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Mirrors > Home > MPE Home > Th. List > 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 27297 | . 2 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈))) |
11 | 1 | fvexi 6687 | . . . . . 6 ⊢ 𝑉 ∈ V |
12 | snex 5335 | . . . . . 6 ⊢ {〈𝐾, 𝐸〉} ∈ V | |
13 | 11, 12 | pm3.2i 473 | . . . . 5 ⊢ (𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) |
14 | opiedgfv 26795 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) → (iEdg‘〈𝑉, {〈𝐾, 𝐸〉}〉) = {〈𝐾, 𝐸〉}) | |
15 | 13, 14 | mp1i 13 | . . . 4 ⊢ (𝜑 → (iEdg‘〈𝑉, {〈𝐾, 𝐸〉}〉) = {〈𝐾, 𝐸〉}) |
16 | opvtxfv 26792 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) → (Vtx‘〈𝑉, {〈𝐾, 𝐸〉}〉) = 𝑉) | |
17 | 13, 16 | mp1i 13 | . . . 4 ⊢ (𝜑 → (Vtx‘〈𝑉, {〈𝐾, 𝐸〉}〉) = 𝑉) |
18 | p1evtxdeq.n | . . . 4 ⊢ (𝜑 → 𝑈 ∉ 𝐸) | |
19 | 15, 17, 6, 8, 9, 18 | 1hevtxdg0 27290 | . . 3 ⊢ (𝜑 → ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈) = 0) |
20 | 19 | oveq2d 7175 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈)) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 0)) |
21 | 1 | vtxdgelxnn0 27257 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0*) |
22 | xnn0xr 11975 | . . . 4 ⊢ (((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0* → ((VtxDeg‘𝐺)‘𝑈) ∈ ℝ*) | |
23 | 8, 21, 22 | 3syl 18 | . . 3 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) ∈ ℝ*) |
24 | 23 | xaddid1d 12639 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑈) +𝑒 0) = ((VtxDeg‘𝐺)‘𝑈)) |
25 | 10, 20, 24 | 3eqtrd 2863 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∉ wnel 3126 Vcvv 3497 ∪ cun 3937 {csn 4570 〈cop 4576 dom cdm 5558 Fun wfun 6352 ‘cfv 6358 (class class class)co 7159 0cc0 10540 ℝ*cxr 10677 ℕ0*cxnn0 11970 +𝑒 cxad 12508 Vtxcvtx 26784 iEdgciedg 26785 VtxDegcvtxdg 27250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-xadd 12511 df-fz 12896 df-hash 13694 df-vtx 26786 df-iedg 26787 df-vtxdg 27251 |
This theorem is referenced by: vdegp1ai 27321 |
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