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Mirrors > Home > MPE Home > Th. List > Mathboxes > cusgrsizeinds | Structured version Visualization version GIF version |
Description: Part 1 of induction step in cusgrsize 40670. The size of a complete simple graph with 𝑛 vertices is (𝑛 − 1) plus the size of the complete graph reduced by one vertex. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) |
Ref | Expression |
---|---|
cusgrsizeindb0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
cusgrsizeindb0.e | ⊢ 𝐸 = (Edg‘𝐺) |
cusgrsizeinds.f | ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} |
Ref | Expression |
---|---|
cusgrsizeinds | ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘𝐸) = (((#‘𝑉) − 1) + (#‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cusgrusgr 40641 | . . . 4 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph ) | |
2 | cusgrsizeindb0.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | isfusgr 40537 | . . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
4 | fusgrfis 40549 | . . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin) | |
5 | 3, 4 | sylbir 224 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (Edg‘𝐺) ∈ Fin) |
6 | 5 | a1d 25 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (𝑁 ∈ 𝑉 → (Edg‘𝐺) ∈ Fin)) |
7 | 6 | ex 449 | . . . 4 ⊢ (𝐺 ∈ USGraph → (𝑉 ∈ Fin → (𝑁 ∈ 𝑉 → (Edg‘𝐺) ∈ Fin))) |
8 | 1, 7 | syl 17 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → (𝑉 ∈ Fin → (𝑁 ∈ 𝑉 → (Edg‘𝐺) ∈ Fin))) |
9 | 8 | 3imp 1249 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (Edg‘𝐺) ∈ Fin) |
10 | eqid 2610 | . . . . . . 7 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} | |
11 | cusgrsizeinds.f | . . . . . . 7 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
12 | 10, 11 | elnelun 3918 | . . . . . 6 ⊢ ({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹) = 𝐸 |
13 | 12 | eqcomi 2619 | . . . . 5 ⊢ 𝐸 = ({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹) |
14 | 13 | fveq2i 6106 | . . . 4 ⊢ (#‘𝐸) = (#‘({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹)) |
15 | 14 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → (#‘𝐸) = (#‘({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹))) |
16 | cusgrsizeindb0.e | . . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺) | |
17 | 16 | eqcomi 2619 | . . . . . . 7 ⊢ (Edg‘𝐺) = 𝐸 |
18 | 17 | eleq1i 2679 | . . . . . 6 ⊢ ((Edg‘𝐺) ∈ Fin ↔ 𝐸 ∈ Fin) |
19 | rabfi 8070 | . . . . . 6 ⊢ (𝐸 ∈ Fin → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) | |
20 | 18, 19 | sylbi 206 | . . . . 5 ⊢ ((Edg‘𝐺) ∈ Fin → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) |
21 | 20 | adantl 481 | . . . 4 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) |
22 | 1 | anim1i 590 | . . . . . . . 8 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
23 | 22, 3 | sylibr 223 | . . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph ) |
24 | 2, 16, 11 | usgrfilem 40546 | . . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ∈ Fin ↔ 𝐹 ∈ Fin)) |
25 | 23, 24 | stoic3 1692 | . . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝐸 ∈ Fin ↔ 𝐹 ∈ Fin)) |
26 | 18, 25 | syl5bb 271 | . . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((Edg‘𝐺) ∈ Fin ↔ 𝐹 ∈ Fin)) |
27 | 26 | biimpa 500 | . . . 4 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → 𝐹 ∈ Fin) |
28 | 10, 11 | elneldisj 3917 | . . . . 5 ⊢ ({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∩ 𝐹) = ∅ |
29 | 28 | a1i 11 | . . . 4 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → ({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∩ 𝐹) = ∅) |
30 | hashun 13032 | . . . 4 ⊢ (({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin ∧ 𝐹 ∈ Fin ∧ ({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∩ 𝐹) = ∅) → (#‘({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹)) = ((#‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) + (#‘𝐹))) | |
31 | 21, 27, 29, 30 | syl3anc 1318 | . . 3 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → (#‘({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹)) = ((#‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) + (#‘𝐹))) |
32 | 2, 16 | cusgrsizeindslem 40667 | . . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) = ((#‘𝑉) − 1)) |
33 | 32 | adantr 480 | . . . 4 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → (#‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) = ((#‘𝑉) − 1)) |
34 | 33 | oveq1d 6564 | . . 3 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → ((#‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) + (#‘𝐹)) = (((#‘𝑉) − 1) + (#‘𝐹))) |
35 | 15, 31, 34 | 3eqtrd 2648 | . 2 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → (#‘𝐸) = (((#‘𝑉) − 1) + (#‘𝐹))) |
36 | 9, 35 | mpdan 699 | 1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘𝐸) = (((#‘𝑉) − 1) + (#‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 {crab 2900 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 1c1 9816 + caddc 9818 − cmin 10145 #chash 12979 Vtxcvtx 25673 Edgcedga 25792 USGraph cusgr 40379 FinUSGraph cfusgr 40535 ComplUSGraphccusgr 40553 |
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-fal 1481 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-2o 7448 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-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-vtx 25675 df-iedg 25676 df-uhgr 25724 df-upgr 25749 df-umgr 25750 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-fusgr 40536 df-nbgr 40554 df-uvtxa 40556 df-cplgr 40557 df-cusgr 40558 |
This theorem is referenced by: cusgrsize2inds 40669 |
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