Theorem ballotlemfval 34792

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 487	. . . . . . . 8 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → 𝑏 = 𝐶)
3	2	ineq2d 4175	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝐶))
4	3	fveq2d 6875	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (♯‘((1...𝑖) ∩ 𝑏)) = (♯‘((1...𝑖) ∩ 𝐶)))
5	2	difeq2d 4083	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝐶))
6	5	fveq2d 6875	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (♯‘((1...𝑖) ∖ 𝑏)) = (♯‘((1...𝑖) ∖ 𝐶)))
7	4, 6	oveq12d 7418	. . . . 5 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏))) = ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶))))
8	7	mpteq2dva 5197	. . . 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 4169	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝑐))
11	10	fveq2d 6875	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (♯‘((1...𝑖) ∩ 𝑏)) = (♯‘((1...𝑖) ∩ 𝑐)))
12		difeq2 4077	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝑐))
13	12	fveq2d 6875	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (♯‘((1...𝑖) ∖ 𝑏)) = (♯‘((1...𝑖) ∖ 𝑐)))
14	11, 13	oveq12d 7418	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏))) = ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))
15	14	mpteq2dv 5198	. . . . . 6 ⊢ (𝑏 = 𝑐 → (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏)))) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
16	15	cbvmptv 5208	. . . . 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 2791	. . . 4 ⊢ 𝐹 = (𝑏 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏)))))
18		zex 12588	. . . . 5 ⊢ ℤ ∈ V
19	18	mptex 7211	. . . 4 ⊢ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))) ∈ V
20	8, 17, 19	fvmpt 6979	. . 3 ⊢ (𝐶 ∈ 𝑂 → (𝐹‘𝐶) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))))
21	1, 20	syl 18	. 2 ⊢ (𝜑 → (𝐹‘𝐶) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))))
22		oveq2 7408	. . . . . 6 ⊢ (𝑖 = 𝐽 → (1...𝑖) = (1...𝐽))
23	22	ineq1d 4174	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∩ 𝐶) = ((1...𝐽) ∩ 𝐶))
24	23	fveq2d 6875	. . . 4 ⊢ (𝑖 = 𝐽 → (♯‘((1...𝑖) ∩ 𝐶)) = (♯‘((1...𝐽) ∩ 𝐶)))
25	22	difeq1d 4082	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∖ 𝐶) = ((1...𝐽) ∖ 𝐶))
26	25	fveq2d 6875	. . . 4 ⊢ (𝑖 = 𝐽 → (♯‘((1...𝑖) ∖ 𝐶)) = (♯‘((1...𝐽) ∖ 𝐶)))
27	24, 26	oveq12d 7418	. . 3 ⊢ (𝑖 = 𝐽 → ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶))) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
28	27	adantl 486	. 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 7435	. 2 ⊢ (𝜑 → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) ∈ V)
31	21, 28, 29, 30	fvmptd 6987	1 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {crab 3417  Vcvv 3457   ∖ cdif 3904   ∩ cin 3906  𝒫 cpw 4558   ↦ cmpt 5185  ‘cfv 6525  (class class class)co 7400  1c1 11089   + caddc 11091   − cmin 11429   / cdiv 11859  ℕcn 12221  ℤcz 12579  ...cfz 13523  ♯chash 14354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pr 5394  ax-cnex 11144  ax-resscn 11145

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-neg 11432  df-z 12580

This theorem is referenced by:  ballotlemfelz  34793  ballotlemfp1  34794  ballotlemfmpn  34797  ballotlemfval0  34798  ballotlemfg  34828  ballotlemfrc  34829