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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 29487. 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 29451 | . . . 4 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) | |
2 | cusgrsizeindb0.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | isfusgr 29350 | . . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
4 | fusgrfis 29362 | . . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin) | |
5 | 3, 4 | sylbir 235 | . . . . . 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 2735 | . . . . . . 7 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} | |
11 | cusgrsizeinds.f | . . . . . . 7 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
12 | 10, 11 | elnelun 4399 | . . . . . 6 ⊢ ({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹) = 𝐸 |
13 | 12 | eqcomi 2744 | . . . . 5 ⊢ 𝐸 = ({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹) |
14 | 13 | fveq2i 6910 | . . . 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 2744 | . . . . . . 7 ⊢ (Edg‘𝐺) = 𝐸 |
18 | 17 | eleq1i 2830 | . . . . . 6 ⊢ ((Edg‘𝐺) ∈ Fin ↔ 𝐸 ∈ Fin) |
19 | rabfi 9301 | . . . . . 6 ⊢ (𝐸 ∈ Fin → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) | |
20 | 18, 19 | sylbi 217 | . . . . 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 615 | . . . . . . . 8 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
23 | 22, 3 | sylibr 234 | . . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph) |
24 | 2, 16, 11 | usgrfilem 29359 | . . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ∈ Fin ↔ 𝐹 ∈ Fin)) |
25 | 23, 24 | stoic3 1773 | . . . . . 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 4398 | . . . . 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 14418 | . . . 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 29484 | . . . . 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 7446 | . . 3 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) + (♯‘𝐹)) = (((♯‘𝑉) − 1) + (♯‘𝐹))) |
35 | 15, 31, 34 | 3eqtrd 2779 | . 2 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (Edg‘𝐺) ∈ Fin) → (♯‘𝐸) = (((♯‘𝑉) − 1) + (♯‘𝐹))) |
36 | 9, 35 | mpdan 687 | 1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘𝐸) = (((♯‘𝑉) − 1) + (♯‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 {crab 3433 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 1c1 11154 + caddc 11156 − cmin 11490 ♯chash 14366 Vtxcvtx 29028 Edgcedg 29079 USGraphcusgr 29181 FinUSGraphcfusgr 29348 ComplUSGraphccusgr 29442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 df-vtx 29030 df-iedg 29031 df-edg 29080 df-uhgr 29090 df-upgr 29114 df-umgr 29115 df-uspgr 29182 df-usgr 29183 df-fusgr 29349 df-nbgr 29365 df-uvtx 29418 df-cplgr 29443 df-cusgr 29444 |
This theorem is referenced by: cusgrsize2inds 29486 |
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