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Mirrors > Home > MPE Home > Th. List > vtxduhgrun | Structured version Visualization version GIF version |
Description: The degree of a vertex in the union of two hypergraphs on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 12-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.) |
Ref | Expression |
---|---|
vtxduhgrun.i | ⊢ 𝐼 = (iEdg‘𝐺) |
vtxduhgrun.j | ⊢ 𝐽 = (iEdg‘𝐻) |
vtxduhgrun.vg | ⊢ 𝑉 = (Vtx‘𝐺) |
vtxduhgrun.vh | ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
vtxduhgrun.vu | ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) |
vtxduhgrun.d | ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅) |
vtxduhgrun.g | ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
vtxduhgrun.h | ⊢ (𝜑 → 𝐻 ∈ UHGraph) |
vtxduhgrun.n | ⊢ (𝜑 → 𝑁 ∈ 𝑉) |
vtxduhgrun.u | ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽)) |
Ref | Expression |
---|---|
vtxduhgrun | ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxduhgrun.i | . 2 ⊢ 𝐼 = (iEdg‘𝐺) | |
2 | vtxduhgrun.j | . 2 ⊢ 𝐽 = (iEdg‘𝐻) | |
3 | vtxduhgrun.vg | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | vtxduhgrun.vh | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | |
5 | vtxduhgrun.vu | . 2 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) | |
6 | vtxduhgrun.d | . 2 ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅) | |
7 | vtxduhgrun.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph) | |
8 | 1 | uhgrfun 26853 | . . 3 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼) |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → Fun 𝐼) |
10 | vtxduhgrun.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ UHGraph) | |
11 | 2 | uhgrfun 26853 | . . 3 ⊢ (𝐻 ∈ UHGraph → Fun 𝐽) |
12 | 10, 11 | syl 17 | . 2 ⊢ (𝜑 → Fun 𝐽) |
13 | vtxduhgrun.n | . 2 ⊢ (𝜑 → 𝑁 ∈ 𝑉) | |
14 | vtxduhgrun.u | . 2 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽)) | |
15 | 1, 2, 3, 4, 5, 6, 9, 12, 13, 14 | vtxdun 27265 | 1 ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 dom cdm 5557 Fun wfun 6351 ‘cfv 6357 (class class class)co 7158 +𝑒 cxad 12508 Vtxcvtx 26783 iEdgciedg 26784 UHGraphcuhgr 26843 VtxDegcvtxdg 27249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-xadd 12511 df-hash 13694 df-uhgr 26845 df-vtxdg 27250 |
This theorem is referenced by: (None) |
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