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Theorem cusgrasizeinds 26004
 Description: Part 1 of induction step in cusgrasize 26006. 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.)
Hypothesis
Ref Expression
cusgrares.f 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
Assertion
Ref Expression
cusgrasizeinds ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘𝐸) = (((#‘𝑉) − 1) + (#‘𝐹)))
Distinct variable groups:   𝑥,𝐸   𝑥,𝑁   𝑥,𝑉
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem cusgrasizeinds
StepHypRef Expression
1 cusisusgra 25987 . . . 4 (𝑉 ComplUSGrph 𝐸𝑉 USGrph 𝐸)
2 usgrafis 25944 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → 𝐸 ∈ Fin)
32a1d 25 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (𝑁𝑉𝐸 ∈ Fin))
43ex 449 . . . 4 (𝑉 USGrph 𝐸 → (𝑉 ∈ Fin → (𝑁𝑉𝐸 ∈ Fin)))
51, 4syl 17 . . 3 (𝑉 ComplUSGrph 𝐸 → (𝑉 ∈ Fin → (𝑁𝑉𝐸 ∈ Fin)))
653imp 1249 . 2 ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → 𝐸 ∈ Fin)
7 usgrafun 25878 . . . . . . 7 (𝑉 USGrph 𝐸 → Fun 𝐸)
81, 7syl 17 . . . . . 6 (𝑉 ComplUSGrph 𝐸 → Fun 𝐸)
983ad2ant1 1075 . . . . 5 ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → Fun 𝐸)
109adantr 480 . . . 4 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → Fun 𝐸)
11 hashfun 13084 . . . . 5 (𝐸 ∈ Fin → (Fun 𝐸 ↔ (#‘𝐸) = (#‘dom 𝐸)))
1211adantl 481 . . . 4 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (Fun 𝐸 ↔ (#‘𝐸) = (#‘dom 𝐸)))
1310, 12mpbid 221 . . 3 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (#‘𝐸) = (#‘dom 𝐸))
14 cusgrares.f . . . . . . 7 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
1514usgrafilem1 25940 . . . . . 6 dom 𝐸 = (dom 𝐹 ∪ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})
1615a1i 11 . . . . 5 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → dom 𝐸 = (dom 𝐹 ∪ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}))
1716fveq2d 6107 . . . 4 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (#‘dom 𝐸) = (#‘(dom 𝐹 ∪ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})))
18 finresfin 8071 . . . . . . . 8 (𝐸 ∈ Fin → (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) ∈ Fin)
1914, 18syl5eqel 2692 . . . . . . 7 (𝐸 ∈ Fin → 𝐹 ∈ Fin)
20 dmfi 8129 . . . . . . 7 (𝐹 ∈ Fin → dom 𝐹 ∈ Fin)
2119, 20syl 17 . . . . . 6 (𝐸 ∈ Fin → dom 𝐹 ∈ Fin)
2221adantl 481 . . . . 5 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → dom 𝐹 ∈ Fin)
23 dmfi 8129 . . . . . . 7 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
24 rabfi 8070 . . . . . . 7 (dom 𝐸 ∈ Fin → {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ Fin)
2523, 24syl 17 . . . . . 6 (𝐸 ∈ Fin → {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ Fin)
2625adantl 481 . . . . 5 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ Fin)
2714cusgrasizeindslem1 26002 . . . . . 6 (dom 𝐹 ∩ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ∅
2827a1i 11 . . . . 5 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (dom 𝐹 ∩ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ∅)
29 hashun 13032 . . . . 5 ((dom 𝐹 ∈ Fin ∧ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ Fin ∧ (dom 𝐹 ∩ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ∅) → (#‘(dom 𝐹 ∪ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})) = ((#‘dom 𝐹) + (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})))
3022, 26, 28, 29syl3anc 1318 . . . 4 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (#‘(dom 𝐹 ∪ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})) = ((#‘dom 𝐹) + (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})))
3114cusgrasizeindslem2 26003 . . . . . . 7 ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ((#‘𝑉) − 1))
3231adantr 480 . . . . . 6 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ((#‘𝑉) − 1))
3332oveq2d 6565 . . . . 5 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → ((#‘dom 𝐹) + (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})) = ((#‘dom 𝐹) + ((#‘𝑉) − 1)))
34 hashcl 13009 . . . . . . . . 9 (dom 𝐹 ∈ Fin → (#‘dom 𝐹) ∈ ℕ0)
3534nn0cnd 11230 . . . . . . . 8 (dom 𝐹 ∈ Fin → (#‘dom 𝐹) ∈ ℂ)
3619, 20, 353syl 18 . . . . . . 7 (𝐸 ∈ Fin → (#‘dom 𝐹) ∈ ℂ)
3736adantl 481 . . . . . 6 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (#‘dom 𝐹) ∈ ℂ)
38 hashcl 13009 . . . . . . . . 9 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
39 nn0cn 11179 . . . . . . . . . 10 ((#‘𝑉) ∈ ℕ0 → (#‘𝑉) ∈ ℂ)
40 1cnd 9935 . . . . . . . . . 10 ((#‘𝑉) ∈ ℕ0 → 1 ∈ ℂ)
4139, 40subcld 10271 . . . . . . . . 9 ((#‘𝑉) ∈ ℕ0 → ((#‘𝑉) − 1) ∈ ℂ)
4238, 41syl 17 . . . . . . . 8 (𝑉 ∈ Fin → ((#‘𝑉) − 1) ∈ ℂ)
43423ad2ant2 1076 . . . . . . 7 ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘𝑉) − 1) ∈ ℂ)
4443adantr 480 . . . . . 6 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → ((#‘𝑉) − 1) ∈ ℂ)
4537, 44addcomd 10117 . . . . 5 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → ((#‘dom 𝐹) + ((#‘𝑉) − 1)) = (((#‘𝑉) − 1) + (#‘dom 𝐹)))
4633, 45eqtrd 2644 . . . 4 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → ((#‘dom 𝐹) + (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})) = (((#‘𝑉) − 1) + (#‘dom 𝐹)))
4717, 30, 463eqtrd 2648 . . 3 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (#‘dom 𝐸) = (((#‘𝑉) − 1) + (#‘dom 𝐹)))
4814cusgrares 26001 . . . . . . . . 9 ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → (𝑉 ∖ {𝑁}) ComplUSGrph 𝐹)
49 cusisusgra 25987 . . . . . . . . 9 ((𝑉 ∖ {𝑁}) ComplUSGrph 𝐹 → (𝑉 ∖ {𝑁}) USGrph 𝐹)
50 usgrafun 25878 . . . . . . . . 9 ((𝑉 ∖ {𝑁}) USGrph 𝐹 → Fun 𝐹)
5148, 49, 503syl 18 . . . . . . . 8 ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → Fun 𝐹)
52513adant2 1073 . . . . . . 7 ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → Fun 𝐹)
5352adantr 480 . . . . . 6 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → Fun 𝐹)
5419adantl 481 . . . . . . 7 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → 𝐹 ∈ Fin)
55 hashfun 13084 . . . . . . 7 (𝐹 ∈ Fin → (Fun 𝐹 ↔ (#‘𝐹) = (#‘dom 𝐹)))
5654, 55syl 17 . . . . . 6 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (Fun 𝐹 ↔ (#‘𝐹) = (#‘dom 𝐹)))
5753, 56mpbid 221 . . . . 5 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (#‘𝐹) = (#‘dom 𝐹))
5857eqcomd 2616 . . . 4 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (#‘dom 𝐹) = (#‘𝐹))
5958oveq2d 6565 . . 3 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (((#‘𝑉) − 1) + (#‘dom 𝐹)) = (((#‘𝑉) − 1) + (#‘𝐹)))
6013, 47, 593eqtrd 2648 . 2 (((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ 𝐸 ∈ Fin) → (#‘𝐸) = (((#‘𝑉) − 1) + (#‘𝐹)))
616, 60mpdan 699 1 ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘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   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  {csn 4125   class class class wbr 4583  dom cdm 5038   ↾ cres 5040  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℂcc 9813  1c1 9816   + caddc 9818   − cmin 10145  ℕ0cn0 11169  #chash 12979   USGrph cusg 25859   ComplUSGrph ccusgra 25947 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-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-usgra 25862  df-nbgra 25949  df-cusgra 25950 This theorem is referenced by:  cusgrasize2inds  26005
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