Theorem ballotlemsval 33996

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 6885	. . . . 5 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼‘𝑑) = (𝐼‘𝐶))
3	2	breq2d 5150	. . . 4 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼‘𝑑) ↔ 𝑖 ≤ (𝐼‘𝐶)))
4	2	oveq1d 7416	. . . . 5 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼‘𝑑) + 1) = ((𝐼‘𝐶) + 1))
5	4	oveq1d 7416	. . . 4 ⊢ ((𝑑 = 𝐶 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼‘𝑑) + 1) − 𝑖) = (((𝐼‘𝐶) + 1) − 𝑖))
6	3, 5	ifbieq1d 4544	. . 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 5238	. 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 6885	. . . . . . 7 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼‘𝑐) = (𝐼‘𝑑))
11	10	breq2d 5150	. . . . . 6 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼‘𝑐) ↔ 𝑖 ≤ (𝐼‘𝑑)))
12	10	oveq1d 7416	. . . . . . 7 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼‘𝑐) + 1) = ((𝐼‘𝑑) + 1))
13	12	oveq1d 7416	. . . . . 6 ⊢ ((𝑐 = 𝑑 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼‘𝑐) + 1) − 𝑖) = (((𝐼‘𝑑) + 1) − 𝑖))
14	11, 13	ifbieq1d 4544	. . . . 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 5238	. . . 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 5251	. . 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 2752	. 2 ⊢ 𝑆 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑑), (((𝐼‘𝑑) + 1) − 𝑖), 𝑖)))
18		ovex 7434	. . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ V
19	18	mptex 7216	. 2 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) ∈ V
20	7, 17, 19	fvmpt 6988	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 1533   ∈ wcel 2098  ∀wral 3053  {crab 3424   ∖ cdif 3937   ∩ cin 3939  ifcif 4520  𝒫 cpw 4594   class class class wbr 5138   ↦ cmpt 5221  ‘cfv 6533  (class class class)co 7401  infcinf 9432  ℝcr 11105  0cc0 11106  1c1 11107   + caddc 11109   < clt 11245   ≤ cle 11246   − cmin 11441   / cdiv 11868  ℕcn 12209  ℤcz 12555  ...cfz 13481  ♯chash 14287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404

This theorem is referenced by:  ballotlemsv  33997  ballotlemsf1o  34001  ballotlemieq  34004