ballotlemfg

Description: Express the value of ( FC ) in terms of .^ . (Contributed by Thierry Arnoux, 21-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	$⊢ S = (c \in (O ∖ E) ⟼ (i \in (1 \dots M + N) ⟼ if (i \leq I (c), I (c) + 1 - i, i)))$
	ballotth.r	$⊢ R = (c \in (O ∖ E) ⟼ S (c) [c])$
	ballotlemg	$⊢ \underset{˙}{\times} = (u \in Fin, v \in Fin ⟼ \|v \cap u\| - \|v ∖ u\|)$
Assertion	ballotlemfg	$⊢ (C \in (O ∖ E) \land J \in (0 \dots M + N)) \to F (C) (J) = C \underset{˙}{\times} (1 \dots J)$

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		ballotth.r	$⊢ R = (c \in (O ∖ E) ⟼ S (c) [c])$
11		ballotlemg	$⊢ \underset{˙}{\times} = (u \in Fin, v \in Fin ⟼ \|v \cap u\| - \|v ∖ u\|)$
12		eldifi	$⊢ C \in (O ∖ E) \to C \in O$
13	12	adantr	$⊢ (C \in (O ∖ E) \land J \in (0 \dots M + N)) \to C \in O$
14		elfzelz	$⊢ J \in (0 \dots M + N) \to J \in ℤ$
15	14	adantl	$⊢ (C \in (O ∖ E) \land J \in (0 \dots M + N)) \to J \in ℤ$
16	1 2 3 4 5 13 15	ballotlemfval	$⊢ (C \in (O ∖ E) \land J \in (0 \dots M + N)) \to F (C) (J) = \|(1 \dots J) \cap C\| - \|(1 \dots J) ∖ C\|$
17		fzfi	$⊢ (1 \dots M + N) \in Fin$
18	1 2 3	ballotlemelo	$⊢ C \in O \leftrightarrow (C \subseteq (1 \dots M + N) \land \|C\| = M)$
19	18	simplbi	$⊢ C \in O \to C \subseteq (1 \dots M + N)$
20		ssfi	$⊢ ((1 \dots M + N) \in Fin \land C \subseteq (1 \dots M + N)) \to C \in Fin$
21	17 19 20	sylancr	$⊢ C \in O \to C \in Fin$
22	13 21	syl	$⊢ (C \in (O ∖ E) \land J \in (0 \dots M + N)) \to C \in Fin$
23		fzfid	$⊢ (C \in (O ∖ E) \land J \in (0 \dots M + N)) \to (1 \dots J) \in Fin$
24	1 2 3 4 5 6 7 8 9 10 11	ballotlemgval	$⊢ (C \in Fin \land (1 \dots J) \in Fin) \to C \underset{˙}{\times} (1 \dots J) = \|(1 \dots J) \cap C\| - \|(1 \dots J) ∖ C\|$
25	22 23 24	syl2anc	$⊢ (C \in (O ∖ E) \land J \in (0 \dots M + N)) \to C \underset{˙}{\times} (1 \dots J) = \|(1 \dots J) \cap C\| - \|(1 \dots J) ∖ C\|$
26	16 25	eqtr4d	$⊢ (C \in (O ∖ E) \land J \in (0 \dots M + N)) \to F (C) (J) = C \underset{˙}{\times} (1 \dots J)$

Metamath Proof Explorer

Theorem ballotlemfg

Proof