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Mirrors > Home > MPE Home > Th. List > cusgrsizeinds | Structured version Visualization version GIF version |
Description: Part 1 of induction step in cusgrsize 29144. 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 29109 | . . . 4 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) | |
2 | cusgrsizeindb0.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | isfusgr 29008 | . . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
4 | fusgrfis 29020 | . . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin) | |
5 | 3, 4 | sylbir 234 | . . . . . 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 412 | . . . 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 1110 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (Edg‘𝐺) ∈ Fin) |
10 | eqid 2731 | . . . . . . 7 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} | |
11 | cusgrsizeinds.f | . . . . . . 7 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
12 | 10, 11 | elnelun 4389 | . . . . . 6 ⊢ ({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹) = 𝐸 |
13 | 12 | eqcomi 2740 | . . . . 5 ⊢ 𝐸 = ({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹) |
14 | 13 | fveq2i 6894 | . . . 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 2740 | . . . . . . 7 ⊢ (Edg‘𝐺) = 𝐸 |
18 | 17 | eleq1i 2823 | . . . . . 6 ⊢ ((Edg‘𝐺) ∈ Fin ↔ 𝐸 ∈ Fin) |
19 | rabfi 9275 | . . . . . 6 ⊢ (𝐸 ∈ Fin → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) | |
20 | 18, 19 | sylbi 216 | . . . . 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 614 | . . . . . . . 8 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
23 | 22, 3 | sylibr 233 | . . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph) |
24 | 2, 16, 11 | usgrfilem 29017 | . . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ∈ Fin ↔ 𝐹 ∈ Fin)) |
25 | 23, 24 | stoic3 1777 | . . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝐸 ∈ Fin ↔ 𝐹 ∈ Fin)) |
26 | 18, 25 | bitrid 283 | . . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((Edg‘𝐺) ∈ Fin ↔ 𝐹 ∈ Fin)) |
27 | 26 | biimpa 476 | . . . 4 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → 𝐹 ∈ Fin) |
28 | 10, 11 | elneldisj 4388 | . . . . 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 14349 | . . . 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 1370 | . . 3 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → (♯‘({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹)) = ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) + (♯‘𝐹))) |
32 | 2, 16 | cusgrsizeindslem 29141 | . . . . 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 7427 | . . 3 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) + (♯‘𝐹)) = (((♯‘𝑉) − 1) + (♯‘𝐹))) |
35 | 15, 31, 34 | 3eqtrd 2775 | . 2 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → (♯‘𝐸) = (((♯‘𝑉) − 1) + (♯‘𝐹))) |
36 | 9, 35 | mpdan 684 | 1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘𝐸) = (((♯‘𝑉) − 1) + (♯‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∉ wnel 3045 {crab 3431 ∪ cun 3946 ∩ cin 3947 ∅c0 4322 ‘cfv 6543 (class class class)co 7412 Fincfn 8945 1c1 11117 + caddc 11119 − cmin 11451 ♯chash 14297 Vtxcvtx 28689 Edgcedg 28740 USGraphcusgr 28842 FinUSGraphcfusgr 29006 ComplUSGraphccusgr 29100 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-oadd 8476 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-n0 12480 df-xnn0 12552 df-z 12566 df-uz 12830 df-fz 13492 df-hash 14298 df-vtx 28691 df-iedg 28692 df-edg 28741 df-uhgr 28751 df-upgr 28775 df-umgr 28776 df-uspgr 28843 df-usgr 28844 df-fusgr 29007 df-nbgr 29023 df-uvtx 29076 df-cplgr 29101 df-cusgr 29102 |
This theorem is referenced by: cusgrsize2inds 29143 |
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