Theorem ballotlemfval 34481

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 482	. . . . . . . 8 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → 𝑏 = 𝐶)
3	2	ineq2d 4183	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝐶))
4	3	fveq2d 6862	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (♯‘((1...𝑖) ∩ 𝑏)) = (♯‘((1...𝑖) ∩ 𝐶)))
5	2	difeq2d 4089	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝐶))
6	5	fveq2d 6862	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (♯‘((1...𝑖) ∖ 𝑏)) = (♯‘((1...𝑖) ∖ 𝐶)))
7	4, 6	oveq12d 7405	. . . . 5 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏))) = ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶))))
8	7	mpteq2dva 5200	. . . 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 4177	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝑐))
11	10	fveq2d 6862	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (♯‘((1...𝑖) ∩ 𝑏)) = (♯‘((1...𝑖) ∩ 𝑐)))
12		difeq2 4083	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝑐))
13	12	fveq2d 6862	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (♯‘((1...𝑖) ∖ 𝑏)) = (♯‘((1...𝑖) ∖ 𝑐)))
14	11, 13	oveq12d 7405	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏))) = ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))
15	14	mpteq2dv 5201	. . . . . 6 ⊢ (𝑏 = 𝑐 → (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏)))) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
16	15	cbvmptv 5211	. . . . 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 2755	. . . 4 ⊢ 𝐹 = (𝑏 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏)))))
18		zex 12538	. . . . 5 ⊢ ℤ ∈ V
19	18	mptex 7197	. . . 4 ⊢ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))) ∈ V
20	8, 17, 19	fvmpt 6968	. . 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 7395	. . . . . 6 ⊢ (𝑖 = 𝐽 → (1...𝑖) = (1...𝐽))
23	22	ineq1d 4182	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∩ 𝐶) = ((1...𝐽) ∩ 𝐶))
24	23	fveq2d 6862	. . . 4 ⊢ (𝑖 = 𝐽 → (♯‘((1...𝑖) ∩ 𝐶)) = (♯‘((1...𝐽) ∩ 𝐶)))
25	22	difeq1d 4088	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∖ 𝐶) = ((1...𝐽) ∖ 𝐶))
26	25	fveq2d 6862	. . . 4 ⊢ (𝑖 = 𝐽 → (♯‘((1...𝑖) ∖ 𝐶)) = (♯‘((1...𝐽) ∖ 𝐶)))
27	24, 26	oveq12d 7405	. . 3 ⊢ (𝑖 = 𝐽 → ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶))) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
28	27	adantl 481	. 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 7422	. 2 ⊢ (𝜑 → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) ∈ V)
31	21, 28, 29, 30	fvmptd 6975	1 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447   ∖ cdif 3911   ∩ cin 3913  𝒫 cpw 4563   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  1c1 11069   + caddc 11071   − cmin 11405   / cdiv 11835  ℕcn 12186  ℤcz 12529  ...cfz 13468  ♯chash 14295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-cnex 11124  ax-resscn 11125

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-neg 11408  df-z 12530

This theorem is referenced by:  ballotlemfelz  34482  ballotlemfp1  34483  ballotlemfmpn  34486  ballotlemfval0  34487  ballotlemfg  34517  ballotlemfrc  34518