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Theorem ballotfilemiex 13302

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 13301	. . 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 10479	. . . . . 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 9673	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
15	14	adantl 277	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → 𝑧 ∈ ℤ)
16		1zzd 9681	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → 1 ∈ ℤ)
17		nnaddcl 9334	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
18	1, 2, 17	mp2an 430	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℕ
19	18	nnzi 9675	. . . . . . . . 9 ⊢ (𝑀 + 𝑁) ∈ ℤ
20	19	a1i 9	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)
21		fzdcel 10461	. . . . . . . 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 13288	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → ((𝐹‘𝐶)‘𝑧) ∈ ℤ)
26		0zd 9666	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑧 ∈ ℕ) → 0 ∈ ℤ)
27		zdceq 9730	. . . . . . . 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 5707	. . . . . . . 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 13300	. . . . 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 12836	. . . 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 5707	. . 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 4131 ↦ cmpt 4193 ‘cfv 5380 (class class class)co 6088 Fincfn 7028 infcinf 7330 ℝcr 8185 0cc0 8186 1c1 8187 + caddc 8189 < clt 8367 − cmin 8505 / cdiv 9011 ℕcn 9314 ℤcz 9654 ...cfz 10428 ♯chash 11236

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304

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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6581 df-irdg 6646 df-frec 6667 df-1o 6692 df-oadd 6696 df-er 6812 df-en 7029 df-dom 7030 df-fin 7031 df-sup 7331 df-inf 7332 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-n0 9576 df-z 9655 df-uz 9938 df-q 10036 df-rp 10072 df-fz 10429 df-fzo 10567 df-ihash 11237

This theorem is used by: ballotfilemi1 13303 ballotfilemii 13304 ballotfilemimin 13307 ballotfilemic 13308 ballotfilem1c 13309 ballotfilemsv 13311 ballotfilemsgt1 13312 ballotfilemsdom 13313 ballotfilemsel1i 13314 ballotfilemsf1o 13315 ballotfilemsi 13316 ballotfilemsima 13317 ballotfilemrv2 13323 ballotfilemfrc 13328 ballotfilemfrci 13329 ballotfilemfrceq 13330 ballotfilemfrcn0 13331 ballotfilemrc 13332 ballotfilemirc 13333 ballotfilem1ri 13336

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