metakunt24 - Mathbox for metakunt

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Theorem metakunt24 40076

Description: Technical condition such that metakunt17 40069 holds. (Contributed by metakunt, 28-May-2024.)

**Hypotheses**
Ref	Expression
metakunt24.1	⊢ (𝜑 → 𝑀 ∈ ℕ)
metakunt24.2	⊢ (𝜑 → 𝐼 ∈ ℕ)
metakunt24.3	⊢ (𝜑 → 𝐼 ≤ 𝑀)

**Assertion**
Ref	Expression
metakunt24	⊢ (𝜑 → ((((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅ ∧ (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}) ∧ (1...𝑀) = (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀})))

**Proof of Theorem metakunt24**
Step	Hyp	Ref	Expression
1		indir 4206	. . . 4 ⊢ (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀}))
2	1	a1i 11	. . 3 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})))
3		metakunt24.1	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4		metakunt24.2	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℕ)
5		metakunt24.3	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ≤ 𝑀)
6	3, 4, 5	metakunt18 40070	. . . . . . 7 ⊢ (𝜑 → ((((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅) ∧ (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∩ (1...(𝑀 − 𝐼))) = ∅ ∧ ((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∩ {𝑀}) = ∅ ∧ ((1...(𝑀 − 𝐼)) ∩ {𝑀}) = ∅)))
7	6	simpld 494	. . . . . 6 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅))
8	7	simp2d 1141	. . . . 5 ⊢ (𝜑 → ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅)
9	7	simp3d 1142	. . . . 5 ⊢ (𝜑 → ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅)
10	8, 9	uneq12d 4094	. . . 4 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = (∅ ∪ ∅))
11		unidm 4082	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
12	11	a1i 11	. . . 4 ⊢ (𝜑 → (∅ ∪ ∅) = ∅)
13	10, 12	eqtrd 2778	. . 3 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = ∅)
14	2, 13	eqtrd 2778	. 2 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅)
15		1zzd 12281	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
16	3	nnzd 12354	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
17	3	nnge1d 11951	. . . . 5 ⊢ (𝜑 → 1 ≤ 𝑀)
18	3	nnred 11918	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
19	18	leidd 11471	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
20	15, 16, 16, 17, 19	elfzd 13176	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
21	20	fzsplitnd 39919	. . 3 ⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
22		oveq1 7262	. . . . . . . . . 10 ⊢ (𝐼 = 𝑀 → (𝐼 − 1) = (𝑀 − 1))
23	22	oveq2d 7271	. . . . . . . . 9 ⊢ (𝐼 = 𝑀 → (1...(𝐼 − 1)) = (1...(𝑀 − 1)))
24		oveq1 7262	. . . . . . . . 9 ⊢ (𝐼 = 𝑀 → (𝐼...(𝑀 − 1)) = (𝑀...(𝑀 − 1)))
25	23, 24	uneq12d 4094	. . . . . . . 8 ⊢ (𝐼 = 𝑀 → ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))))
26	25	uneq1d 4092	. . . . . . 7 ⊢ (𝐼 = 𝑀 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
27	26	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
28	18	ltm1d 11837	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
29	16, 15	zsubcld 12360	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
30		fzn 13201	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
31	16, 29, 30	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
32	28, 31	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
33	32	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (𝑀...(𝑀 − 1)) = ∅)
34	33	uneq2d 4093	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ ∅))
35		un0 4321	. . . . . . . . 9 ⊢ ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1))
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1)))
37	34, 36	eqtrd 2778	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = (1...(𝑀 − 1)))
38	37	uneq1d 4092	. . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
39	27, 38	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
40	39	eqcomd 2744	. . . 4 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
41	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℤ)
42	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℤ)
43	42, 41	zsubcld 12360	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 1) ∈ ℤ)
44	4	nnzd 12354	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℤ)
45	44	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℤ)
46	4	nnge1d 11951	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 𝐼)
47	46	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ 𝐼)
48		eqid 2738	. . . . . . . . . . 11 ⊢ 𝑀 = 𝑀
49		eqeq1 2742	. . . . . . . . . . 11 ⊢ (𝑀 = 𝐼 → (𝑀 = 𝑀 ↔ 𝐼 = 𝑀))
50	48, 49	mpbii 232	. . . . . . . . . 10 ⊢ (𝑀 = 𝐼 → 𝐼 = 𝑀)
51	50	necon3bi 2969	. . . . . . . . 9 ⊢ (¬ 𝐼 = 𝑀 → 𝑀 ≠ 𝐼)
52	51	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ≠ 𝐼)
53	4	nnred 11918	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℝ)
54	53, 18, 5	leltned 11058	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 < 𝑀 ↔ 𝑀 ≠ 𝐼))
55	54	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀 ↔ 𝑀 ≠ 𝐼))
56	52, 55	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 < 𝑀)
57		zltlem1 12303	. . . . . . . . 9 ⊢ ((𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐼 < 𝑀 ↔ 𝐼 ≤ (𝑀 − 1)))
58	44, 16, 57	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐼 < 𝑀 ↔ 𝐼 ≤ (𝑀 − 1)))
59	58	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀 ↔ 𝐼 ≤ (𝑀 − 1)))
60	56, 59	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ≤ (𝑀 − 1))
61	41, 43, 45, 47, 60	fzsplitnr 39920	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
62	61	uneq1d 4092	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
63	40, 62	pm2.61dan 809	. . 3 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
64		fzsn 13227	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
65	16, 64	syl 17	. . . 4 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
66	65	uneq2d 4093	. . 3 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
67	21, 63, 66	3eqtrd 2782	. 2 ⊢ (𝜑 → (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
68		uncom 4083	. . . . . 6 ⊢ ((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1)))
69	68	a1i 11	. . . . 5 ⊢ (𝜑 → ((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
70	69	uneq1d 4092	. . . 4 ⊢ (𝜑 → (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀}) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
71	65	uneq2d 4093	. . . . . 6 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
72	71	eqcomd 2744	. . . . 5 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
73		fz10 13206	. . . . . . . . . . . . . . 15 ⊢ (1...0) = ∅
74	73	uneq1i 4089	. . . . . . . . . . . . . 14 ⊢ ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1)))
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1))))
76	75	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1))))
77		uncom 4083	. . . . . . . . . . . . . . 15 ⊢ ((1...(𝑀 − 1)) ∪ ∅) = (∅ ∪ (1...(𝑀 − 1)))
78	77	eqeq1i 2743	. . . . . . . . . . . . . 14 ⊢ (((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1)) ↔ (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
79	78	imbi2i 335	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1))) ↔ ((𝜑 ∧ 𝐼 = 𝑀) → (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1))))
80	36, 79	mpbi 229	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
81	76, 80	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
82	81	eqcomd 2744	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...0) ∪ (1...(𝑀 − 1))))
83		oveq2 7263	. . . . . . . . . . . . . . 15 ⊢ (𝐼 = 𝑀 → (𝑀 − 𝐼) = (𝑀 − 𝑀))
84	83	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (𝑀 − 𝐼) = (𝑀 − 𝑀))
85	18	recnd 10934	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℂ)
86	85	subidd 11250	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 − 𝑀) = 0)
87	86	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (𝑀 − 𝑀) = 0)
88	84, 87	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (𝑀 − 𝐼) = 0)
89	88	oveq2d 7271	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (1...(𝑀 − 𝐼)) = (1...0))
90	83	oveq1d 7270	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 = 𝑀 → ((𝑀 − 𝐼) + 1) = ((𝑀 − 𝑀) + 1))
91	90	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((𝑀 − 𝐼) + 1) = ((𝑀 − 𝑀) + 1))
92	87	oveq1d 7270	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((𝑀 − 𝑀) + 1) = (0 + 1))
93	91, 92	eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((𝑀 − 𝐼) + 1) = (0 + 1))
94		1e0p1 12408	. . . . . . . . . . . . . 14 ⊢ 1 = (0 + 1)
95	93, 94	eqtr4di 2797	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((𝑀 − 𝐼) + 1) = 1)
96	95	oveq1d 7270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (((𝑀 − 𝐼) + 1)...(𝑀 − 1)) = (1...(𝑀 − 1)))
97	89, 96	uneq12d 4094	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) = ((1...0) ∪ (1...(𝑀 − 1))))
98	97	eqcomd 2744	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
99	82, 98	eqtrd 2778	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
100	42, 45	zsubcld 12360	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 𝐼) ∈ ℤ)
101	53	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℝ)
102	18	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℝ)
103		1red 10907	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℝ)
104	101, 102, 103, 60	lesubd 11509	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ (𝑀 − 𝐼))
105	103, 101, 102, 47	lesub2dd 11522	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 𝐼) ≤ (𝑀 − 1))
106	41, 43, 100, 104, 105	elfzd 13176	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 𝐼) ∈ (1...(𝑀 − 1)))
107		fzsplit 13211	. . . . . . . . . 10 ⊢ ((𝑀 − 𝐼) ∈ (1...(𝑀 − 1)) → (1...(𝑀 − 1)) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
108	106, 107	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
109	99, 108	pm2.61dan 809	. . . . . . . 8 ⊢ (𝜑 → (1...(𝑀 − 1)) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
110	109	uneq1d 4092	. . . . . . 7 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
111	21, 110	eqtrd 2778	. . . . . 6 ⊢ (𝜑 → (1...𝑀) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
112	111	eqcomd 2744	. . . . 5 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (1...𝑀))
113	72, 112	eqtrd 2778	. . . 4 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (1...𝑀))
114	70, 113	eqtrd 2778	. . 3 ⊢ (𝜑 → (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀}) = (1...𝑀))
115	114	eqcomd 2744	. 2 ⊢ (𝜑 → (1...𝑀) = (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀}))
116	14, 67, 115	3jca 1126	1 ⊢ (𝜑 → ((((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅ ∧ (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}) ∧ (1...𝑀) = (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∪ cun 3881 ∩ cin 3882 ∅c0 4253 {csn 4558 class class class wbr 5070 (class class class)co 7255 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 < clt 10940 ≤ cle 10941 − cmin 11135 ℕcn 11903 ℤcz 12249 ...cfz 13168

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169

This theorem is referenced by: metakunt25 40077

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