ballotlemsima - Mathbox for Thierry Arnoux

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Theorem ballotlemsima 34673

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 6030	. . . . . 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 34671	. . . . . . . 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 6774	. . . . . . 7 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))⟶(1...(𝑀 + 𝑁)))
14		frn 6669	. . . . . . 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 3945	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ (1...𝐽)) ⊆ (1...(𝑀 + 𝑁)))
17		fzssuz 13481	. . . . . 6 ⊢ (1...(𝑀 + 𝑁)) ⊆ (ℤ_≥‘1)
18		uzssz 12772	. . . . . 6 ⊢ (ℤ_≥‘1) ⊆ ℤ
19	17, 18	sstri 3943	. . . . 5 ⊢ (1...(𝑀 + 𝑁)) ⊆ ℤ
20	16, 19	sstrdi 3946	. . . 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 3933	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ((𝑆‘𝐶) “ (1...𝐽))) → 𝑘 ∈ ℤ)
23		elfzelz 13440	. . 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 6775	. . . . . . 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 34659	. . . . . . . . . 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 13437	. . . . . . . 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 13437	. . . . . . . 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 12769	. . . . . . 7 ⊢ (((𝑀 + 𝑁) ∈ (ℤ_≥‘(𝐼‘𝐶)) ∧ (𝐼‘𝐶) ∈ (ℤ_≥‘𝐽)) → (𝑀 + 𝑁) ∈ (ℤ_≥‘𝐽))
36	32, 34, 35	syl2anc 584	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ (ℤ_≥‘𝐽))
37		fzss2 13480	. . . . . 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 6906	. . . . 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 584	. . . 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 12522	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 1 ∈ ℤ)
43	2	nnzi 12515	. . . . . . . . . . . . 13 ⊢ 𝑀 ∈ ℤ
44	3	nnzi 12515	. . . . . . . . . . . . 13 ⊢ 𝑁 ∈ ℤ
45		zaddcl 12531	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
46	43, 44, 45	mp2an 692	. . . . . . . . . . . 12 ⊢ (𝑀 + 𝑁) ∈ ℤ
47	46	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ ℤ)
48		elfzelz 13440	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ)
49	48	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
50		elfzle1 13443	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 1 ≤ 𝐽)
51	50	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 1 ≤ 𝐽)
52	49	zred 12596	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℝ)
53		elfzelz 13440	. . . . . . . . . . . . . . 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 12596	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ ℝ)
57	47	zred 12596	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ ℝ)
58		elfzle2 13444	. . . . . . . . . . . . 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 13444	. . . . . . . . . . . . . 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 11290	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ≤ (𝑀 + 𝑁))
64	42, 47, 49, 51, 63	elfzd 13431	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
65	2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemsv 34667	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) = if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽))
66	64, 65	syldan 591	. . . . . . . . 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 4485	. . . . . . . . . 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 2771	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) = (((𝐼‘𝐶) + 1) − 𝐽))
71	70	oveq1d 7373	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) = ((((𝐼‘𝐶) + 1) − 𝐽)...(𝐼‘𝐶)))
72	71	eleq2d 2822	. . . . . 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 726	. . . . . . . . 9 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (𝐼‘𝐶) ∈ ℤ)
75	74	zcnd 12597	. . . . . . . 8 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (𝐼‘𝐶) ∈ ℂ)
76		1cnd 11127	. . . . . . . 8 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → 1 ∈ ℂ)
77	75, 76	pncand 11493	. . . . . . 7 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → (((𝐼‘𝐶) + 1) − 1) = (𝐼‘𝐶))
78	77	oveq2d 7374	. . . . . 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 2822	. . . . 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 12522	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → 1 ∈ ℤ)
81	48	ad2antlr 727	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) → 𝐽 ∈ ℤ)
82	74	peano2zd 12599	. . . . . 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 13503	. . . . . 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 3211	. . . . 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 2743	. . . . . . 7 ⊢ ((((𝐼‘𝐶) + 1) − 𝑘) = 𝑗 ↔ 𝑗 = (((𝐼‘𝐶) + 1) − 𝑘))
90	54	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼‘𝐶) ∈ ℤ)
91	90	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼‘𝐶) ∈ ℤ)
92	91	zcnd 12597	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼‘𝐶) ∈ ℂ)
93		1cnd 11127	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 1 ∈ ℂ)
94	92, 93	addcld 11151	. . . . . . . 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 12597	. . . . . . . 8 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 𝑘 ∈ ℂ)
97		elfzelz 13440	. . . . . . . . . 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 12597	. . . . . . . 8 ⊢ ((((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ ℂ)
100		subsub23 11385	. . . . . . . 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 1373	. . . . . . 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 3933	. . . . . . . . . 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 34667	. . . . . . . . . 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 584	. . . . . . . . 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 12596	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ ℝ)
109	48	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝐽 ∈ ℤ)
110	109	zred 12596	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝐽 ∈ ℝ)
111	90	zred 12596	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼‘𝐶) ∈ ℝ)
112		elfzle2 13444	. . . . . . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝐽 ≤ (𝐼‘𝐶))
115	108, 110, 111, 113, 114	letrd 11290	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ≤ (𝐼‘𝐶))
116		iftrue 4485	. . . . . . . . . 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 2771	. . . . . . . 8 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → ((𝑆‘𝐶)‘𝑗) = (((𝐼‘𝐶) + 1) − 𝑗))
119	118	eqeq1d 2738	. . . . . . 7 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → (((𝑆‘𝐶)‘𝑗) = 𝑘 ↔ (((𝐼‘𝐶) + 1) − 𝑗) = 𝑘))
120	119	adantlr 715	. . . . . 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 3158	. . . 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 2735	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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ∃wrex 3060 {crab 3399 ∖ cdif 3898 ∩ cin 3900 ⊆ wss 3901 ifcif 4479 𝒫 cpw 4554 class class class wbr 5098 ↦ cmpt 5179 ^◡ccnv 5623 ran crn 5625 “ cima 5627 Fn wfn 6487 ⟶wf 6488 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 infcinf 9344 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 < clt 11166 ≤ cle 11167 − cmin 11364 / cdiv 11794 ℕcn 12145 ℤcz 12488 ℤ_≥cuz 12751 ...cfz 13423 ♯chash 14253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-oadd 8401 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-n0 12402 df-z 12489 df-uz 12752 df-rp 12906 df-fz 13424 df-hash 14254

This theorem is referenced by: ballotlemfrc 34684

W3C validator