Theorem ballotfilemfval 13150

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

Hypotheses

Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotfi.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfi.p	⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘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
ballotfilemfval	⊢ (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 ballotfilemfval

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotlemfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑂)
2		simpl 109	. . . . . . . 8 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → 𝑏 = 𝐶)
3	2	ineq2d 3424	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝐶))
4	3	fveq2d 5676	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (♯‘((1...𝑖) ∩ 𝑏)) = (♯‘((1...𝑖) ∩ 𝐶)))
5	2	difeq2d 3339	. . . . . . 7 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝐶))
6	5	fveq2d 5676	. . . . . 6 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → (♯‘((1...𝑖) ∖ 𝑏)) = (♯‘((1...𝑖) ∖ 𝐶)))
7	4, 6	oveq12d 6070	. . . . 5 ⊢ ((𝑏 = 𝐶 ∧ 𝑖 ∈ ℤ) → ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏))) = ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶))))
8	7	mpteq2dva 4202	. . . 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 3418	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∩ 𝑏) = ((1...𝑖) ∩ 𝑐))
11	10	fveq2d 5676	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (♯‘((1...𝑖) ∩ 𝑏)) = (♯‘((1...𝑖) ∩ 𝑐)))
12		difeq2 3333	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → ((1...𝑖) ∖ 𝑏) = ((1...𝑖) ∖ 𝑐))
13	12	fveq2d 5676	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (♯‘((1...𝑖) ∖ 𝑏)) = (♯‘((1...𝑖) ∖ 𝑐)))
14	11, 13	oveq12d 6070	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏))) = ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))
15	14	mpteq2dv 4203	. . . . . 6 ⊢ (𝑏 = 𝑐 → (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏)))) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
16	15	cbvmptv 4208	. . . . 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 2258	. . . 4 ⊢ 𝐹 = (𝑏 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑏)) − (♯‘((1...𝑖) ∖ 𝑏)))))
18		zex 9588	. . . . 5 ⊢ ℤ ∈ V
19	18	mptex 5914	. . . 4 ⊢ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))) ∈ V
20	8, 17, 19	fvmpt 5756	. . 3 ⊢ (𝐶 ∈ 𝑂 → (𝐹‘𝐶) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))))
21	1, 20	syl 14	. 2 ⊢ (𝜑 → (𝐹‘𝐶) = (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶)))))
22		oveq2 6060	. . . . . 6 ⊢ (𝑖 = 𝐽 → (1...𝑖) = (1...𝐽))
23	22	ineq1d 3423	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∩ 𝐶) = ((1...𝐽) ∩ 𝐶))
24	23	fveq2d 5676	. . . 4 ⊢ (𝑖 = 𝐽 → (♯‘((1...𝑖) ∩ 𝐶)) = (♯‘((1...𝐽) ∩ 𝐶)))
25	22	difeq1d 3338	. . . . 5 ⊢ (𝑖 = 𝐽 → ((1...𝑖) ∖ 𝐶) = ((1...𝐽) ∖ 𝐶))
26	25	fveq2d 5676	. . . 4 ⊢ (𝑖 = 𝐽 → (♯‘((1...𝑖) ∖ 𝐶)) = (♯‘((1...𝐽) ∖ 𝐶)))
27	24, 26	oveq12d 6070	. . 3 ⊢ (𝑖 = 𝐽 → ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶))) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
28	27	adantl 277	. 2 ⊢ ((𝜑 ∧ 𝑖 = 𝐽) → ((♯‘((1...𝑖) ∩ 𝐶)) − (♯‘((1...𝑖) ∖ 𝐶))) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
29		ballotlemfval.j	. 2 ⊢ (𝜑 → 𝐽 ∈ ℤ)
30		ballotth.m	. . . . . 6 ⊢ 𝑀 ∈ ℕ
31		ballotth.n	. . . . . 6 ⊢ 𝑁 ∈ ℕ
32		ballotfi.o	. . . . . 6 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
33	30, 31, 32, 1, 29	ballotfilemcinfi 13147	. . . . 5 ⊢ (𝜑 → ((1...𝐽) ∩ 𝐶) ∈ Fin)
34		hashcl 11148	. . . . 5 ⊢ (((1...𝐽) ∩ 𝐶) ∈ Fin → (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0)
35	33, 34	syl 14	. . . 4 ⊢ (𝜑 → (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0)
36	35	nn0zd 9701	. . 3 ⊢ (𝜑 → (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℤ)
37	30, 31, 32, 1, 29	ballotfilemdifcfi 13148	. . . . 5 ⊢ (𝜑 → ((1...𝐽) ∖ 𝐶) ∈ Fin)
38		hashcl 11148	. . . . 5 ⊢ (((1...𝐽) ∖ 𝐶) ∈ Fin → (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0)
39	37, 38	syl 14	. . . 4 ⊢ (𝜑 → (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0)
40	39	nn0zd 9701	. . 3 ⊢ (𝜑 → (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℤ)
41	36, 40	zsubcld 9708	. 2 ⊢ (𝜑 → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) ∈ ℤ)
42	21, 28, 29, 41	fvmptd 5760	1 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  {crab 2526   ∖ cdif 3210   ∩ cin 3212  𝒫 cpw 3671   ↦ cmpt 4173  ‘cfv 5354  (class class class)co 6052  Fincfn 6977  1c1 8130   + caddc 8132   − cmin 8446   / cdiv 8948  ℕcn 9239  ℕ₀cn0 9498  ℤcz 9579  ...cfz 10345  ♯chash 11142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346  df-ihash 11143

This theorem is referenced by:  ballotfilemfelz  13151  ballotfilemfp1  13152  ballotfilemfmpn  13155  ballotfilemfval0  13156