Theorem ballotlemsval 34507

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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))

Assertion

Ref	Expression
ballotlemsval	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 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.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . 6 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → 𝑑 = 𝐶)
2	1	fveq2d 6865	. . . . 5 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼‘𝑑) = (𝐼‘𝐶))
3	2	breq2d 5122	. . . 4 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼‘𝑑) ↔ 𝑖 ≤ (𝐼‘𝐶)))
4	2	oveq1d 7405	. . . . 5 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼‘𝑑) + 1) = ((𝐼‘𝐶) + 1))
5	4	oveq1d 7405	. . . 4 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼‘𝑑) + 1) − 𝑖) = (((𝐼‘𝐶) + 1) − 𝑖))
6	3, 5	ifbieq1d 4516	. . 3 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → if(𝑖 ≤ (𝐼‘𝑑), (((𝐼‘𝑑) + 1) − 𝑖), 𝑖) = if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖))
7	6	mpteq2dva 5203	. 2 ⊢ (𝑑 = 𝐶 → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑑), (((𝐼‘𝑑) + 1) − 𝑖), 𝑖)) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)))
8		ballotth.s	. . 3 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
9		simpl 482	. . . . . . . 8 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → 𝑐 = 𝑑)
10	9	fveq2d 6865	. . . . . . 7 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼‘𝑐) = (𝐼‘𝑑))
11	10	breq2d 5122	. . . . . 6 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼‘𝑐) ↔ 𝑖 ≤ (𝐼‘𝑑)))
12	10	oveq1d 7405	. . . . . . 7 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼‘𝑐) + 1) = ((𝐼‘𝑑) + 1))
13	12	oveq1d 7405	. . . . . 6 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼‘𝑐) + 1) − 𝑖) = (((𝐼‘𝑑) + 1) − 𝑖))
14	11, 13	ifbieq1d 4516	. . . . 5 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖) = if(𝑖 ≤ (𝐼‘𝑑), (((𝐼‘𝑑) + 1) − 𝑖), 𝑖))
15	14	mpteq2dva 5203	. . . 4 ⊢ (𝑐 = 𝑑 → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑑), (((𝐼‘𝑑) + 1) − 𝑖), 𝑖)))
16	15	cbvmptv 5214	. . 3 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖))) = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑑), (((𝐼‘𝑑) + 1) − 𝑖), 𝑖)))
17	8, 16	eqtri 2753	. 2 ⊢ 𝑆 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑑), (((𝐼‘𝑑) + 1) − 𝑖), 𝑖)))
18		ovex 7423	. . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ V
19	18	mptex 7200	. 2 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) ∈ V
20	7, 17, 19	fvmpt 6971	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408   ∖ cdif 3914   ∩ cin 3916  ifcif 4491  𝒫 cpw 4566   class class class wbr 5110   ↦ cmpt 5191  ‘cfv 6514  (class class class)co 7390  infcinf 9399  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078   < clt 11215   ≤ cle 11216   − cmin 11412   / cdiv 11842  ℕcn 12193  ℤcz 12536  ...cfz 13475  ♯chash 14302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393

This theorem is referenced by:  ballotlemsv  34508  ballotlemsf1o  34512  ballotlemieq  34515