| Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > ILE Home > Th. List > p1evtxdeqfi | 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) |
| p1evtxdeqfi.vfi | ⊢ (𝜑 → 𝑉 ∈ Fin) |
| p1evtxdeqfi.u | ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
| p1evtxdeqfi.ifi | ⊢ (𝜑 → dom 𝐼 ∈ Fin) |
| p1evtxdeqfi.e | ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) |
| p1evtxdeqfi.2o | ⊢ (𝜑 → 𝐸 ≈ 2o) |
| p1evtxdeq.e | ⊢ (𝜑 → 𝐸 ∈ 𝑌) |
| p1evtxdeq.n | ⊢ (𝜑 → 𝑈 ∉ 𝐸) |
| Ref | Expression |
|---|---|
| p1evtxdeqfi | ⊢ (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 | p1evtxdeqfi.vfi | . . 3 ⊢ (𝜑 → 𝑉 ∈ Fin) | |
| 10 | p1evtxdeqfi.u | . . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph) | |
| 11 | p1evtxdeqfi.ifi | . . 3 ⊢ (𝜑 → dom 𝐼 ∈ Fin) | |
| 12 | p1evtxdeqfi.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) | |
| 13 | p1evtxdeqfi.2o | . . 3 ⊢ (𝜑 → 𝐸 ≈ 2o) | |
| 14 | p1evtxdeq.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑌) | |
| 15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | p1evtxdeqfilem 16535 | . 2 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) + ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈))) |
| 16 | 9 | elexd 2835 | . . . . 5 ⊢ (𝜑 → 𝑉 ∈ V) |
| 17 | opexg 4366 | . . . . . . 7 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝐸 ∈ 𝒫 𝑉) → 〈𝐾, 𝐸〉 ∈ V) | |
| 18 | 6, 12, 17 | syl2anc 415 | . . . . . 6 ⊢ (𝜑 → 〈𝐾, 𝐸〉 ∈ V) |
| 19 | snexg 4319 | . . . . . 6 ⊢ (〈𝐾, 𝐸〉 ∈ V → {〈𝐾, 𝐸〉} ∈ V) | |
| 20 | 18, 19 | syl 14 | . . . . 5 ⊢ (𝜑 → {〈𝐾, 𝐸〉} ∈ V) |
| 21 | opiedgfv 16249 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) → (iEdg‘〈𝑉, {〈𝐾, 𝐸〉}〉) = {〈𝐾, 𝐸〉}) | |
| 22 | 16, 20, 21 | syl2anc 415 | . . . 4 ⊢ (𝜑 → (iEdg‘〈𝑉, {〈𝐾, 𝐸〉}〉) = {〈𝐾, 𝐸〉}) |
| 23 | opvtxfv 16246 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) → (Vtx‘〈𝑉, {〈𝐾, 𝐸〉}〉) = 𝑉) | |
| 24 | 16, 20, 23 | syl2anc 415 | . . . 4 ⊢ (𝜑 → (Vtx‘〈𝑉, {〈𝐾, 𝐸〉}〉) = 𝑉) |
| 25 | 6, 9, 12, 13 | upgr1een 16348 | . . . 4 ⊢ (𝜑 → 〈𝑉, {〈𝐾, 𝐸〉}〉 ∈ UPGraph) |
| 26 | p1evtxdeq.n | . . . 4 ⊢ (𝜑 → 𝑈 ∉ 𝐸) | |
| 27 | 22, 24, 6, 8, 9, 25, 14, 26 | 1hevtxdg0fi 16531 | . . 3 ⊢ (𝜑 → ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈) = 0) |
| 28 | 27 | oveq2d 6095 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑈) + ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈)) = (((VtxDeg‘𝐺)‘𝑈) + 0)) |
| 29 | eqid 2238 | . . . . . 6 ⊢ dom 𝐼 = dom 𝐼 | |
| 30 | 1, 2, 29, 11, 9, 10 | vtxdgfif 16517 | . . . . 5 ⊢ (𝜑 → (VtxDeg‘𝐺):𝑉⟶ℕ0) |
| 31 | 30, 8 | ffvelcdmd 5838 | . . . 4 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0) |
| 32 | 31 | nn0cnd 9605 | . . 3 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) ∈ ℂ) |
| 33 | 32 | addridd 8469 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑈) + 0) = ((VtxDeg‘𝐺)‘𝑈)) |
| 34 | 15, 28, 33 | 3eqtrd 2275 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 = wceq 1402 ∈ wcel 2209 ∉ wnel 2515 Vcvv 2821 ∪ cun 3218 𝒫 cpw 3688 {csn 3708 〈cop 3711 class class class wbr 4128 dom cdm 4772 Fun wfun 5369 ‘cfv 5375 (class class class)co 6079 2oc2o 6675 ≈ cen 7014 Fincfn 7016 0cc0 8173 + caddc 8176 ℕ0cn0 9546 Vtxcvtx 16236 iEdgciedg 16237 UPGraphcupgr 16315 VtxDegcvtxdg 16510 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 |
| This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-irdg 6635 df-frec 6656 df-1o 6681 df-2o 6682 df-oadd 6685 df-er 6801 df-en 7017 df-dom 7018 df-fin 7019 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-5 9349 df-6 9350 df-7 9351 df-8 9352 df-9 9353 df-n0 9547 df-z 9628 df-dec 9761 df-uz 9905 df-xadd 10158 df-fz 10395 df-ihash 11198 df-ndx 13338 df-slot 13339 df-base 13341 df-edgf 16229 df-vtx 16238 df-iedg 16239 df-upgren 16317 df-vtxdg 16511 |
| This theorem is referenced by: vdegp1aid 16538 |
| Copyright terms: Public domain | W3C validator |