Theorem ballotlemfval 30551

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 473	. . . . . . . 8 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → 𝑏 = 𝐶)
3	2	ineq2d 3814	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝐶))
4	3	fveq2d 6195	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (#‘((1...𝑖) ∩ 𝑏)) = (#‘((1...𝑖) ∩ 𝐶)))
5	2	difeq2d 3728	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝐶))
6	5	fveq2d 6195	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (#‘((1...𝑖) ∖ 𝑏)) = (#‘((1...𝑖) ∖ 𝐶)))
7	4, 6	oveq12d 6668	. . . . 5 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏))) = ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶))))
8	7	mpteq2dva 4744	. . . 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 3808	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝑐))
11	10	fveq2d 6195	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (#‘((1...𝑖) ∩ 𝑏)) = (#‘((1...𝑖) ∩ 𝑐)))
12		difeq2 3722	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝑐))
13	12	fveq2d 6195	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (#‘((1...𝑖) ∖ 𝑏)) = (#‘((1...𝑖) ∖ 𝑐)))
14	11, 13	oveq12d 6668	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏))) = ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐))))
15	14	mpteq2dv 4745	. . . . . 6 ⊢ (𝑏 = 𝑐 → (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏)))) = (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))
16	15	cbvmptv 4750	. . . . 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 2647	. . . 4 ⊢ 𝐹 = (𝑏 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑏)) − (#‘((1...𝑖) ∖ 𝑏)))))
18		zex 11386	. . . . 5 ⊢ ℤ ∈ V
19	18	mptex 6486	. . . 4 ⊢ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶)))) ∈ V
20	8, 17, 19	fvmpt 6282	. . 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 6658	. . . . . 6 ⊢ (𝑖 = 𝐽 → (1...𝑖) = (1...𝐽))
23	22	ineq1d 3813	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∩ 𝐶) = ((1...𝐽) ∩ 𝐶))
24	23	fveq2d 6195	. . . 4 ⊢ (𝑖 = 𝐽 → (#‘((1...𝑖) ∩ 𝐶)) = (#‘((1...𝐽) ∩ 𝐶)))
25	22	difeq1d 3727	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∖ 𝐶) = ((1...𝐽) ∖ 𝐶))
26	25	fveq2d 6195	. . . 4 ⊢ (𝑖 = 𝐽 → (#‘((1...𝑖) ∖ 𝐶)) = (#‘((1...𝐽) ∖ 𝐶)))
27	24, 26	oveq12d 6668	. . 3 ⊢ (𝑖 = 𝐽 → ((#‘((1...𝑖) ∩ 𝐶)) − (#‘((1...𝑖) ∖ 𝐶))) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))
28	27	adantl 482	. 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 6680	. 2 ⊢ (𝜑 → ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))) ∈ V)
31	21, 28, 29, 30	fvmptd 6288	1 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573  𝒫 cpw 4158   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  1c1 9937   + caddc 9939   − cmin 10266   / cdiv 10684  ℕcn 11020  ℤcz 11377  ...cfz 12326  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-neg 10269  df-z 11378

This theorem is referenced by:  ballotlemfelz  30552  ballotlemfp1  30553  ballotlemfmpn  30556  ballotlemfval0  30557  ballotlemfg  30587  ballotlemfrc  30588