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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 29530 | . 2 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈))) |
| 11 | 1 | fvexi 6920 | . . . . . 6 ⊢ 𝑉 ∈ V |
| 12 | snex 5436 | . . . . . 6 ⊢ {〈𝐾, 𝐸〉} ∈ V | |
| 13 | 11, 12 | pm3.2i 470 | . . . . 5 ⊢ (𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) |
| 14 | opiedgfv 29024 | . . . . 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 29021 | . . . . 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 29523 | . . 3 ⊢ (𝜑 → ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈) = 0) |
| 20 | 19 | oveq2d 7447 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈)) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 0)) |
| 21 | 1 | vtxdgelxnn0 29490 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0*) |
| 22 | xnn0xr 12604 | . . . 4 ⊢ (((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0* → ((VtxDeg‘𝐺)‘𝑈) ∈ ℝ*) | |
| 23 | 8, 21, 22 | 3syl 18 | . . 3 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) ∈ ℝ*) |
| 24 | 23 | xaddridd 13285 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑈) +𝑒 0) = ((VtxDeg‘𝐺)‘𝑈)) |
| 25 | 10, 20, 24 | 3eqtrd 2781 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 Vcvv 3480 ∪ cun 3949 {csn 4626 〈cop 4632 dom cdm 5685 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℝ*cxr 11294 ℕ0*cxnn0 12599 +𝑒 cxad 13152 Vtxcvtx 29013 iEdgciedg 29014 VtxDegcvtxdg 29483 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-xadd 13155 df-fz 13548 df-hash 14370 df-vtx 29015 df-iedg 29016 df-vtxdg 29484 |
| This theorem is referenced by: vdegp1ai 29554 |
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