Theorem ballotlemsval 35188

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 488	. . . . . 6 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → 𝑑 = 𝐶)
2	1	fveq2d 6890	. . . . 5 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼‘𝑑) = (𝐼‘𝐶))
3	2	breq2d 5114	. . . 4 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼‘𝑑) ↔ 𝑖 ≤ (𝐼‘𝐶)))
4	2	oveq1d 7436	. . . . 5 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼‘𝑑) + 1) = ((𝐼‘𝐶) + 1))
5	4	oveq1d 7436	. . . 4 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼‘𝑑) + 1) − 𝑖) = (((𝐼‘𝐶) + 1) − 𝑖))
6	3, 5	ifbieq1d 4506	. . 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 5197	. 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 488	. . . . . . . 8 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → 𝑐 = 𝑑)
10	9	fveq2d 6890	. . . . . . 7 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼‘𝑐) = (𝐼‘𝑑))
11	10	breq2d 5114	. . . . . 6 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼‘𝑐) ↔ 𝑖 ≤ (𝐼‘𝑑)))
12	10	oveq1d 7436	. . . . . . 7 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼‘𝑐) + 1) = ((𝐼‘𝑑) + 1))
13	12	oveq1d 7436	. . . . . 6 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼‘𝑐) + 1) − 𝑖) = (((𝐼‘𝑑) + 1) − 𝑖))
14	11, 13	ifbieq1d 4506	. . . . 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 5197	. . . 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 5208	. . 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 2783	. 2 ⊢ 𝑆 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑑), (((𝐼‘𝑑) + 1) − 𝑖), 𝑖)))
18		ovex 7454	. . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ V
19	18	mptex 7230	. 2 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) ∈ V
20	7, 17, 19	fvmpt 6994	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ∖ cdif 4045   ∩ cin 4089  ifcif 4481  𝒫 cpw 4556   class class class wbr 5102   ↦ cmpt 5185  ‘cfv 6539  (class class class)co 7421  infcinf 9437  ℝcr 11203  0cc0 11204  1c1 11205   + caddc 11207   < clt 11347   ≤ cle 11348   − cmin 11545   / cdiv 11977  ℕcn 12339  ℤcz 12697  ...cfz 13643  ♯chash 14478

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424

This theorem is used by:  ballotlemsv  35189  ballotlemsf1o  35193  ballotlemieq  35196