ballotfilemgval - Intuitionistic Logic Explorer

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Theorem ballotfilemgval 13211

Description: Expand the value of ↑. (Contributed by Thierry Arnoux, 21-Apr-2017.) (Revised by Jim Kingdon, 15-Jun-2026.)

**Hypotheses**
Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotfilem.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfilem.p	⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotth.f	⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r	⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
ballotlemg	⊢ ↑ = (𝑢 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
ballotfilemgval.u	⊢ (𝜑 → 𝑈 ∈ 𝑂)
ballotfilemgval.j	⊢ (𝜑 → 𝐽 ∈ ℤ)
ballotfilemgval.k	⊢ (𝜑 → 𝐾 ∈ ℤ)
ballotfilemgval.fz	⊢ (𝜑 → 𝑉 = (𝐽...𝐾))

**Assertion**
Ref	Expression
ballotfilemgval	⊢ (𝜑 → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))

Distinct variable groups: 𝑀,𝑐 𝑁,𝑐 𝑂,𝑐 𝑖,𝑀 𝑖,𝑁 𝑖,𝑂 𝑘,𝑀 𝑘,𝑁 𝑘,𝑂 𝑖,𝑐,𝐹,𝑘 𝑖,𝐸,𝑘 𝑘,𝐼,𝑐 𝐸,𝑐 𝑖,𝐼,𝑐 𝑘,𝐽 𝑆,𝑘,𝑖,𝑐 𝑅,𝑖 𝑣,𝑢,𝐼 𝑢,𝐽,𝑣 𝑢,𝑅,𝑣 𝑢,𝑆,𝑣 𝑢,𝑈,𝑣 𝑢,𝑉,𝑣 𝑢,𝑂,𝑣 Allowed substitution hints: 𝜑(𝑥,𝑣,𝑢,𝑖,𝑘,𝑐) 𝑃(𝑥,𝑣,𝑢,𝑖,𝑘,𝑐) 𝑅(𝑥,𝑘,𝑐) 𝑆(𝑥) 𝑈(𝑥,𝑖,𝑘,𝑐) 𝐸(𝑥,𝑣,𝑢) ↑ (𝑥,𝑣,𝑢,𝑖,𝑘,𝑐) 𝐹(𝑥,𝑣,𝑢) 𝐼(𝑥) 𝐽(𝑥,𝑖,𝑐) 𝐾(𝑥,𝑣,𝑢,𝑖,𝑘,𝑐) 𝑀(𝑥,𝑣,𝑢) 𝑁(𝑥,𝑣,𝑢) 𝑂(𝑥) 𝑉(𝑥,𝑖,𝑘,𝑐)

**Proof of Theorem ballotfilemgval**
Step	Hyp	Ref	Expression
1		ballotfilemgval.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑂)
2		ballotfilemgval.fz	. . 3 ⊢ (𝜑 → 𝑉 = (𝐽...𝐾))
3		ballotfilemgval.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ ℤ)
4		ballotfilemgval.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ)
5	3, 4	fzfigd 10817	. . 3 ⊢ (𝜑 → (𝐽...𝐾) ∈ Fin)
6	2, 5	eqeltrd 2311	. 2 ⊢ (𝜑 → 𝑉 ∈ Fin)
7	2	ineq1d 3425	. . . . . 6 ⊢ (𝜑 → (𝑉 ∩ 𝑈) = ((𝐽...𝐾) ∩ 𝑈))
8		ballotth.m	. . . . . . 7 ⊢ 𝑀 ∈ ℕ
9		ballotth.n	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
10		ballotfilem.o	. . . . . . 7 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
11	8, 9, 10, 1, 3, 4	ballotfilemcinfz 13170	. . . . . 6 ⊢ (𝜑 → ((𝐽...𝐾) ∩ 𝑈) ∈ Fin)
12	7, 11	eqeltrd 2311	. . . . 5 ⊢ (𝜑 → (𝑉 ∩ 𝑈) ∈ Fin)
13		hashcl 11169	. . . . 5 ⊢ ((𝑉 ∩ 𝑈) ∈ Fin → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
14	12, 13	syl 14	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
15	14	nn0zd 9716	. . 3 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℤ)
16	2	difeq1d 3340	. . . . . 6 ⊢ (𝜑 → (𝑉 ∖ 𝑈) = ((𝐽...𝐾) ∖ 𝑈))
17	8, 9, 10, 1, 3, 4	ballotfilemdifcfz 13171	. . . . . 6 ⊢ (𝜑 → ((𝐽...𝐾) ∖ 𝑈) ∈ Fin)
18	16, 17	eqeltrd 2311	. . . . 5 ⊢ (𝜑 → (𝑉 ∖ 𝑈) ∈ Fin)
19		hashcl 11169	. . . . 5 ⊢ ((𝑉 ∖ 𝑈) ∈ Fin → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
20	18, 19	syl 14	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
21	20	nn0zd 9716	. . 3 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℤ)
22	15, 21	zsubcld 9723	. 2 ⊢ (𝜑 → ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) ∈ ℤ)
23		ineq2 3420	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑣 ∩ 𝑢) = (𝑣 ∩ 𝑈))
24	23	fveq2d 5679	. . . 4 ⊢ (𝑢 = 𝑈 → (♯‘(𝑣 ∩ 𝑢)) = (♯‘(𝑣 ∩ 𝑈)))
25		difeq2 3335	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑣 ∖ 𝑢) = (𝑣 ∖ 𝑈))
26	25	fveq2d 5679	. . . 4 ⊢ (𝑢 = 𝑈 → (♯‘(𝑣 ∖ 𝑢)) = (♯‘(𝑣 ∖ 𝑈)))
27	24, 26	oveq12d 6076	. . 3 ⊢ (𝑢 = 𝑈 → ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))) = ((♯‘(𝑣 ∩ 𝑈)) − (♯‘(𝑣 ∖ 𝑈))))
28		ineq1 3419	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑣 ∩ 𝑈) = (𝑉 ∩ 𝑈))
29	28	fveq2d 5679	. . . 4 ⊢ (𝑣 = 𝑉 → (♯‘(𝑣 ∩ 𝑈)) = (♯‘(𝑉 ∩ 𝑈)))
30		difeq1 3334	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑣 ∖ 𝑈) = (𝑉 ∖ 𝑈))
31	30	fveq2d 5679	. . . 4 ⊢ (𝑣 = 𝑉 → (♯‘(𝑣 ∖ 𝑈)) = (♯‘(𝑉 ∖ 𝑈)))
32	29, 31	oveq12d 6076	. . 3 ⊢ (𝑣 = 𝑉 → ((♯‘(𝑣 ∩ 𝑈)) − (♯‘(𝑣 ∖ 𝑈))) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
33		ballotlemg	. . 3 ⊢ ↑ = (𝑢 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
34	27, 32, 33	ovmpog 6196	. 2 ⊢ ((𝑈 ∈ 𝑂 ∧ 𝑉 ∈ Fin ∧ ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) ∈ ℤ) → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
35	1, 6, 22, 34	syl3anc 1274	1 ⊢ (𝜑 → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1398 ∈ wcel 2205 ∀wral 2522 {crab 2526 ∖ cdif 3211 ∩ cin 3213 ifcif 3624 𝒫 cpw 3674 class class class wbr 4114 ↦ cmpt 4176 “ cima 4757 ‘cfv 5357 (class class class)co 6058 ∈ cmpo 6060 Fincfn 6988 infcinf 7287 ℝcr 8142 0cc0 8143 1c1 8144 + caddc 8146 < clt 8324 ≤ cle 8325 − cmin 8460 / cdiv 8963 ℕcn 9254 ℕ₀cn0 9513 ℤcz 9594 ...cfz 10361 ♯chash 11163

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 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-addass 8245 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-0id 8251 ax-rnegex 8252 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259

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 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-1o 6660 df-er 6780 df-en 6989 df-dom 6990 df-fin 6991 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-inn 9255 df-n0 9514 df-z 9595 df-uz 9872 df-fz 10362 df-ihash 11164

This theorem is referenced by: ballotfilemgun 13212 ballotfilemfg 13213 ballotfilemfrc 13214

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