metakunt24 - Mathbox for metakunt

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Theorem metakunt24 39822

Description: Technical condition such that metakunt17 39815 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 4180	. . . 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 39816	. . . . . . 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 498	. . . . . 6 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅))
8	7	simp2d 1145	. . . . 5 ⊢ (𝜑 → ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅)
9	7	simp3d 1146	. . . . 5 ⊢ (𝜑 → ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅)
10	8, 9	uneq12d 4068	. . . 4 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = (∅ ∪ ∅))
11		unidm 4056	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
12	11	a1i 11	. . . 4 ⊢ (𝜑 → (∅ ∪ ∅) = ∅)
13	10, 12	eqtrd 2774	. . 3 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = ∅)
14	2, 13	eqtrd 2774	. 2 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅)
15		1zzd 12191	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
16	3	nnzd 12264	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
17	3	nnge1d 11861	. . . . 5 ⊢ (𝜑 → 1 ≤ 𝑀)
18	3	nnred 11828	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
19	18	leidd 11381	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
20	15, 16, 16, 17, 19	elfzd 13086	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
21	20	fzsplitnd 39682	. . 3 ⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
22		oveq1 7209	. . . . . . . . . 10 ⊢ (𝐼 = 𝑀 → (𝐼 − 1) = (𝑀 − 1))
23	22	oveq2d 7218	. . . . . . . . 9 ⊢ (𝐼 = 𝑀 → (1...(𝐼 − 1)) = (1...(𝑀 − 1)))
24		oveq1 7209	. . . . . . . . 9 ⊢ (𝐼 = 𝑀 → (𝐼...(𝑀 − 1)) = (𝑀...(𝑀 − 1)))
25	23, 24	uneq12d 4068	. . . . . . . 8 ⊢ (𝐼 = 𝑀 → ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))))
26	25	uneq1d 4066	. . . . . . 7 ⊢ (𝐼 = 𝑀 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
27	26	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
28	18	ltm1d 11747	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
29	16, 15	zsubcld 12270	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
30		fzn 13111	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
31	16, 29, 30	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
32	28, 31	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
33	32	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (𝑀...(𝑀 − 1)) = ∅)
34	33	uneq2d 4067	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ ∅))
35		un0 4295	. . . . . . . . 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 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = (1...(𝑀 − 1)))
38	37	uneq1d 4066	. . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
39	27, 38	eqtrd 2774	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
40	39	eqcomd 2740	. . . 4 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
41	15	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℤ)
42	16	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℤ)
43	42, 41	zsubcld 12270	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 1) ∈ ℤ)
44	4	nnzd 12264	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℤ)
45	44	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℤ)
46	4	nnge1d 11861	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 𝐼)
47	46	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ 𝐼)
48		eqid 2734	. . . . . . . . . . 11 ⊢ 𝑀 = 𝑀
49		eqeq1 2738	. . . . . . . . . . 11 ⊢ (𝑀 = 𝐼 → (𝑀 = 𝑀 ↔ 𝐼 = 𝑀))
50	48, 49	mpbii 236	. . . . . . . . . 10 ⊢ (𝑀 = 𝐼 → 𝐼 = 𝑀)
51	50	necon3bi 2961	. . . . . . . . 9 ⊢ (¬ 𝐼 = 𝑀 → 𝑀 ≠ 𝐼)
52	51	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ≠ 𝐼)
53	4	nnred 11828	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℝ)
54	53, 18, 5	leltned 10968	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 < 𝑀 ↔ 𝑀 ≠ 𝐼))
55	54	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀 ↔ 𝑀 ≠ 𝐼))
56	52, 55	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 < 𝑀)
57		zltlem1 12213	. . . . . . . . 9 ⊢ ((𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐼 < 𝑀 ↔ 𝐼 ≤ (𝑀 − 1)))
58	44, 16, 57	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝐼 < 𝑀 ↔ 𝐼 ≤ (𝑀 − 1)))
59	58	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀 ↔ 𝐼 ≤ (𝑀 − 1)))
60	56, 59	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ≤ (𝑀 − 1))
61	41, 43, 45, 47, 60	fzsplitnr 39683	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
62	61	uneq1d 4066	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
63	40, 62	pm2.61dan 813	. . 3 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
64		fzsn 13137	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
65	16, 64	syl 17	. . . 4 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
66	65	uneq2d 4067	. . 3 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
67	21, 63, 66	3eqtrd 2778	. 2 ⊢ (𝜑 → (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
68		uncom 4057	. . . . . 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 4066	. . . 4 ⊢ (𝜑 → (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀}) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
71	65	uneq2d 4067	. . . . . 6 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
72	71	eqcomd 2740	. . . . 5 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
73		fz10 13116	. . . . . . . . . . . . . . 15 ⊢ (1...0) = ∅
74	73	uneq1i 4063	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1))))
77		uncom 4057	. . . . . . . . . . . . . . 15 ⊢ ((1...(𝑀 − 1)) ∪ ∅) = (∅ ∪ (1...(𝑀 − 1)))
78	77	eqeq1i 2739	. . . . . . . . . . . . . 14 ⊢ (((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1)) ↔ (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
79	78	imbi2i 339	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1))) ↔ ((𝜑 ∧ 𝐼 = 𝑀) → (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1))))
80	36, 79	mpbi 233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
81	76, 80	eqtrd 2774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
82	81	eqcomd 2740	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...0) ∪ (1...(𝑀 − 1))))
83		oveq2 7210	. . . . . . . . . . . . . . 15 ⊢ (𝐼 = 𝑀 → (𝑀 − 𝐼) = (𝑀 − 𝑀))
84	83	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (𝑀 − 𝐼) = (𝑀 − 𝑀))
85	18	recnd 10844	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℂ)
86	85	subidd 11160	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 − 𝑀) = 0)
87	86	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (𝑀 − 𝑀) = 0)
88	84, 87	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (𝑀 − 𝐼) = 0)
89	88	oveq2d 7218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (1...(𝑀 − 𝐼)) = (1...0))
90	83	oveq1d 7217	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 = 𝑀 → ((𝑀 − 𝐼) + 1) = ((𝑀 − 𝑀) + 1))
91	90	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((𝑀 − 𝐼) + 1) = ((𝑀 − 𝑀) + 1))
92	87	oveq1d 7217	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((𝑀 − 𝑀) + 1) = (0 + 1))
93	91, 92	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((𝑀 − 𝐼) + 1) = (0 + 1))
94		1e0p1 12318	. . . . . . . . . . . . . 14 ⊢ 1 = (0 + 1)
95	93, 94	eqtr4di 2792	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((𝑀 − 𝐼) + 1) = 1)
96	95	oveq1d 7217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (((𝑀 − 𝐼) + 1)...(𝑀 − 1)) = (1...(𝑀 − 1)))
97	89, 96	uneq12d 4068	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) = ((1...0) ∪ (1...(𝑀 − 1))))
98	97	eqcomd 2740	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
99	82, 98	eqtrd 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
100	42, 45	zsubcld 12270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 𝐼) ∈ ℤ)
101	53	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℝ)
102	18	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℝ)
103		1red 10817	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℝ)
104	101, 102, 103, 60	lesubd 11419	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ (𝑀 − 𝐼))
105	103, 101, 102, 47	lesub2dd 11432	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 𝐼) ≤ (𝑀 − 1))
106	41, 43, 100, 104, 105	elfzd 13086	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 𝐼) ∈ (1...(𝑀 − 1)))
107		fzsplit 13121	. . . . . . . . . 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 813	. . . . . . . 8 ⊢ (𝜑 → (1...(𝑀 − 1)) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
110	109	uneq1d 4066	. . . . . . 7 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
111	21, 110	eqtrd 2774	. . . . . 6 ⊢ (𝜑 → (1...𝑀) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
112	111	eqcomd 2740	. . . . 5 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (1...𝑀))
113	72, 112	eqtrd 2774	. . . 4 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (1...𝑀))
114	70, 113	eqtrd 2774	. . 3 ⊢ (𝜑 → (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀}) = (1...𝑀))
115	114	eqcomd 2740	. 2 ⊢ (𝜑 → (1...𝑀) = (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀}))
116	14, 67, 115	3jca 1130	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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∪ cun 3855 ∩ cin 3856 ∅c0 4227 {csn 4531 class class class wbr 5043 (class class class)co 7202 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 < clt 10850 ≤ cle 10851 − cmin 11045 ℕcn 11813 ℤcz 12159 ...cfz 13078

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-fz 13079

This theorem is referenced by: metakunt25 39823

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