | opiedgfv 25684 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) → (iEdg‘〈𝑉, {〈𝐾, 𝐸〉}〉) = {〈𝐾, 𝐸〉}) | |
16 | 14, 15 | mp1i 13 | . . . 4 ⊢ (𝜑 → (iEdg‘〈𝑉, {〈𝐾, 𝐸〉}〉) = {〈𝐾, 𝐸〉}) |
17 | opvtxfv 25681 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) → (Vtx‘〈𝑉, {〈𝐾, 𝐸〉}〉) = 𝑉) | |
18 | 14, 17 | mp1i 13 | . . . 4 ⊢ (𝜑 → (Vtx‘〈𝑉, {〈𝐾, 𝐸〉}〉) = 𝑉) |
19 | p1evtxdeq.n | . . . 4 ⊢ (𝜑 → 𝑈 ∉ 𝐸) | |
20 | 16, 18, 6, 8, 9, 19 | 1hevtxdg0 40720 | . . 3 ⊢ (𝜑 → ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈) = 0) |
21 | 20 | oveq2d 6565 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈)) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 0)) |
22 | 1 | vtxdgelxnn0 40687 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0*) |
23 | xnn0xr 11245 | . . . 4 ⊢ (((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0* → ((VtxDeg‘𝐺)‘𝑈) ∈ ℝ*) | |
24 | 8, 22, 23 | 3syl 18 | . . 3 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) ∈ ℝ*) |
25 | 24 | xaddid1d 11948 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑈) +𝑒 0) = ((VtxDeg‘𝐺)‘𝑈)) |
26 | 10, 21, 25 | 3eqtrd 2648 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 Vcvv 3173 ∪ cun 3538 {csn 4125 〈cop 4131 dom cdm 5038 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℝ*cxr 9952 ℕ0*cxnn0 11240 +𝑒 cxad 11820 Vtxcvtx 25673 iEdgciedg 25674 VtxDegcvtxdg 40681 |
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-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-hash 12980 df-vtx 25675 df-iedg 25676 df-vtxdg 40682 |
This theorem is referenced by: vdegp1ai-av 40752 |
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