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Theorem ballotlemsima 31025
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 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
ballotth.mgtn 𝑁 < 𝑀
ballotth.i 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
Assertion
Ref Expression
ballotlemsima ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (1...𝐽)) = (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))
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.
StepHypRef Expression
1 imassrn 5659 . . . . . 6 ((𝑆𝐶) “ (1...𝐽)) ⊆ ran (𝑆𝐶)
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 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
7 ballotth.e . . . . . . . . 9 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
8 ballotth.mgtn . . . . . . . . 9 𝑁 < 𝑀
9 ballotth.i . . . . . . . . 9 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
10 ballotth.s . . . . . . . . 9 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
112, 3, 4, 5, 6, 7, 8, 9, 10ballotlemsf1o 31023 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ (𝑆𝐶) = (𝑆𝐶)))
1211simpld 488 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
13 f1of 6320 . . . . . . 7 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆𝐶):(1...(𝑀 + 𝑁))⟶(1...(𝑀 + 𝑁)))
14 frn 6229 . . . . . . 7 ((𝑆𝐶):(1...(𝑀 + 𝑁))⟶(1...(𝑀 + 𝑁)) → ran (𝑆𝐶) ⊆ (1...(𝑀 + 𝑁)))
1512, 13, 143syl 18 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → ran (𝑆𝐶) ⊆ (1...(𝑀 + 𝑁)))
161, 15syl5ss 3772 . . . . 5 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ (1...𝐽)) ⊆ (1...(𝑀 + 𝑁)))
17 fzssuz 12589 . . . . . 6 (1...(𝑀 + 𝑁)) ⊆ (ℤ‘1)
18 uzssz 11906 . . . . . 6 (ℤ‘1) ⊆ ℤ
1917, 18sstri 3770 . . . . 5 (1...(𝑀 + 𝑁)) ⊆ ℤ
2016, 19syl6ss 3773 . . . 4 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ (1...𝐽)) ⊆ ℤ)
2120adantr 472 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (1...𝐽)) ⊆ ℤ)
2221sselda 3761 . 2 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ((𝑆𝐶) “ (1...𝐽))) → 𝑘 ∈ ℤ)
23 elfzelz 12549 . . 3 (𝑘 ∈ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) → 𝑘 ∈ ℤ)
2423adantl 473 . 2 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) → 𝑘 ∈ ℤ)
25 f1ofn 6321 . . . . . . 7 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆𝐶) Fn (1...(𝑀 + 𝑁)))
2612, 25syl 17 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) Fn (1...(𝑀 + 𝑁)))
2726adantr 472 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑆𝐶) Fn (1...(𝑀 + 𝑁)))
282, 3, 4, 5, 6, 7, 8, 9ballotlemiex 31011 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
2928simpld 488 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
3029adantr 472 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
31 elfzuz3 12546 . . . . . . . 8 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
3230, 31syl 17 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
33 elfzuz3 12546 . . . . . . . 8 (𝐽 ∈ (1...(𝐼𝐶)) → (𝐼𝐶) ∈ (ℤ𝐽))
3433adantl 473 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (ℤ𝐽))
35 uztrn 11903 . . . . . . 7 (((𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)) ∧ (𝐼𝐶) ∈ (ℤ𝐽)) → (𝑀 + 𝑁) ∈ (ℤ𝐽))
3632, 34, 35syl2anc 579 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ (ℤ𝐽))
37 fzss2 12588 . . . . . 6 ((𝑀 + 𝑁) ∈ (ℤ𝐽) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
3836, 37syl 17 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
39 fvelimab 6442 . . . . 5 (((𝑆𝐶) Fn (1...(𝑀 + 𝑁)) ∧ (1...𝐽) ⊆ (1...(𝑀 + 𝑁))) → (𝑘 ∈ ((𝑆𝐶) “ (1...𝐽)) ↔ ∃𝑗 ∈ (1...𝐽)((𝑆𝐶)‘𝑗) = 𝑘))
4027, 38, 39syl2anc 579 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑘 ∈ ((𝑆𝐶) “ (1...𝐽)) ↔ ∃𝑗 ∈ (1...𝐽)((𝑆𝐶)‘𝑗) = 𝑘))
4140adantr 472 . . 3 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ((𝑆𝐶) “ (1...𝐽)) ↔ ∃𝑗 ∈ (1...𝐽)((𝑆𝐶)‘𝑗) = 𝑘))
42 1zzd 11655 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 1 ∈ ℤ)
432nnzi 11648 . . . . . . . . . . . . 13 𝑀 ∈ ℤ
443nnzi 11648 . . . . . . . . . . . . 13 𝑁 ∈ ℤ
45 zaddcl 11664 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
4643, 44, 45mp2an 683 . . . . . . . . . . . 12 (𝑀 + 𝑁) ∈ ℤ
4746a1i 11 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ ℤ)
48 elfzelz 12549 . . . . . . . . . . . 12 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
4948adantl 473 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
50 elfzle1 12551 . . . . . . . . . . . 12 (𝐽 ∈ (1...(𝐼𝐶)) → 1 ≤ 𝐽)
5150adantl 473 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 1 ≤ 𝐽)
5249zred 11729 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℝ)
53 elfzelz 12549 . . . . . . . . . . . . . . 15 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
5429, 53syl 17 . . . . . . . . . . . . . 14 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℤ)
5554adantr 472 . . . . . . . . . . . . 13 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ ℤ)
5655zred 11729 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ ℝ)
5747zred 11729 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ ℝ)
58 elfzle2 12552 . . . . . . . . . . . . 13 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ≤ (𝐼𝐶))
5958adantl 473 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ≤ (𝐼𝐶))
60 elfzle2 12552 . . . . . . . . . . . . . 14 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
6129, 60syl 17 . . . . . . . . . . . . 13 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
6261adantr 472 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
6352, 56, 57, 59, 62letrd 10448 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ≤ (𝑀 + 𝑁))
64 elfz4 12542 . . . . . . . . . . 11 (((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (1 ≤ 𝐽𝐽 ≤ (𝑀 + 𝑁))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
6542, 47, 49, 51, 63, 64syl32anc 1497 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
662, 3, 4, 5, 6, 7, 8, 9, 10ballotlemsv 31019 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) = if(𝐽 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝐽), 𝐽))
6765, 66syldan 585 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) = if(𝐽 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝐽), 𝐽))
68 simpr 477 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝐼𝐶)))
69 iftrue 4249 . . . . . . . . . 10 (𝐽 ≤ (𝐼𝐶) → if(𝐽 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝐽), 𝐽) = (((𝐼𝐶) + 1) − 𝐽))
7068, 58, 693syl 18 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → if(𝐽 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝐽), 𝐽) = (((𝐼𝐶) + 1) − 𝐽))
7167, 70eqtrd 2799 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) = (((𝐼𝐶) + 1) − 𝐽))
7271oveq1d 6857 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) = ((((𝐼𝐶) + 1) − 𝐽)...(𝐼𝐶)))
7372eleq2d 2830 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑘 ∈ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ↔ 𝑘 ∈ ((((𝐼𝐶) + 1) − 𝐽)...(𝐼𝐶))))
7473adantr 472 . . . . 5 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ↔ 𝑘 ∈ ((((𝐼𝐶) + 1) − 𝐽)...(𝐼𝐶))))
7554ad2antrr 717 . . . . . . . . 9 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → (𝐼𝐶) ∈ ℤ)
7675zcnd 11730 . . . . . . . 8 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → (𝐼𝐶) ∈ ℂ)
77 1cnd 10288 . . . . . . . 8 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → 1 ∈ ℂ)
7876, 77pncand 10647 . . . . . . 7 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → (((𝐼𝐶) + 1) − 1) = (𝐼𝐶))
7978oveq2d 6858 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → ((((𝐼𝐶) + 1) − 𝐽)...(((𝐼𝐶) + 1) − 1)) = ((((𝐼𝐶) + 1) − 𝐽)...(𝐼𝐶)))
8079eleq2d 2830 . . . . 5 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ((((𝐼𝐶) + 1) − 𝐽)...(((𝐼𝐶) + 1) − 1)) ↔ 𝑘 ∈ ((((𝐼𝐶) + 1) − 𝐽)...(𝐼𝐶))))
81 1zzd 11655 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → 1 ∈ ℤ)
8248ad2antlr 718 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → 𝐽 ∈ ℤ)
8375peano2zd 11732 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → ((𝐼𝐶) + 1) ∈ ℤ)
84 simpr 477 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ)
85 fzrev 12610 . . . . . 6 (((1 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (((𝐼𝐶) + 1) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 ∈ ((((𝐼𝐶) + 1) − 𝐽)...(((𝐼𝐶) + 1) − 1)) ↔ (((𝐼𝐶) + 1) − 𝑘) ∈ (1...𝐽)))
8681, 82, 83, 84, 85syl22anc 867 . . . . 5 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ((((𝐼𝐶) + 1) − 𝐽)...(((𝐼𝐶) + 1) − 1)) ↔ (((𝐼𝐶) + 1) − 𝑘) ∈ (1...𝐽)))
8774, 80, 863bitr2d 298 . . . 4 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ↔ (((𝐼𝐶) + 1) − 𝑘) ∈ (1...𝐽)))
88 risset 3209 . . . . 5 ((((𝐼𝐶) + 1) − 𝑘) ∈ (1...𝐽) ↔ ∃𝑗 ∈ (1...𝐽)𝑗 = (((𝐼𝐶) + 1) − 𝑘))
8988a1i 11 . . . 4 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → ((((𝐼𝐶) + 1) − 𝑘) ∈ (1...𝐽) ↔ ∃𝑗 ∈ (1...𝐽)𝑗 = (((𝐼𝐶) + 1) − 𝑘)))
90 eqcom 2772 . . . . . . 7 ((((𝐼𝐶) + 1) − 𝑘) = 𝑗𝑗 = (((𝐼𝐶) + 1) − 𝑘))
9154ad2antrr 717 . . . . . . . . . . 11 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼𝐶) ∈ ℤ)
9291adantlr 706 . . . . . . . . . 10 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼𝐶) ∈ ℤ)
9392zcnd 11730 . . . . . . . . 9 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼𝐶) ∈ ℂ)
94 1cnd 10288 . . . . . . . . 9 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 1 ∈ ℂ)
9593, 94addcld 10313 . . . . . . . 8 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → ((𝐼𝐶) + 1) ∈ ℂ)
96 simplr 785 . . . . . . . . 9 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 𝑘 ∈ ℤ)
9796zcnd 11730 . . . . . . . 8 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 𝑘 ∈ ℂ)
98 elfzelz 12549 . . . . . . . . . 10 (𝑗 ∈ (1...𝐽) → 𝑗 ∈ ℤ)
9998adantl 473 . . . . . . . . 9 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ ℤ)
10099zcnd 11730 . . . . . . . 8 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ ℂ)
101 subsub23 10540 . . . . . . . 8 ((((𝐼𝐶) + 1) ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((((𝐼𝐶) + 1) − 𝑘) = 𝑗 ↔ (((𝐼𝐶) + 1) − 𝑗) = 𝑘))
10295, 97, 100, 101syl3anc 1490 . . . . . . 7 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → ((((𝐼𝐶) + 1) − 𝑘) = 𝑗 ↔ (((𝐼𝐶) + 1) − 𝑗) = 𝑘))
10390, 102syl5bbr 276 . . . . . 6 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (𝑗 = (((𝐼𝐶) + 1) − 𝑘) ↔ (((𝐼𝐶) + 1) − 𝑗) = 𝑘))
104 simpll 783 . . . . . . . . . 10 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝐶 ∈ (𝑂𝐸))
10538sselda 3761 . . . . . . . . . 10 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ (1...(𝑀 + 𝑁)))
1062, 3, 4, 5, 6, 7, 8, 9, 10ballotlemsv 31019 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑗) = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
107104, 105, 106syl2anc 579 . . . . . . . . 9 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → ((𝑆𝐶)‘𝑗) = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
10898adantl 473 . . . . . . . . . . . 12 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ ℤ)
109108zred 11729 . . . . . . . . . . 11 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ∈ ℝ)
11048ad2antlr 718 . . . . . . . . . . . 12 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝐽 ∈ ℤ)
111110zred 11729 . . . . . . . . . . 11 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝐽 ∈ ℝ)
11291zred 11729 . . . . . . . . . . 11 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → (𝐼𝐶) ∈ ℝ)
113 elfzle2 12552 . . . . . . . . . . . 12 (𝑗 ∈ (1...𝐽) → 𝑗𝐽)
114113adantl 473 . . . . . . . . . . 11 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗𝐽)
11558ad2antlr 718 . . . . . . . . . . 11 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝐽 ≤ (𝐼𝐶))
116109, 111, 112, 114, 115letrd 10448 . . . . . . . . . 10 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → 𝑗 ≤ (𝐼𝐶))
117 iftrue 4249 . . . . . . . . . 10 (𝑗 ≤ (𝐼𝐶) → if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗) = (((𝐼𝐶) + 1) − 𝑗))
118116, 117syl 17 . . . . . . . . 9 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗) = (((𝐼𝐶) + 1) − 𝑗))
119107, 118eqtrd 2799 . . . . . . . 8 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → ((𝑆𝐶)‘𝑗) = (((𝐼𝐶) + 1) − 𝑗))
120119eqeq1d 2767 . . . . . . 7 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑗 ∈ (1...𝐽)) → (((𝑆𝐶)‘𝑗) = 𝑘 ↔ (((𝐼𝐶) + 1) − 𝑗) = 𝑘))
121120adantlr 706 . . . . . 6 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (((𝑆𝐶)‘𝑗) = 𝑘 ↔ (((𝐼𝐶) + 1) − 𝑗) = 𝑘))
122103, 121bitr4d 273 . . . . 5 ((((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) ∧ 𝑗 ∈ (1...𝐽)) → (𝑗 = (((𝐼𝐶) + 1) − 𝑘) ↔ ((𝑆𝐶)‘𝑗) = 𝑘))
123122rexbidva 3196 . . . 4 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → (∃𝑗 ∈ (1...𝐽)𝑗 = (((𝐼𝐶) + 1) − 𝑘) ↔ ∃𝑗 ∈ (1...𝐽)((𝑆𝐶)‘𝑗) = 𝑘))
12487, 89, 1233bitrd 296 . . 3 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ↔ ∃𝑗 ∈ (1...𝐽)((𝑆𝐶)‘𝑗) = 𝑘))
12541, 124bitr4d 273 . 2 (((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ((𝑆𝐶) “ (1...𝐽)) ↔ 𝑘 ∈ (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
12622, 24, 125eqrdav 2764 1 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (1...𝐽)) = (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  {crab 3059  cdif 3729  cin 3731  wss 3732  ifcif 4243  𝒫 cpw 4315   class class class wbr 4809  cmpt 4888  ccnv 5276  ran crn 5278  cima 5280   Fn wfn 6063  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  infcinf 8554  cc 10187  cr 10188  0cc0 10189  1c1 10190   + caddc 10192   < clt 10328  cle 10329  cmin 10520   / cdiv 10938  cn 11274  cz 11624  cuz 11886  ...cfz 12533  chash 13321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-hash 13322
This theorem is referenced by:  ballotlemfrc  31036
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