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Theorem ballotlemrinv 30400
Description: 𝑅 is its own inverse : it is an involution. (Contributed by Thierry Arnoux, 10-Apr-2017.)
Hypotheses
Ref Expression
ballotth.m 𝑀 ∈ ℕ
ballotth.n 𝑁 ∈ ℕ
ballotth.o 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}
ballotth.p 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))
ballotth.f 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
ballotth.e 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
ballotth.mgtn 𝑁 < 𝑀
ballotth.i 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
Assertion
Ref Expression
ballotlemrinv 𝑅 = 𝑅
Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂   𝑘,𝑀   𝑘,𝑁   𝑘,𝑂   𝑖,𝑐,𝐹,𝑘   𝑖,𝐸,𝑘   𝑘,𝐼,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐   𝑆,𝑘,𝑖,𝑐   𝑅,𝑖,𝑘   𝑥,𝑐,𝐹   𝑥,𝑀   𝑥,𝑁,𝑖,𝑘
Allowed substitution hints:   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑐)   𝑆(𝑥)   𝐸(𝑥)   𝐼(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemrinv
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 ballotth.m . . . . . . . 8 𝑀 ∈ ℕ
2 ballotth.n . . . . . . . 8 𝑁 ∈ ℕ
3 ballotth.o . . . . . . . 8 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}
4 ballotth.p . . . . . . . 8 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))
5 ballotth.f . . . . . . . 8 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
6 ballotth.e . . . . . . . 8 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
7 ballotth.mgtn . . . . . . . 8 𝑁 < 𝑀
8 ballotth.i . . . . . . . 8 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
9 ballotth.s . . . . . . . 8 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
10 ballotth.r . . . . . . . 8 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemrinv0 30399 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ 𝑑 = ((𝑆𝑐) “ 𝑐)) → (𝑑 ∈ (𝑂𝐸) ∧ 𝑐 = ((𝑆𝑑) “ 𝑑)))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemrinv0 30399 . . . . . . 7 ((𝑑 ∈ (𝑂𝐸) ∧ 𝑐 = ((𝑆𝑑) “ 𝑑)) → (𝑐 ∈ (𝑂𝐸) ∧ 𝑑 = ((𝑆𝑐) “ 𝑐)))
1311, 12impbii 199 . . . . . 6 ((𝑐 ∈ (𝑂𝐸) ∧ 𝑑 = ((𝑆𝑐) “ 𝑐)) ↔ (𝑑 ∈ (𝑂𝐸) ∧ 𝑐 = ((𝑆𝑑) “ 𝑑)))
1413a1i 11 . . . . 5 (⊤ → ((𝑐 ∈ (𝑂𝐸) ∧ 𝑑 = ((𝑆𝑐) “ 𝑐)) ↔ (𝑑 ∈ (𝑂𝐸) ∧ 𝑐 = ((𝑆𝑑) “ 𝑑))))
1514mptcnv 5498 . . . 4 (⊤ → (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐)) = (𝑑 ∈ (𝑂𝐸) ↦ ((𝑆𝑑) “ 𝑑)))
1615trud 1490 . . 3 (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐)) = (𝑑 ∈ (𝑂𝐸) ↦ ((𝑆𝑑) “ 𝑑))
17 fveq2 6153 . . . . 5 (𝑑 = 𝑐 → (𝑆𝑑) = (𝑆𝑐))
18 id 22 . . . . 5 (𝑑 = 𝑐𝑑 = 𝑐)
1917, 18imaeq12d 5431 . . . 4 (𝑑 = 𝑐 → ((𝑆𝑑) “ 𝑑) = ((𝑆𝑐) “ 𝑐))
2019cbvmptv 4715 . . 3 (𝑑 ∈ (𝑂𝐸) ↦ ((𝑆𝑑) “ 𝑑)) = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
2116, 20eqtri 2643 . 2 (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐)) = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
2210cnveqi 5262 . 2 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
2321, 22, 103eqtr4i 2653 1 𝑅 = 𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wtru 1481  wcel 1987  wral 2907  {crab 2911  cdif 3556  cin 3558  ifcif 4063  𝒫 cpw 4135   class class class wbr 4618  cmpt 4678  ccnv 5078  cima 5082  cfv 5852  (class class class)co 6610  infcinf 8299  cr 9887  0cc0 9888  1c1 9889   + caddc 9891   < clt 10026  cle 10027  cmin 10218   / cdiv 10636  cn 10972  cz 11329  ...cfz 12276  #chash 13065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-inf 8301  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-n0 11245  df-z 11330  df-uz 11640  df-rp 11785  df-fz 12277  df-hash 13066
This theorem is referenced by:  ballotlem7  30402
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