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Theorem ballotfilemiex 13246

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 13245	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))
10		ssrab2 3333	. . . . . 6 ⊢ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ⊆ (1...(𝑀 + 𝑁))
11		fz1ssnn 10464	. . . . . 6 ⊢ (1...(𝑀 + 𝑁)) ⊆ ℕ
12	10, 11	sstri 3257	. . . . 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 9665	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
15	14	adantl 277	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → 𝑧 ∈ ℤ)
16		1zzd 9673	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → 1 ∈ ℤ)
17		nnaddcl 9325	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
18	1, 2, 17	mp2an 430	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℕ
19	18	nnzi 9667	. . . . . . . . 9 ⊢ (𝑀 + 𝑁) ∈ ℤ
20	19	a1i 9	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)
21		fzdcel 10446	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → DECID 𝑧 ∈ (1...(𝑀 + 𝑁)))
22	15, 16, 20, 21	syl3anc 1278	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → DECID 𝑧 ∈ (1...(𝑀 + 𝑁)))
23		eldifi 3351	. . . . . . . . . 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 13232	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → ((𝐹‘𝐶)‘𝑧) ∈ ℤ)
26		0zd 9658	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → 0 ∈ ℤ)
27		zdceq 9722	. . . . . . . 8 ⊢ ((((𝐹‘𝐶)‘𝑧) ∈ ℤ ∧ 0 ∈ ℤ) → DECID ((𝐹‘𝐶)‘𝑧) = 0)
28	25, 26, 27	syl2anc 415	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → DECID ((𝐹‘𝐶)‘𝑧) = 0)
29	22, 28	dcand 945	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → DECID (𝑧 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘𝑧) = 0))
30		fveqeq2 5704	. . . . . . . 8 ⊢ (𝑘 = 𝑧 → (((𝐹‘𝐶)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘𝑧) = 0))
31	30	elrab 2982	. . . . . . 7 ⊢ (𝑧 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ↔ (𝑧 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘𝑧) = 0))
32	31	dcbii 852	. . . . . 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 2623	. . . 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 13244	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ∃𝑘 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑘) = 0)
36		rabn0m 3549	. . . . 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 12813	. . . 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 1278	. . 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 2315	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
41		fveqeq2 5704	. . 3 ⊢ (𝑘 = (𝐼‘𝐶) → (((𝐹‘𝐶)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
42	41	elrab 2982	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 DECID wdc 846 = wceq 1402 ∃wex 1545 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 {crab 2532 ∖ cdif 3217 ∩ cin 3219 ⊆ wss 3220 𝒫 cpw 3688 class class class wbr 4130 ↦ cmpt 4192 ‘cfv 5377 (class class class)co 6085 Fincfn 7022 infcinf 7323 ℝcr 8178 0cc0 8179 1c1 8180 + caddc 8182 < clt 8360 − cmin 8497 / cdiv 9003 ℕcn 9305 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-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297

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-rmo 2536 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-po 4441 df-iso 4442 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-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7324 df-inf 7325 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8904 df-ap 8911 df-div 9004 df-inn 9306 df-2 9364 df-n0 9566 df-z 9647 df-uz 9924 df-q 10022 df-rp 10057 df-fz 10414 df-fzo 10552 df-ihash 11217

This theorem is used by: ballotfilemi1 13247 ballotfilemii 13248 ballotfilemimin 13251 ballotfilemic 13252 ballotfilem1c 13253 ballotfilemsv 13255 ballotfilemsgt1 13256 ballotfilemsdom 13257 ballotfilemsel1i 13258 ballotfilemsf1o 13259 ballotfilemsi 13260 ballotfilemsima 13261 ballotfilemrv2 13267 ballotfilemfrc 13272 ballotfilemfrci 13273 ballotfilemfrceq 13274 ballotfilemfrcn0 13275 ballotfilemrc 13276 ballotfilemirc 13277 ballotfilem1ri 13280

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