ballotlemro - Mathbox for Thierry Arnoux

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Theorem ballotlemro 34065

Description: Range of 𝑅 is included in 𝑂. (Contributed by Thierry Arnoux, 17-Apr-2017.)

**Hypotheses**
Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotth.o	⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
ballotth.p	⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘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))

**Assertion**
Ref	Expression
ballotlemro	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)

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

**Proof of Theorem ballotlemro**
Step	Hyp	Ref	Expression
1		ballotth.m	. . . 4 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . 4 ⊢ 𝑁 ∈ ℕ
3		ballotth.o	. . . 4 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4		ballotth.p	. . . 4 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘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		ballotth.s	. . . 4 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
10		ballotth.r	. . . 4 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemrval 34060	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶))
12		imassrn 6068	. . . 4 ⊢ ((𝑆‘𝐶) “ 𝐶) ⊆ ran (𝑆‘𝐶)
13	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsf1o 34056	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ^◡(𝑆‘𝐶) = (𝑆‘𝐶)))
14	13	simpld 494	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
15		f1ofo 6840	. . . . 5 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)))
16		forn 6808	. . . . 5 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)) → ran (𝑆‘𝐶) = (1...(𝑀 + 𝑁)))
17	14, 15, 16	3syl 18	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ran (𝑆‘𝐶) = (1...(𝑀 + 𝑁)))
18	12, 17	sseqtrid 4030	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ 𝐶) ⊆ (1...(𝑀 + 𝑁)))
19	11, 18	eqsstrd 4016	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ⊆ (1...(𝑀 + 𝑁)))
20		f1of1 6832	. . . . . . 7 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
21	14, 20	syl 17	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
22		eldifi 4122	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
23	1, 2, 3	ballotlemelo 34030	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
24	22, 23	sylib 217	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
25	24	simpld 494	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
26		id 22	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ (𝑂 ∖ 𝐸))
27		f1imaeng 9024	. . . . . 6 ⊢ (((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)) ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ (𝑂 ∖ 𝐸)) → ((𝑆‘𝐶) “ 𝐶) ≈ 𝐶)
28	21, 25, 26, 27	syl3anc 1369	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ 𝐶) ≈ 𝐶)
29	11, 28	eqbrtrd 5164	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ≈ 𝐶)
30		hasheni 14325	. . . 4 ⊢ ((𝑅‘𝐶) ≈ 𝐶 → (♯‘(𝑅‘𝐶)) = (♯‘𝐶))
31	29, 30	syl 17	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (♯‘(𝑅‘𝐶)) = (♯‘𝐶))
32	24	simprd 495	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (♯‘𝐶) = 𝑀)
33	31, 32	eqtrd 2767	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (♯‘(𝑅‘𝐶)) = 𝑀)
34	1, 2, 3	ballotlemelo 34030	. 2 ⊢ ((𝑅‘𝐶) ∈ 𝑂 ↔ ((𝑅‘𝐶) ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘(𝑅‘𝐶)) = 𝑀))
35	19, 33, 34	sylanbrc 582	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 {crab 3427 ∖ cdif 3941 ∩ cin 3943 ⊆ wss 3944 ifcif 4524 𝒫 cpw 4598 class class class wbr 5142 ↦ cmpt 5225 ^◡ccnv 5671 ran crn 5673 “ cima 5675 –1-1→wf1 6539 –onto→wfo 6540 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7414 ≈ cen 8950 infcinf 9450 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 < clt 11264 ≤ cle 11265 − cmin 11460 / cdiv 11887 ℕcn 12228 ℤcz 12574 ...cfz 13502 ♯chash 14307

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-oadd 8482 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-dju 9910 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-fz 13503 df-hash 14308

This theorem is referenced by: ballotlemfrc 34069 ballotlemfrceq 34071 ballotlemfrcn0 34072 ballotlemrc 34073

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