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Theorem ballotlemsval 29699
Description: Value of 𝑆. (Contributed by Thierry Arnoux, 12-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)))
Assertion
Ref Expression
ballotlemsval (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
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   𝐶,𝑖,𝑘   𝑖,𝐸,𝑘   𝐶,𝑘   𝑘,𝐼,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐
Allowed substitution hints:   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑆(𝑥,𝑖,𝑘,𝑐)   𝐸(𝑥)   𝐹(𝑥)   𝐼(𝑥)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemsval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 simpl 471 . . . . . 6 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → 𝑑 = 𝐶)
21fveq2d 6088 . . . . 5 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼𝑑) = (𝐼𝐶))
32breq2d 4585 . . . 4 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼𝑑) ↔ 𝑖 ≤ (𝐼𝐶)))
42oveq1d 6538 . . . . 5 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼𝑑) + 1) = ((𝐼𝐶) + 1))
54oveq1d 6538 . . . 4 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼𝑑) + 1) − 𝑖) = (((𝐼𝐶) + 1) − 𝑖))
63, 5ifbieq1d 4054 . . 3 ((𝑑 = 𝐶𝑖 ∈ (1...(𝑀 + 𝑁))) → if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖) = if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖))
76mpteq2dva 4662 . 2 (𝑑 = 𝐶 → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖)) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
8 ballotth.s . . 3 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
9 simpl 471 . . . . . . . 8 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → 𝑐 = 𝑑)
109fveq2d 6088 . . . . . . 7 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼𝑐) = (𝐼𝑑))
1110breq2d 4585 . . . . . 6 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼𝑐) ↔ 𝑖 ≤ (𝐼𝑑)))
1210oveq1d 6538 . . . . . . 7 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼𝑐) + 1) = ((𝐼𝑑) + 1))
1312oveq1d 6538 . . . . . 6 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼𝑐) + 1) − 𝑖) = (((𝐼𝑑) + 1) − 𝑖))
1411, 13ifbieq1d 4054 . . . . 5 ((𝑐 = 𝑑𝑖 ∈ (1...(𝑀 + 𝑁))) → if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖) = if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖))
1514mpteq2dva 4662 . . . 4 (𝑐 = 𝑑 → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖)))
1615cbvmptv 4668 . . 3 (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖))) = (𝑑 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖)))
178, 16eqtri 2627 . 2 𝑆 = (𝑑 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑑), (((𝐼𝑑) + 1) − 𝑖), 𝑖)))
18 ovex 6551 . . 3 (1...(𝑀 + 𝑁)) ∈ V
1918mptex 6364 . 2 (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)) ∈ V
207, 17, 19fvmpt 6172 1 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  wral 2891  {crab 2895  cdif 3532  cin 3534  ifcif 4031  𝒫 cpw 4103   class class class wbr 4573  cmpt 4633  cfv 5786  (class class class)co 6523  infcinf 8203  cr 9787  0cc0 9788  1c1 9789   + caddc 9791   < clt 9926  cle 9927  cmin 10113   / cdiv 10529  cn 10863  cz 11206  ...cfz 12148  #chash 12930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526
This theorem is referenced by:  ballotlemsv  29700  ballotlemsf1o  29704  ballotlemieq  29707
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