Theorem ballotlemfval 34747

Description: The value of 𝐹. (Contributed by Thierry Arnoux, 23-Nov-2016.)

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)))))
ballotlemfval.c	⊢ (𝜑 → 𝐶 ∈ 𝑂)
ballotlemfval.j	⊢ (𝜑 → 𝐽 ∈ ℤ)

Assertion

Ref	Expression
ballotlemfval	⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))

Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂,𝑐   𝐹,𝑐,𝑖   𝐶,𝑖   𝑖,𝐽   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑥,𝑐)   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑐)   𝐹(𝑥)   𝐽(𝑥,𝑐)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemfval

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotlemfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑂)
2		simpl 486	. . . . . . . 8 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → 𝑏 = 𝐶)
3	2	ineq2d 4170	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝐶))
4	3	fveq2d 6865	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (♯‘((1...𝑖) ∩ 𝑏)) = (♯‘((1...𝑖) ∩ 𝐶)))
5	2	difeq2d 4078	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝐶))
6	5	fveq2d 6865	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (♯‘((1...𝑖) ∖ 𝑏)) = (♯‘((1...𝑖) ∖ 𝐶)))
7	4, 6	oveq12d 7408	. . . . 5 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏))) = ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶))))
8	7	mpteq2dva 5190	. . . 4 ⊢ (𝑏 = 𝐶 → (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏)))) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))))
9		ballotth.f	. . . . 5 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
10		ineq2 4164	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝑐))
11	10	fveq2d 6865	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (♯‘((1...𝑖) ∩ 𝑏)) = (♯‘((1...𝑖) ∩ 𝑐)))
12		difeq2 4072	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝑐))
13	12	fveq2d 6865	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (♯‘((1...𝑖) ∖ 𝑏)) = (♯‘((1...𝑖) ∖ 𝑐)))
14	11, 13	oveq12d 7408	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏))) = ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))
15	14	mpteq2dv 5191	. . . . . 6 ⊢ (𝑏 = 𝑐 → (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏)))) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
16	15	cbvmptv 5201	. . . . 5 ⊢ (𝑏 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏))))) = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
17	9, 16	eqtr4i 2787	. . . 4 ⊢ 𝐹 = (𝑏 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏)))))
18		zex 12570	. . . . 5 ⊢ ℤ ∈ V
19	18	mptex 7201	. . . 4 ⊢ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))) ∈ V
20	8, 17, 19	fvmpt 6969	. . 3 ⊢ (𝐶 ∈ 𝑂 → (𝐹‘𝐶) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))))
21	1, 20	syl 17	. 2 ⊢ (𝜑 → (𝐹‘𝐶) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))))
22		oveq2 7398	. . . . . 6 ⊢ (𝑖 = 𝐽 → (1...𝑖) = (1...𝐽))
23	22	ineq1d 4169	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∩ 𝐶) = ((1...𝐽) ∩ 𝐶))
24	23	fveq2d 6865	. . . 4 ⊢ (𝑖 = 𝐽 → (♯‘((1...𝑖) ∩ 𝐶)) = (♯‘((1...𝐽) ∩ 𝐶)))
25	22	difeq1d 4077	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∖ 𝐶) = ((1...𝐽) ∖ 𝐶))
26	25	fveq2d 6865	. . . 4 ⊢ (𝑖 = 𝐽 → (♯‘((1...𝑖) ∖ 𝐶)) = (♯‘((1...𝐽) ∖ 𝐶)))
27	24, 26	oveq12d 7408	. . 3 ⊢ (𝑖 = 𝐽 → ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶))) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
28	27	adantl 485	. 2 ⊢ ((𝜑 ∧ 𝑖 = 𝐽) → ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶))) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
29		ballotlemfval.j	. 2 ⊢ (𝜑 → 𝐽 ∈ ℤ)
30		ovexd 7425	. 2 ⊢ (𝜑 → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) ∈ V)
31	21, 28, 29, 30	fvmptd 6977	1 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453   ∖ cdif 3899   ∩ cin 3901  𝒫 cpw 4552   ↦ cmpt 5178  ‘cfv 6515  (class class class)co 7390  1c1 11067   + caddc 11069   − cmin 11407   / cdiv 11837  ℕcn 12203  ℤcz 12561  ...cfz 13505  ♯chash 14336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-cnex 11122  ax-resscn 11123

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-neg 11410  df-z 12562

This theorem is referenced by:  ballotlemfelz  34748  ballotlemfp1  34749  ballotlemfmpn  34752  ballotlemfval0  34753  ballotlemfg  34783  ballotlemfrc  34784