ballotlemsima - Mathbox for Thierry Arnoux

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Theorem ballotlemsima 34071

Description: The image by 𝑆 of an interval before the first pick. (Contributed by Thierry Arnoux, 5-May-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)))

**Assertion**
Ref	Expression
ballotlemsima	⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ (1...𝐽)) = (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))

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

Proof of Theorem ballotlemsima Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 6068	. . . . . 6 ⊢ ((𝑆‘𝐶) “ (1...𝐽)) ⊆ ran (𝑆‘𝐶)
2		ballotth.m	. . . . . . . . 9 ⊢ 𝑀 ∈ ℕ
3		ballotth.n	. . . . . . . . 9 ⊢ 𝑁 ∈ ℕ
4		ballotth.o	. . . . . . . . 9 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
5		ballotth.p	. . . . . . . . 9 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
6		ballotth.f	. . . . . . . . 9 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
7		ballotth.e	. . . . . . . . 9 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
8		ballotth.mgtn	. . . . . . . . 9 ⊢ 𝑁 < 𝑀
9		ballotth.i	. . . . . . . . 9 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
10		ballotth.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
11	2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemsf1o 34069	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ^◡(𝑆‘𝐶) = (𝑆‘𝐶)))
12	11	simpld 494	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
13		f1of 6833	. . . . . . 7 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))⟶(1...(𝑀 + 𝑁)))
14		frn 6723	. . . . . . 7 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))⟶(1...(𝑀 + 𝑁)) → ran (𝑆‘𝐶) ⊆ (1...(𝑀 + 𝑁)))
15	12, 13, 14	3syl 18	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ran (𝑆‘𝐶) ⊆ (1...(𝑀 + 𝑁)))
16	1, 15	sstrid 3989	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ (1...𝐽)) ⊆ (1...(𝑀 + 𝑁)))
17		fzssuz 13566	. . . . . 6 ⊢ (1...(𝑀 + 𝑁)) ⊆ (ℤ_≥‘1)
18		uzssz 12865	. . . . . 6 ⊢ (ℤ_≥‘1) ⊆ ℤ
19	17, 18	sstri 3987	. . . . 5 ⊢ (1...(𝑀 + 𝑁)) ⊆ ℤ
20	16, 19	sstrdi 3990	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ (1...𝐽)) ⊆ ℤ)
21	20	adantr 480	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ (1...𝐽)) ⊆ ℤ)
22	21	sselda 3978	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ((𝑆‘𝐶) “ (1...𝐽))) → 𝑘 ∈ ℤ)
23		elfzelz 13525	. . 3 ⊢ (𝑘 ∈ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) → 𝑘 ∈ ℤ)
24	23	adantl 481	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))) → 𝑘 ∈ ℤ)
25		f1ofn 6834	. . . . . . 7 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆‘𝐶) Fn (1...(𝑀 + 𝑁)))
26	12, 25	syl 17	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶) Fn (1...(𝑀 + 𝑁)))
27	26	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑆‘𝐶) Fn (1...(𝑀 + 𝑁)))
28	2, 3, 4, 5, 6, 7, 8, 9	ballotlemiex 34057	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
29	28	simpld 494	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
30	29	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
31		elfzuz3 13522	. . . . . . . 8 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ_≥‘(𝐼‘𝐶)))
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ (ℤ_≥‘(𝐼‘𝐶)))
33		elfzuz3 13522	. . . . . . . 8 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → (𝐼‘𝐶) ∈ (ℤ_≥‘𝐽))
34	33	adantl 481	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (ℤ_≥‘𝐽))
35		uztrn 12862	. . . . . . 7 ⊢ (((𝑀 + 𝑁) ∈ (ℤ_≥‘(𝐼‘𝐶)) ∧ (𝐼‘𝐶) ∈ (ℤ_≥‘𝐽)) → (𝑀 + 𝑁) ∈ (ℤ_≥‘𝐽))
36	32, 34, 35	syl2anc 583	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ (ℤ_≥‘𝐽))
37		fzss2 13565	. . . . . 6 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘𝐽) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
38	36, 37	syl 17	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
39		fvelimab 6965	. . . . 5 ⊢ (((𝑆‘𝐶) Fn (1...(𝑀 + 𝑁)) ∧ (1...𝐽) ⊆ (1...(𝑀 + 𝑁))) → (𝑘 ∈ ((𝑆‘𝐶) “ (1...𝐽)) ↔ ∃𝑗 ∈ (1...𝐽)((𝑆‘𝐶)‘𝑗) = 𝑘))
40	27, 38, 39	syl2anc 583	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑘 ∈ ((𝑆‘𝐶) “ (1...𝐽)) ↔ ∃𝑗 ∈ (1...𝐽)((𝑆‘𝐶)‘𝑗) = 𝑘))
41	40	adantr 480	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ((𝑆‘𝐶) “ (1...𝐽)) ↔ ∃𝑗 ∈ (1...𝐽)((𝑆‘𝐶)‘𝑗) = 𝑘))
42		1zzd 12615	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 1 ∈ ℤ)
43	2	nnzi 12608	. . . . . . . . . . . . 13 ⊢ 𝑀 ∈ ℤ
44	3	nnzi 12608	. . . . . . . . . . . . 13 ⊢ 𝑁 ∈ ℤ
45		zaddcl 12624	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
46	43, 44, 45	mp2an 691	. . . . . . . . . . . 12 ⊢ (𝑀 + 𝑁) ∈ ℤ
47	46	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ ℤ)
48		elfzelz 13525	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ)
49	48	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
50		elfzle1 13528	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 1 ≤ 𝐽)
51	50	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 1 ≤ 𝐽)
52	49	zred 12688	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℝ)
53		elfzelz 13525	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℤ)
54	29, 53	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
55	54	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ ℤ)
56	55	zred 12688	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ ℝ)
57	47	zred 12688	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ ℝ)
58		elfzle2 13529	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ≤ (𝐼‘𝐶))
59	58	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ≤ (𝐼‘𝐶))
60		elfzle2 13529	. . . . . . . . . . . . . 14 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
61	29, 60	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
62	61	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
63	52, 56, 57, 59, 62	letrd 11393	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ≤ (𝑀 + 𝑁))
64	42, 47, 49, 51, 63	elfzd 13516	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
65	2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemsv 34065	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) = if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽))
66	64, 65	syldan 590	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) = if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽))
67		simpr 484	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ (1...(𝐼‘𝐶)))
68		iftrue 4530	. . . . . . . . . 10 ⊢ (𝐽 ≤ (𝐼‘𝐶) → if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽) = (((𝐼‘𝐶) + 1) − 𝐽))
69	67, 58, 68	3syl 18	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽) = (((𝐼‘𝐶) + 1) − 𝐽))
70	66, 69	eqtrd 2767	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) = (((𝐼‘𝐶) + 1) − 𝐽))
71	70	oveq1d 7429	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) = ((((𝐼‘𝐶) + 1) − 𝐽)...(𝐼‘𝐶)))
72	71	eleq2d 2814	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑘 ∈ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ↔ 𝑘 ∈ ((((𝐼‘𝐶) + 1) − 𝐽)...(𝐼‘𝐶))))
73	72	adantr 480	. . . . 5 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ↔ 𝑘 ∈ ((((𝐼‘𝐶) + 1) − 𝐽)...(𝐼‘𝐶))))
74	54	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (𝐼‘𝐶) ∈ ℤ)
75	74	zcnd 12689	. . . . . . . 8 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (𝐼‘𝐶) ∈ ℂ)
76		1cnd 11231	. . . . . . . 8 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → 1 ∈ ℂ)
77	75, 76	pncand 11594	. . . . . . 7 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (((𝐼‘𝐶) + 1) − 1) = (𝐼‘𝐶))
78	77	oveq2d 7430	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → ((((𝐼‘𝐶) + 1) − 𝐽)...(((𝐼‘𝐶) + 1) − 1)) = ((((𝐼‘𝐶) + 1) − 𝐽)...(𝐼‘𝐶)))
79	78	eleq2d 2814	. . . . 5 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ((((𝐼‘𝐶) + 1) − 𝐽)...(((𝐼‘𝐶) + 1) − 1)) ↔ 𝑘 ∈ ((((𝐼‘𝐶) + 1) − 𝐽)...(𝐼‘𝐶))))
80		1zzd 12615	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → 1 ∈ ℤ)
81	48	ad2antlr 726	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → 𝐽 ∈ ℤ)
82	74	peano2zd 12691	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → ((𝐼‘𝐶) + 1) ∈ ℤ)
83		simpr 484	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ)
84		fzrev 13588	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (((𝐼‘𝐶) + 1) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 ∈ ((((𝐼‘𝐶) + 1) − 𝐽)...(((𝐼‘𝐶) + 1) − 1)) ↔ (((𝐼‘𝐶) + 1) − 𝑘) ∈ (1...𝐽)))
85	80, 81, 82, 83, 84	syl22anc 838	. . . . 5 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ((((𝐼‘𝐶) + 1) − 𝐽)...(((𝐼‘𝐶) + 1) − 1)) ↔ (((𝐼‘𝐶) + 1) − 𝑘) ∈ (1...𝐽)))
86	73, 79, 85	3bitr2d 307	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ↔ (((𝐼‘𝐶) + 1) − 𝑘) ∈ (1...𝐽)))
87		risset 3225	. . . . 5 ⊢ ((((𝐼‘𝐶) + 1) − 𝑘) ∈ (1...𝐽) ↔ ∃𝑗 ∈ (1...𝐽)𝑗 = (((𝐼‘𝐶) + 1) − 𝑘))
88	87	a1i 11	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → ((((𝐼‘𝐶) + 1) − 𝑘) ∈ (1...𝐽) ↔ ∃𝑗 ∈ (1...𝐽)𝑗 = (((𝐼‘𝐶) + 1) − 𝑘)))
89		eqcom 2734	. . . . . . 7 ⊢ ((((𝐼‘𝐶) + 1) − 𝑘) = 𝑗 ↔ 𝑗 = (((𝐼‘𝐶) + 1) − 𝑘))
90	54	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼‘𝐶) ∈ ℤ)
91	90	adantlr 714	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼‘𝐶) ∈ ℤ)
92	91	zcnd 12689	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼‘𝐶) ∈ ℂ)
93		1cnd 11231	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 1 ∈ ℂ)
94	92, 93	addcld 11255	. . . . . . . 8 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → ((𝐼‘𝐶) + 1) ∈ ℂ)
95		simplr 768	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 𝑘 ∈ ℤ)
96	95	zcnd 12689	. . . . . . . 8 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 𝑘 ∈ ℂ)
97		elfzelz 13525	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...𝐽) → 𝑗 ∈ ℤ)
98	97	adantl 481	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ ℤ)
99	98	zcnd 12689	. . . . . . . 8 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ ℂ)
100		subsub23 11487	. . . . . . . 8 ⊢ ((((𝐼‘𝐶) + 1) ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((((𝐼‘𝐶) + 1) − 𝑘) = 𝑗 ↔ (((𝐼‘𝐶) + 1) − 𝑗) = 𝑘))
101	94, 96, 99, 100	syl3anc 1369	. . . . . . 7 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → ((((𝐼‘𝐶) + 1) − 𝑘) = 𝑗 ↔ (((𝐼‘𝐶) + 1) − 𝑗) = 𝑘))
102	89, 101	bitr3id 285	. . . . . 6 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (𝑗 = (((𝐼‘𝐶) + 1) − 𝑘) ↔ (((𝐼‘𝐶) + 1) − 𝑗) = 𝑘))
103		simpll 766	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝐶 ∈ (𝑂 ∖ 𝐸))
104	38	sselda 3978	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ (1...(𝑀 + 𝑁)))
105	2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemsv 34065	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝑗) = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
106	103, 104, 105	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → ((𝑆‘𝐶)‘𝑗) = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
107	97	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ ℤ)
108	107	zred 12688	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ ℝ)
109	48	ad2antlr 726	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝐽 ∈ ℤ)
110	109	zred 12688	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝐽 ∈ ℝ)
111	90	zred 12688	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼‘𝐶) ∈ ℝ)
112		elfzle2 13529	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝐽) → 𝑗 ≤ 𝐽)
113	112	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ≤ 𝐽)
114	58	ad2antlr 726	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝐽 ≤ (𝐼‘𝐶))
115	108, 110, 111, 113, 114	letrd 11393	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ≤ (𝐼‘𝐶))
116		iftrue 4530	. . . . . . . . . 10 ⊢ (𝑗 ≤ (𝐼‘𝐶) → if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗) = (((𝐼‘𝐶) + 1) − 𝑗))
117	115, 116	syl 17	. . . . . . . . 9 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗) = (((𝐼‘𝐶) + 1) − 𝑗))
118	106, 117	eqtrd 2767	. . . . . . . 8 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → ((𝑆‘𝐶)‘𝑗) = (((𝐼‘𝐶) + 1) − 𝑗))
119	118	eqeq1d 2729	. . . . . . 7 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → (((𝑆‘𝐶)‘𝑗) = 𝑘 ↔ (((𝐼‘𝐶) + 1) − 𝑗) = 𝑘))
120	119	adantlr 714	. . . . . 6 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (((𝑆‘𝐶)‘𝑗) = 𝑘 ↔ (((𝐼‘𝐶) + 1) − 𝑗) = 𝑘))
121	102, 120	bitr4d 282	. . . . 5 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (𝑗 = (((𝐼‘𝐶) + 1) − 𝑘) ↔ ((𝑆‘𝐶)‘𝑗) = 𝑘))
122	121	rexbidva 3171	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (∃𝑗 ∈ (1...𝐽)𝑗 = (((𝐼‘𝐶) + 1) − 𝑘) ↔ ∃𝑗 ∈ (1...𝐽)((𝑆‘𝐶)‘𝑗) = 𝑘))
123	86, 88, 122	3bitrd 305	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ↔ ∃𝑗 ∈ (1...𝐽)((𝑆‘𝐶)‘𝑗) = 𝑘))
124	41, 123	bitr4d 282	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ((𝑆‘𝐶) “ (1...𝐽)) ↔ 𝑘 ∈ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))
125	22, 24, 124	eqrdav 2726	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ (1...𝐽)) = (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 {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 Fn wfn 6537 ⟶wf 6538 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7414 infcinf 9456 ℂcc 11128 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 < clt 11270 ≤ cle 11271 − cmin 11466 / cdiv 11893 ℕcn 12234 ℤcz 12580 ℤ_≥cuz 12844 ...cfz 13508 ♯chash 14313

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 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

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 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-hash 14314

This theorem is referenced by: ballotlemfrc 34082

W3C validator