ballotfilemgval - Intuitionistic Logic Explorer

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Theorem ballotfilemgval 13269

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 10870	. . 3 ⊢ (𝜑 → (𝐽...𝐾) ∈ Fin)
6	2, 5	eqeltrd 2315	. 2 ⊢ (𝜑 → 𝑉 ∈ Fin)
7	2	ineq1d 3431	. . . . . 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 13228	. . . . . 6 ⊢ (𝜑 → ((𝐽...𝐾) ∩ 𝑈) ∈ Fin)
12	7, 11	eqeltrd 2315	. . . . 5 ⊢ (𝜑 → (𝑉 ∩ 𝑈) ∈ Fin)
13		hashcl 11222	. . . . 5 ⊢ ((𝑉 ∩ 𝑈) ∈ Fin → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
14	12, 13	syl 14	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
15	14	nn0zd 9768	. . 3 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℤ)
16	2	difeq1d 3346	. . . . . 6 ⊢ (𝜑 → (𝑉 ∖ 𝑈) = ((𝐽...𝐾) ∖ 𝑈))
17	8, 9, 10, 1, 3, 4	ballotfilemdifcfz 13229	. . . . . 6 ⊢ (𝜑 → ((𝐽...𝐾) ∖ 𝑈) ∈ Fin)
18	16, 17	eqeltrd 2315	. . . . 5 ⊢ (𝜑 → (𝑉 ∖ 𝑈) ∈ Fin)
19		hashcl 11222	. . . . 5 ⊢ ((𝑉 ∖ 𝑈) ∈ Fin → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
20	18, 19	syl 14	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
21	20	nn0zd 9768	. . 3 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℤ)
22	15, 21	zsubcld 9775	. 2 ⊢ (𝜑 → ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) ∈ ℤ)
23		ineq2 3426	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑣 ∩ 𝑢) = (𝑣 ∩ 𝑈))
24	23	fveq2d 5699	. . . 4 ⊢ (𝑢 = 𝑈 → (♯‘(𝑣 ∩ 𝑢)) = (♯‘(𝑣 ∩ 𝑈)))
25		difeq2 3341	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑣 ∖ 𝑢) = (𝑣 ∖ 𝑈))
26	25	fveq2d 5699	. . . 4 ⊢ (𝑢 = 𝑈 → (♯‘(𝑣 ∖ 𝑢)) = (♯‘(𝑣 ∖ 𝑈)))
27	24, 26	oveq12d 6103	. . 3 ⊢ (𝑢 = 𝑈 → ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))) = ((♯‘(𝑣 ∩ 𝑈)) − (♯‘(𝑣 ∖ 𝑈))))
28		ineq1 3425	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑣 ∩ 𝑈) = (𝑉 ∩ 𝑈))
29	28	fveq2d 5699	. . . 4 ⊢ (𝑣 = 𝑉 → (♯‘(𝑣 ∩ 𝑈)) = (♯‘(𝑉 ∩ 𝑈)))
30		difeq1 3340	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑣 ∖ 𝑈) = (𝑉 ∖ 𝑈))
31	30	fveq2d 5699	. . . 4 ⊢ (𝑣 = 𝑉 → (♯‘(𝑣 ∖ 𝑈)) = (♯‘(𝑉 ∖ 𝑈)))
32	29, 31	oveq12d 6103	. . 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 6223	. 2 ⊢ ((𝑈 ∈ 𝑂 ∧ 𝑉 ∈ Fin ∧ ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) ∈ ℤ) → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
35	1, 6, 22, 34	syl3anc 1278	1 ⊢ (𝜑 → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 = wceq 1402 ∈ wcel 2209 ∀wral 2528 {crab 2532 ∖ cdif 3217 ∩ cin 3219 ifcif 3638 𝒫 cpw 3688 class class class wbr 4130 ↦ cmpt 4192 “ cima 4777 ‘cfv 5377 (class class class)co 6085 ∈ cmpo 6087 Fincfn 7022 infcinf 7323 ℝcr 8178 0cc0 8179 1c1 8180 + caddc 8182 < clt 8360 ≤ cle 8361 − cmin 8497 / cdiv 9003 ℕcn 9305 ℕ₀cn0 9565 ℤcz 9646 ...cfz 10413 ♯chash 11216

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-addass 8281 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-1o 6687 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-inn 9306 df-n0 9566 df-z 9647 df-uz 9924 df-fz 10414 df-ihash 11217

This theorem is used by: ballotfilemgun 13270 ballotfilemfg 13271 ballotfilemfrc 13272

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