Theorem ballotlemfval 34650

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 4161	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝐶))
4	3	fveq2d 6838	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (♯‘((1...𝑖) ∩ 𝑏)) = (♯‘((1...𝑖) ∩ 𝐶)))
5	2	difeq2d 4067	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝐶))
6	5	fveq2d 6838	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (♯‘((1...𝑖) ∖ 𝑏)) = (♯‘((1...𝑖) ∖ 𝐶)))
7	4, 6	oveq12d 7378	. . . . 5 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏))) = ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶))))
8	7	mpteq2dva 5179	. . . 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 4155	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝑐))
11	10	fveq2d 6838	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (♯‘((1...𝑖) ∩ 𝑏)) = (♯‘((1...𝑖) ∩ 𝑐)))
12		difeq2 4061	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝑐))
13	12	fveq2d 6838	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (♯‘((1...𝑖) ∖ 𝑏)) = (♯‘((1...𝑖) ∖ 𝑐)))
14	11, 13	oveq12d 7378	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏))) = ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))
15	14	mpteq2dv 5180	. . . . . 6 ⊢ (𝑏 = 𝑐 → (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏)))) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
16	15	cbvmptv 5190	. . . . 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 2763	. . . 4 ⊢ 𝐹 = (𝑏 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏)))))
18		zex 12524	. . . . 5 ⊢ ℤ ∈ V
19	18	mptex 7171	. . . 4 ⊢ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))) ∈ V
20	8, 17, 19	fvmpt 6941	. . 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 7368	. . . . . 6 ⊢ (𝑖 = 𝐽 → (1...𝑖) = (1...𝐽))
23	22	ineq1d 4160	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∩ 𝐶) = ((1...𝐽) ∩ 𝐶))
24	23	fveq2d 6838	. . . 4 ⊢ (𝑖 = 𝐽 → (♯‘((1...𝑖) ∩ 𝐶)) = (♯‘((1...𝐽) ∩ 𝐶)))
25	22	difeq1d 4066	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∖ 𝐶) = ((1...𝐽) ∖ 𝐶))
26	25	fveq2d 6838	. . . 4 ⊢ (𝑖 = 𝐽 → (♯‘((1...𝑖) ∖ 𝐶)) = (♯‘((1...𝐽) ∖ 𝐶)))
27	24, 26	oveq12d 7378	. . 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 7395	. 2 ⊢ (𝜑 → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) ∈ V)
31	21, 28, 29, 30	fvmptd 6949	1 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3390  Vcvv 3430   ∖ cdif 3887   ∩ cin 3889  𝒫 cpw 4542   ↦ cmpt 5167  ‘cfv 6492  (class class class)co 7360  1c1 11030   + caddc 11032   − cmin 11368   / cdiv 11798  ℕcn 12165  ℤcz 12515  ...cfz 13452  ♯chash 14283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-cnex 11085  ax-resscn 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-neg 11371  df-z 12516

This theorem is referenced by:  ballotlemfelz  34651  ballotlemfp1  34652  ballotlemfmpn  34655  ballotlemfval0  34656  ballotlemfg  34686  ballotlemfrc  34687