Theorem ballotlemsv 32160

Description: Value of 𝑆 evaluated at 𝐽 for a given counting 𝐶. (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
ballotlemsv	⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) = if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽))

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)   𝐽(𝑥,𝑖,𝑘,𝑐)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemsv

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . 5 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . . 5 ⊢ 𝑁 ∈ ℕ
3		ballotth.o	. . . . 5 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4		ballotth.p	. . . . 5 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotth.f	. . . . 5 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		ballotth.e	. . . . 5 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
7		ballotth.mgtn	. . . . 5 ⊢ 𝑁 < 𝑀
8		ballotth.i	. . . . 5 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
9		ballotth.s	. . . . 5 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsval 32159	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)))
11		breq1 5046	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑖 ≤ (𝐼‘𝐶) ↔ 𝑗 ≤ (𝐼‘𝐶)))
12		oveq2 7210	. . . . . 6 ⊢ (𝑖 = 𝑗 → (((𝐼‘𝐶) + 1) − 𝑖) = (((𝐼‘𝐶) + 1) − 𝑗))
13		id 22	. . . . . 6 ⊢ (𝑖 = 𝑗 → 𝑖 = 𝑗)
14	11, 12, 13	ifbieq12d 4457	. . . . 5 ⊢ (𝑖 = 𝑗 → if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖) = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
15	14	cbvmptv 5147	. . . 4 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
16	10, 15	eqtrdi 2790	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗)))
17	16	adantr 484	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → (𝑆‘𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗)))
18		simpr 488	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
19	18	breq1d 5053	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 = 𝐽) → (𝑗 ≤ (𝐼‘𝐶) ↔ 𝐽 ≤ (𝐼‘𝐶)))
20	18	oveq2d 7218	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 = 𝐽) → (((𝐼‘𝐶) + 1) − 𝑗) = (((𝐼‘𝐶) + 1) − 𝐽))
21	19, 20, 18	ifbieq12d 4457	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 = 𝐽) → if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗) = if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽))
22	21	adantlr 715	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑗 = 𝐽) → if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗) = if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽))
23		simpr 488	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
24		ovexd 7237	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) ∧ 𝐽 ≤ (𝐼‘𝐶)) → (((𝐼‘𝐶) + 1) − 𝐽) ∈ V)
25		elex 3419	. . . 4 ⊢ (𝐽 ∈ (1...(𝑀 + 𝑁)) → 𝐽 ∈ V)
26	25	ad2antlr 727	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) ∧ ¬ 𝐽 ≤ (𝐼‘𝐶)) → 𝐽 ∈ V)
27	24, 26	ifclda 4464	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽) ∈ V)
28	17, 22, 23, 27	fvmptd 6814	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) = if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3054  {crab 3058  Vcvv 3401   ∖ cdif 3854   ∩ cin 3856  ifcif 4429  𝒫 cpw 4503   class class class wbr 5043   ↦ cmpt 5124  ‘cfv 6369  (class class class)co 7202  infcinf 9046  ℝcr 10711  0cc0 10712  1c1 10713   + caddc 10715   < clt 10850   ≤ cle 10851   − cmin 11045   / cdiv 11472  ℕcn 11813  ℤcz 12159  ...cfz 13078  ♯chash 13879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pr 5311

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-ov 7205

This theorem is referenced by:  ballotlemsgt1  32161  ballotlemsdom  32162  ballotlemsel1i  32163  ballotlemsf1o  32164  ballotlemsi  32165  ballotlemsima  32166  ballotlemrv  32170