Theorem ballotlemsval 34653

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 6844	. . . . 5 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼‘𝑑) = (𝐼‘𝐶))
3	2	breq2d 5097	. . . 4 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼‘𝑑) ↔ 𝑖 ≤ (𝐼‘𝐶)))
4	2	oveq1d 7382	. . . . 5 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼‘𝑑) + 1) = ((𝐼‘𝐶) + 1))
5	4	oveq1d 7382	. . . 4 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼‘𝑑) + 1) − 𝑖) = (((𝐼‘𝐶) + 1) − 𝑖))
6	3, 5	ifbieq1d 4491	. . 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 5178	. 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 6844	. . . . . . 7 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼‘𝑐) = (𝐼‘𝑑))
11	10	breq2d 5097	. . . . . 6 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼‘𝑐) ↔ 𝑖 ≤ (𝐼‘𝑑)))
12	10	oveq1d 7382	. . . . . . 7 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼‘𝑐) + 1) = ((𝐼‘𝑑) + 1))
13	12	oveq1d 7382	. . . . . 6 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼‘𝑐) + 1) − 𝑖) = (((𝐼‘𝑑) + 1) − 𝑖))
14	11, 13	ifbieq1d 4491	. . . . 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 5178	. . . 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 5189	. . 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 2759	. 2 ⊢ 𝑆 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑑), (((𝐼‘𝑑) + 1) − 𝑖), 𝑖)))
18		ovex 7400	. . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ V
19	18	mptex 7178	. 2 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) ∈ V
20	7, 17, 19	fvmpt 6947	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 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389   ∖ cdif 3886   ∩ cin 3888  ifcif 4466  𝒫 cpw 4541   class class class wbr 5085   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367  infcinf 9354  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   < clt 11179   ≤ cle 11180   − cmin 11377   / cdiv 11807  ℕcn 12174  ℤcz 12524  ...cfz 13461  ♯chash 14292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370

This theorem is referenced by:  ballotlemsv  34654  ballotlemsf1o  34658  ballotlemieq  34661