metakunt27

Description: Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024)

	Ref	Expression
Hypotheses	metakunt27.1	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
	metakunt27.2	⊢ ( 𝜑 → 𝐼 ∈ ℕ )
	metakunt27.3	⊢ ( 𝜑 → 𝐼 ≤ 𝑀 )
	metakunt27.4	⊢ ( 𝜑 → 𝑋 ∈ ( 1 ... 𝑀 ) )
	metakunt27.5	⊢ 𝐴 = ( 𝑥 ∈ ( 1 ... 𝑀 ) ↦ if ( 𝑥 = 𝐼 , 𝑀 , if ( 𝑥 < 𝐼 , 𝑥 , ( 𝑥 − 1 ) ) ) )
	metakunt27.6	⊢ 𝐵 = ( 𝑧 ∈ ( 1 ... 𝑀 ) ↦ if ( 𝑧 = 𝑀 , 𝑀 , if ( 𝑧 < 𝐼 , ( 𝑧 + ( 𝑀 − 𝐼 ) ) , ( 𝑧 + ( 1 − 𝐼 ) ) ) ) )
	metakunt27.7	⊢ ( 𝜑 → ¬ 𝑋 = 𝐼 )
	metakunt27.8	⊢ ( 𝜑 → 𝑋 < 𝐼 )
Assertion	metakunt27	⊢ ( 𝜑 → ( 𝐵 ‘ ( 𝐴 ‘ 𝑋 ) ) = ( 𝑋 + ( 𝑀 − 𝐼 ) ) )

Proof

Step	Hyp	Ref	Expression
1		metakunt27.1	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
2		metakunt27.2	⊢ ( 𝜑 → 𝐼 ∈ ℕ )
3		metakunt27.3	⊢ ( 𝜑 → 𝐼 ≤ 𝑀 )
4		metakunt27.4	⊢ ( 𝜑 → 𝑋 ∈ ( 1 ... 𝑀 ) )
5		metakunt27.5	⊢ 𝐴 = ( 𝑥 ∈ ( 1 ... 𝑀 ) ↦ if ( 𝑥 = 𝐼 , 𝑀 , if ( 𝑥 < 𝐼 , 𝑥 , ( 𝑥 − 1 ) ) ) )
6		metakunt27.6	⊢ 𝐵 = ( 𝑧 ∈ ( 1 ... 𝑀 ) ↦ if ( 𝑧 = 𝑀 , 𝑀 , if ( 𝑧 < 𝐼 , ( 𝑧 + ( 𝑀 − 𝐼 ) ) , ( 𝑧 + ( 1 − 𝐼 ) ) ) ) )
7		metakunt27.7	⊢ ( 𝜑 → ¬ 𝑋 = 𝐼 )
8		metakunt27.8	⊢ ( 𝜑 → 𝑋 < 𝐼 )
9	5	a1i	⊢ ( 𝜑 → 𝐴 = ( 𝑥 ∈ ( 1 ... 𝑀 ) ↦ if ( 𝑥 = 𝐼 , 𝑀 , if ( 𝑥 < 𝐼 , 𝑥 , ( 𝑥 − 1 ) ) ) ) )
10	7	adantr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ¬ 𝑋 = 𝐼 )
11		simpr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 𝑥 = 𝑋 )
12	11	eqeq1d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝑥 = 𝐼 ↔ 𝑋 = 𝐼 ) )
13	12	notbid	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( ¬ 𝑥 = 𝐼 ↔ ¬ 𝑋 = 𝐼 ) )
14	10 13	mpbird	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ¬ 𝑥 = 𝐼 )
15	14	iffalsed	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → if ( 𝑥 = 𝐼 , 𝑀 , if ( 𝑥 < 𝐼 , 𝑥 , ( 𝑥 − 1 ) ) ) = if ( 𝑥 < 𝐼 , 𝑥 , ( 𝑥 − 1 ) ) )
16	8	adantr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 𝑋 < 𝐼 )
17	11	breq1d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝑥 < 𝐼 ↔ 𝑋 < 𝐼 ) )
18	16 17	mpbird	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 𝑥 < 𝐼 )
19	18	iftrued	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → if ( 𝑥 < 𝐼 , 𝑥 , ( 𝑥 − 1 ) ) = 𝑥 )
20	19 11	eqtrd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → if ( 𝑥 < 𝐼 , 𝑥 , ( 𝑥 − 1 ) ) = 𝑋 )
21	15 20	eqtrd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → if ( 𝑥 = 𝐼 , 𝑀 , if ( 𝑥 < 𝐼 , 𝑥 , ( 𝑥 − 1 ) ) ) = 𝑋 )
22	9 21 4 4	fvmptd	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑋 ) = 𝑋 )
23	22	fveq2d	⊢ ( 𝜑 → ( 𝐵 ‘ ( 𝐴 ‘ 𝑋 ) ) = ( 𝐵 ‘ 𝑋 ) )
24	6	a1i	⊢ ( 𝜑 → 𝐵 = ( 𝑧 ∈ ( 1 ... 𝑀 ) ↦ if ( 𝑧 = 𝑀 , 𝑀 , if ( 𝑧 < 𝐼 , ( 𝑧 + ( 𝑀 − 𝐼 ) ) , ( 𝑧 + ( 1 − 𝐼 ) ) ) ) ) )
25		elfznn	⊢ ( 𝑋 ∈ ( 1 ... 𝑀 ) → 𝑋 ∈ ℕ )
26	4 25	syl	⊢ ( 𝜑 → 𝑋 ∈ ℕ )
27	26	nnred	⊢ ( 𝜑 → 𝑋 ∈ ℝ )
28	2	nnred	⊢ ( 𝜑 → 𝐼 ∈ ℝ )
29	1	nnred	⊢ ( 𝜑 → 𝑀 ∈ ℝ )
30	27 28 29 8 3	ltletrd	⊢ ( 𝜑 → 𝑋 < 𝑀 )
31	27 30	ltned	⊢ ( 𝜑 → 𝑋 ≠ 𝑀 )
32	31	neneqd	⊢ ( 𝜑 → ¬ 𝑋 = 𝑀 )
33	32	adantr	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → ¬ 𝑋 = 𝑀 )
34		simpr	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → 𝑧 = 𝑋 )
35	34	eqeq1d	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → ( 𝑧 = 𝑀 ↔ 𝑋 = 𝑀 ) )
36	35	notbid	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → ( ¬ 𝑧 = 𝑀 ↔ ¬ 𝑋 = 𝑀 ) )
37	33 36	mpbird	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → ¬ 𝑧 = 𝑀 )
38	37	iffalsed	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → if ( 𝑧 = 𝑀 , 𝑀 , if ( 𝑧 < 𝐼 , ( 𝑧 + ( 𝑀 − 𝐼 ) ) , ( 𝑧 + ( 1 − 𝐼 ) ) ) ) = if ( 𝑧 < 𝐼 , ( 𝑧 + ( 𝑀 − 𝐼 ) ) , ( 𝑧 + ( 1 − 𝐼 ) ) ) )
39	8	adantr	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → 𝑋 < 𝐼 )
40	34	breq1d	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → ( 𝑧 < 𝐼 ↔ 𝑋 < 𝐼 ) )
41	39 40	mpbird	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → 𝑧 < 𝐼 )
42	41	iftrued	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → if ( 𝑧 < 𝐼 , ( 𝑧 + ( 𝑀 − 𝐼 ) ) , ( 𝑧 + ( 1 − 𝐼 ) ) ) = ( 𝑧 + ( 𝑀 − 𝐼 ) ) )
43	34	oveq1d	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → ( 𝑧 + ( 𝑀 − 𝐼 ) ) = ( 𝑋 + ( 𝑀 − 𝐼 ) ) )
44	42 43	eqtrd	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → if ( 𝑧 < 𝐼 , ( 𝑧 + ( 𝑀 − 𝐼 ) ) , ( 𝑧 + ( 1 − 𝐼 ) ) ) = ( 𝑋 + ( 𝑀 − 𝐼 ) ) )
45	38 44	eqtrd	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑋 ) → if ( 𝑧 = 𝑀 , 𝑀 , if ( 𝑧 < 𝐼 , ( 𝑧 + ( 𝑀 − 𝐼 ) ) , ( 𝑧 + ( 1 − 𝐼 ) ) ) ) = ( 𝑋 + ( 𝑀 − 𝐼 ) ) )
46	4	elfzelzd	⊢ ( 𝜑 → 𝑋 ∈ ℤ )
47	1	nnzd	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
48	2	nnzd	⊢ ( 𝜑 → 𝐼 ∈ ℤ )
49	47 48	zsubcld	⊢ ( 𝜑 → ( 𝑀 − 𝐼 ) ∈ ℤ )
50	46 49	zaddcld	⊢ ( 𝜑 → ( 𝑋 + ( 𝑀 − 𝐼 ) ) ∈ ℤ )
51	24 45 4 50	fvmptd	⊢ ( 𝜑 → ( 𝐵 ‘ 𝑋 ) = ( 𝑋 + ( 𝑀 − 𝐼 ) ) )
52	23 51	eqtrd	⊢ ( 𝜑 → ( 𝐵 ‘ ( 𝐴 ‘ 𝑋 ) ) = ( 𝑋 + ( 𝑀 − 𝐼 ) ) )

Metamath Proof Explorer

Theorem metakunt27

Proof