ballotlemsdom

Description: Domain of S for a given counting C . (Contributed by Thierry Arnoux, 12-Apr-2017)

	Ref	Expression
Hypotheses	ballotth.m	$⊢ M \in ℕ$
	ballotth.n	$⊢ N \in ℕ$
	ballotth.o	$⊢ O = \{c \in 𝒫 (1 \dots M + N) \| \|c\| = M\}$
	ballotth.p	$⊢ P = (x \in 𝒫 O ⟼ \frac{\|x\|}{\|O\|})$
	ballotth.f	$⊢ F = (c \in O ⟼ (i \in ℤ ⟼ \|(1 \dots i) \cap c\| - \|(1 \dots i) ∖ c\|))$
	ballotth.e	$⊢ E = \{c \in O \| \forall i \in (1 \dots M + N) 0 < F (c) (i)\}$
	ballotth.mgtn	$⊢ N < M$
	ballotth.i	$⊢ I = (c \in (O ∖ E) ⟼ sup (\{k \in (1 \dots M + N) \| F (c) (k) = 0\} ℝ <))$
	ballotth.s	$⊢ S = (c \in (O ∖ E) ⟼ (i \in (1 \dots M + N) ⟼ if (i \leq I (c), I (c) + 1 - i, i)))$
Assertion	ballotlemsdom	$⊢ (C \in (O ∖ E) \land J \in (1 \dots M + N)) \to S (C) (J) \in (1 \dots M + N)$

Proof

Step	Hyp	Ref	Expression
1		ballotth.m	$⊢ M \in ℕ$
2		ballotth.n	$⊢ N \in ℕ$
3		ballotth.o	$⊢ O = \{c \in 𝒫 (1 \dots M + N) \| \|c\| = M\}$
4		ballotth.p	$⊢ P = (x \in 𝒫 O ⟼ \frac{\|x\|}{\|O\|})$
5		ballotth.f	$⊢ F = (c \in O ⟼ (i \in ℤ ⟼ \|(1 \dots i) \cap c\| - \|(1 \dots i) ∖ c\|))$
6		ballotth.e	$⊢ E = \{c \in O \| \forall i \in (1 \dots M + N) 0 < F (c) (i)\}$
7		ballotth.mgtn	$⊢ N < M$
8		ballotth.i	$⊢ I = (c \in (O ∖ E) ⟼ sup (\{k \in (1 \dots M + N) \| F (c) (k) = 0\} ℝ <))$
9		ballotth.s	$⊢ S = (c \in (O ∖ E) ⟼ (i \in (1 \dots M + N) ⟼ if (i \leq I (c), I (c) + 1 - i, i)))$
10	1 2 3 4 5 6 7 8 9	ballotlemsv	$⊢ (C \in (O ∖ E) \land J \in (1 \dots M + N)) \to S (C) (J) = if (J \leq I (C), I (C) + 1 - J, J)$
11	1 2 3 4 5 6 7 8	ballotlemiex	$⊢ C \in (O ∖ E) \to (I (C) \in (1 \dots M + N) \land F (C) (I (C)) = 0)$
12	11	simpld	$⊢ C \in (O ∖ E) \to I (C) \in (1 \dots M + N)$
13	12	elfzelzd	$⊢ C \in (O ∖ E) \to I (C) \in ℤ$
14	13	ad2antrr	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to I (C) \in ℤ$
15		nnaddcl	$⊢ (M \in ℕ \land N \in ℕ) \to M + N \in ℕ$
16	1 2 15	mp2an	$⊢ M + N \in ℕ$
17	16	nnzi	$⊢ M + N \in ℤ$
18	17	a1i	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to M + N \in ℤ$
19	12	ad2antrr	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to I (C) \in (1 \dots M + N)$
20		elfzle2	$⊢ I (C) \in (1 \dots M + N) \to I (C) \leq M + N$
21	19 20	syl	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to I (C) \leq M + N$
22		eluz2	$⊢ M + N \in ℤ_{\geq I (C)} \leftrightarrow (I (C) \in ℤ \land M + N \in ℤ \land I (C) \leq M + N)$
23		fzss2	$⊢ M + N \in ℤ_{\geq I (C)} \to (1 \dots I (C)) \subseteq (1 \dots M + N)$
24	22 23	sylbir	$⊢ (I (C) \in ℤ \land M + N \in ℤ \land I (C) \leq M + N) \to (1 \dots I (C)) \subseteq (1 \dots M + N)$
25	14 18 21 24	syl3anc	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to (1 \dots I (C)) \subseteq (1 \dots M + N)$
26		1zzd	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to 1 \in ℤ$
27		simplr	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to J \in (1 \dots M + N)$
28	27	elfzelzd	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to J \in ℤ$
29		elfzle1	$⊢ J \in (1 \dots M + N) \to 1 \leq J$
30	27 29	syl	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to 1 \leq J$
31		simpr	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to J \leq I (C)$
32	26 14 28 30 31	elfzd	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to J \in (1 \dots I (C))$
33		fzrev3i	$⊢ J \in (1 \dots I (C)) \to 1 + I (C) - J \in (1 \dots I (C))$
34	32 33	syl	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to 1 + I (C) - J \in (1 \dots I (C))$
35		1cnd	$⊢ C \in (O ∖ E) \to 1 \in ℂ$
36	13	zcnd	$⊢ C \in (O ∖ E) \to I (C) \in ℂ$
37	35 36	addcomd	$⊢ C \in (O ∖ E) \to 1 + I (C) = I (C) + 1$
38	37	oveq1d	$⊢ C \in (O ∖ E) \to 1 + I (C) - J = I (C) + 1 - J$
39	38	eleq1d	$⊢ C \in (O ∖ E) \to (1 + I (C) - J \in (1 \dots I (C)) \leftrightarrow I (C) + 1 - J \in (1 \dots I (C)))$
40	39	ad2antrr	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to (1 + I (C) - J \in (1 \dots I (C)) \leftrightarrow I (C) + 1 - J \in (1 \dots I (C)))$
41	34 40	mpbid	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to I (C) + 1 - J \in (1 \dots I (C))$
42	25 41	sseldd	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land J \leq I (C)) \to I (C) + 1 - J \in (1 \dots M + N)$
43		simplr	$⊢ ((C \in (O ∖ E) \land J \in (1 \dots M + N)) \land \neg J \leq I (C)) \to J \in (1 \dots M + N)$
44	42 43	ifclda	$⊢ (C \in (O ∖ E) \land J \in (1 \dots M + N)) \to if (J \leq I (C), I (C) + 1 - J, J) \in (1 \dots M + N)$
45	10 44	eqeltrd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots M + N)) \to S (C) (J) \in (1 \dots M + N)$

Metamath Proof Explorer

Theorem ballotlemsdom

Proof