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Theorem ballotfilemiex 13188

Description: Properties of (𝐼‘𝐶). (Contributed by Thierry Arnoux, 12-Dec-2016.) (Revised by AV, 6-Oct-2020.)

**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, < ))

**Assertion**
Ref	Expression
ballotfilemiex	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))

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 𝐶,𝑖,𝑘 𝑖,𝐸,𝑘 𝐶,𝑘 𝑘,𝐼 𝑘,𝑐,𝐸 Allowed substitution hints: 𝐶(𝑥,𝑐) 𝑃(𝑥,𝑖,𝑘,𝑐) 𝐸(𝑥) 𝐹(𝑥) 𝐼(𝑥,𝑖,𝑐) 𝑀(𝑥) 𝑁(𝑥) 𝑂(𝑥)

Proof of Theorem ballotfilemiex Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . . 4 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . 4 ⊢ 𝑁 ∈ ℕ
3		ballotfilem.o	. . . 4 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
4		ballotfilem.p	. . . 4 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotth.f	. . . 4 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		ballotth.e	. . . 4 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
7		ballotth.mgtn	. . . 4 ⊢ 𝑁 < 𝑀
8		ballotth.i	. . . 4 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
9	1, 2, 3, 4, 5, 6, 7, 8	ballotfilemi 13187	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))
10		ssrab2 3327	. . . . . 6 ⊢ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ⊆ (1...(𝑀 + 𝑁))
11		fz1ssnn 10411	. . . . . 6 ⊢ (1...(𝑀 + 𝑁)) ⊆ ℕ
12	10, 11	sstri 3251	. . . . 5 ⊢ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ⊆ ℕ
13	12	a1i 9	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ⊆ ℕ)
14		nnz 9613	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
15	14	adantl 277	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → 𝑧 ∈ ℤ)
16		1zzd 9621	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → 1 ∈ ℤ)
17		nnaddcl 9274	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
18	1, 2, 17	mp2an 426	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℕ
19	18	nnzi 9615	. . . . . . . . 9 ⊢ (𝑀 + 𝑁) ∈ ℤ
20	19	a1i 9	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)
21		fzdcel 10394	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → DECID 𝑧 ∈ (1...(𝑀 + 𝑁)))
22	15, 16, 20, 21	syl3anc 1274	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → DECID 𝑧 ∈ (1...(𝑀 + 𝑁)))
23		eldifi 3345	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
24	23	adantr 276	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → 𝐶 ∈ 𝑂)
25	1, 2, 3, 4, 5, 24, 15	ballotfilemfelz 13174	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → ((𝐹‘𝐶)‘𝑧) ∈ ℤ)
26		0zd 9606	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → 0 ∈ ℤ)
27		zdceq 9670	. . . . . . . 8 ⊢ ((((𝐹‘𝐶)‘𝑧) ∈ ℤ ∧ 0 ∈ ℤ) → DECID ((𝐹‘𝐶)‘𝑧) = 0)
28	25, 26, 27	syl2anc 411	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → DECID ((𝐹‘𝐶)‘𝑧) = 0)
29	22, 28	dcand 941	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → DECID (𝑧 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘𝑧) = 0))
30		fveqeq2 5684	. . . . . . . 8 ⊢ (𝑘 = 𝑧 → (((𝐹‘𝐶)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘𝑧) = 0))
31	30	elrab 2976	. . . . . . 7 ⊢ (𝑧 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ↔ (𝑧 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘𝑧) = 0))
32	31	dcbii 848	. . . . . 6 ⊢ (DECID 𝑧 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ↔ DECID (𝑧 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘𝑧) = 0))
33	29, 32	sylibr 134	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → DECID 𝑧 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
34	33	ralrimiva 2617	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ∀𝑧 ∈ ℕ DECID 𝑧 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
35	1, 2, 3, 4, 5, 6, 7	ballotfilem5 13186	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ∃𝑘 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑘) = 0)
36		rabn0m 3540	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ↔ ∃𝑘 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑘) = 0)
37	35, 36	sylibr 134	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ∃𝑦 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
38		nnmindc 12755	. . . 4 ⊢ (({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ∧ ∃𝑦 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}) → inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
39	13, 34, 37, 38	syl3anc 1274	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
40	9, 39	eqeltrd 2311	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
41		fveqeq2 5684	. . 3 ⊢ (𝑘 = (𝐼‘𝐶) → (((𝐹‘𝐶)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
42	41	elrab 2976	. 2 ⊢ ((𝐼‘𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ↔ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
43	40, 42	sylib 122	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 DECID wdc 842 = wceq 1398 ∃wex 1541 ∈ wcel 2205 ∀wral 2522 ∃wrex 2523 {crab 2526 ∖ cdif 3211 ∩ cin 3213 ⊆ wss 3214 𝒫 cpw 3674 class class class wbr 4114 ↦ cmpt 4176 ‘cfv 5357 (class class class)co 6058 Fincfn 6988 infcinf 7287 ℝcr 8142 0cc0 8143 1c1 8144 + caddc 8146 < clt 8324 − cmin 8460 / cdiv 8963 ℕcn 9254 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-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261

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-rmo 2530 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-po 4422 df-iso 4423 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-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-fz 10362 df-fzo 10499 df-ihash 11164

This theorem is referenced by: ballotfilemi1 13189 ballotfilemii 13190 ballotfilemimin 13193 ballotfilemic 13194 ballotfilem1c 13195 ballotfilemsv 13197 ballotfilemsgt1 13198 ballotfilemsdom 13199 ballotfilemsel1i 13200 ballotfilemsf1o 13201 ballotfilemsi 13202 ballotfilemsima 13203 ballotfilemrv2 13209 ballotfilemfrc 13214 ballotfilemfrci 13215 ballotfilemfrceq 13216 ballotfilemfrcn0 13217 ballotfilemrc 13218 ballotfilemirc 13219 ballotfilem1ri 13222

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