ballotlemsel1i

Description: The range ( 1 ... ( IC ) ) is invariant under ( SC ) . (Contributed by Thierry Arnoux, 28-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	ballotlemsel1i	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to S (C) (J) \in (1 \dots I (C))$

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		1zzd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to 1 \in ℤ$
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	adantr	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to I (C) \in ℤ$
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	$⊢ C \in (O ∖ E) \to M + N \in ℤ$
19		elfzle2	$⊢ I (C) \in (1 \dots M + N) \to I (C) \leq M + N$
20	12 19	syl	$⊢ C \in (O ∖ E) \to I (C) \leq M + N$
21		eluz2	$⊢ M + N \in ℤ_{\geq I (C)} \leftrightarrow (I (C) \in ℤ \land M + N \in ℤ \land I (C) \leq M + N)$
22	13 18 20 21	syl3anbrc	$⊢ C \in (O ∖ E) \to M + N \in ℤ_{\geq I (C)}$
23		fzss2	$⊢ M + N \in ℤ_{\geq I (C)} \to (1 \dots I (C)) \subseteq (1 \dots M + N)$
24	22 23	syl	$⊢ C \in (O ∖ E) \to (1 \dots I (C)) \subseteq (1 \dots M + N)$
25	24	sselda	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to J \in (1 \dots M + N)$
26	1 2 3 4 5 6 7 8 9	ballotlemsdom	$⊢ (C \in (O ∖ E) \land J \in (1 \dots M + N)) \to S (C) (J) \in (1 \dots M + N)$
27	25 26	syldan	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to S (C) (J) \in (1 \dots M + N)$
28	27	elfzelzd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to S (C) (J) \in ℤ$
29		elfzelz	$⊢ J \in (1 \dots I (C)) \to J \in ℤ$
30	29	adantl	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to J \in ℤ$
31	30	zred	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to J \in ℝ$
32	14	zred	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to I (C) \in ℝ$
33		1red	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to 1 \in ℝ$
34	32 33	readdcld	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to I (C) + 1 \in ℝ$
35		elfzle2	$⊢ J \in (1 \dots I (C)) \to J \leq I (C)$
36	35	adantl	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to J \leq I (C)$
37	14	zcnd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to I (C) \in ℂ$
38		1cnd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to 1 \in ℂ$
39	37 38	pncand	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to I (C) + 1 - 1 = I (C)$
40	36 39	breqtrrd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to J \leq I (C) + 1 - 1$
41	31 34 33 40	lesubd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to 1 \leq I (C) + 1 - J$
42	1 2 3 4 5 6 7 8 9	ballotlemsv	$⊢ (C \in (O ∖ E) \land J \in (1 \dots M + N)) \to S (C) (J) = if (J \leq I (C), I (C) + 1 - J, J)$
43	25 42	syldan	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to S (C) (J) = if (J \leq I (C), I (C) + 1 - J, J)$
44	36	iftrued	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to if (J \leq I (C), I (C) + 1 - J, J) = I (C) + 1 - J$
45	43 44	eqtrd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to S (C) (J) = I (C) + 1 - J$
46	41 45	breqtrrd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to 1 \leq S (C) (J)$
47	13	adantr	$⊢ (C \in (O ∖ E) \land J \in (1 \dots M + N)) \to I (C) \in ℤ$
48		elfznn	$⊢ J \in (1 \dots M + N) \to J \in ℕ$
49	48	adantl	$⊢ (C \in (O ∖ E) \land J \in (1 \dots M + N)) \to J \in ℕ$
50	47 49	ltesubnnd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots M + N)) \to I (C) + 1 - J \leq I (C)$
51	25 50	syldan	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to I (C) + 1 - J \leq I (C)$
52	45 51	eqbrtrd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to S (C) (J) \leq I (C)$
53	10 14 28 46 52	elfzd	$⊢ (C \in (O ∖ E) \land J \in (1 \dots I (C))) \to S (C) (J) \in (1 \dots I (C))$

Metamath Proof Explorer

Theorem ballotlemsel1i

Proof