metakunt24 - Mathbox for metakunt

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Theorem metakunt24 41680

Description: Technical condition such that metakunt17 41673 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 4276	. . . 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 41674	. . . . . . 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 4163	. . . 4 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = (∅ ∪ ∅))
11		unidm 4151	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
12	11	a1i 11	. . . 4 ⊢ (𝜑 → (∅ ∪ ∅) = ∅)
13	10, 12	eqtrd 2768	. . 3 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = ∅)
14	2, 13	eqtrd 2768	. 2 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅)
15		1zzd 12624	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
16	3	nnzd 12616	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
17	3	nnge1d 12291	. . . . 5 ⊢ (𝜑 → 1 ≤ 𝑀)
18	3	nnred 12258	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
19	18	leidd 11811	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
20	15, 16, 16, 17, 19	elfzd 13525	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
21	20	fzsplitnd 41453	. . 3 ⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
22		oveq1 7427	. . . . . . . . . 10 ⊢ (𝐼 = 𝑀 → (𝐼 − 1) = (𝑀 − 1))
23	22	oveq2d 7436	. . . . . . . . 9 ⊢ (𝐼 = 𝑀 → (1...(𝐼 − 1)) = (1...(𝑀 − 1)))
24		oveq1 7427	. . . . . . . . 9 ⊢ (𝐼 = 𝑀 → (𝐼...(𝑀 − 1)) = (𝑀...(𝑀 − 1)))
25	23, 24	uneq12d 4163	. . . . . . . 8 ⊢ (𝐼 = 𝑀 → ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))))
26	25	uneq1d 4161	. . . . . . 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 12177	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
29	16, 15	zsubcld 12702	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
30		fzn 13550	. . . . . . . . . . . 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 4162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ ∅))
35		un0 4391	. . . . . . . . 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 2768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = (1...(𝑀 − 1)))
38	37	uneq1d 4161	. . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
39	27, 38	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
40	39	eqcomd 2734	. . . 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 12702	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 1) ∈ ℤ)
44	4	nnzd 12616	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℤ)
45	44	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℤ)
46	4	nnge1d 12291	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 𝐼)
47	46	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ 𝐼)
48		eqid 2728	. . . . . . . . . . 11 ⊢ 𝑀 = 𝑀
49		eqeq1 2732	. . . . . . . . . . 11 ⊢ (𝑀 = 𝐼 → (𝑀 = 𝑀 ↔ 𝐼 = 𝑀))
50	48, 49	mpbii 232	. . . . . . . . . 10 ⊢ (𝑀 = 𝐼 → 𝐼 = 𝑀)
51	50	necon3bi 2964	. . . . . . . . 9 ⊢ (¬ 𝐼 = 𝑀 → 𝑀 ≠ 𝐼)
52	51	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ≠ 𝐼)
53	4	nnred 12258	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℝ)
54	53, 18, 5	leltned 11398	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 < 𝑀 ↔ 𝑀 ≠ 𝐼))
55	54	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀 ↔ 𝑀 ≠ 𝐼))
56	52, 55	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 < 𝑀)
57		zltlem1 12646	. . . . . . . . 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 41454	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
62	61	uneq1d 4161	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
63	40, 62	pm2.61dan 812	. . 3 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
64		fzsn 13576	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
65	16, 64	syl 17	. . . 4 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
66	65	uneq2d 4162	. . 3 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
67	21, 63, 66	3eqtrd 2772	. 2 ⊢ (𝜑 → (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
68		uncom 4152	. . . . . 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 4161	. . . 4 ⊢ (𝜑 → (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀}) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
71	65	uneq2d 4162	. . . . . 6 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
72	71	eqcomd 2734	. . . . 5 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
73		fz10 13555	. . . . . . . . . . . . . . 15 ⊢ (1...0) = ∅
74	73	uneq1i 4158	. . . . . . . . . . . . . 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 4152	. . . . . . . . . . . . . . 15 ⊢ ((1...(𝑀 − 1)) ∪ ∅) = (∅ ∪ (1...(𝑀 − 1)))
78	77	eqeq1i 2733	. . . . . . . . . . . . . 14 ⊢ (((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1)) ↔ (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
79	78	imbi2i 336	. . . . . . . . . . . . 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 2768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
82	81	eqcomd 2734	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...0) ∪ (1...(𝑀 − 1))))
83		oveq2 7428	. . . . . . . . . . . . . . 15 ⊢ (𝐼 = 𝑀 → (𝑀 − 𝐼) = (𝑀 − 𝑀))
84	83	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (𝑀 − 𝐼) = (𝑀 − 𝑀))
85	18	recnd 11273	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℂ)
86	85	subidd 11590	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 − 𝑀) = 0)
87	86	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (𝑀 − 𝑀) = 0)
88	84, 87	eqtrd 2768	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (𝑀 − 𝐼) = 0)
89	88	oveq2d 7436	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (1...(𝑀 − 𝐼)) = (1...0))
90	83	oveq1d 7435	. . . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((𝑀 − 𝑀) + 1) = (0 + 1))
93	91, 92	eqtrd 2768	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((𝑀 − 𝐼) + 1) = (0 + 1))
94		1e0p1 12750	. . . . . . . . . . . . . 14 ⊢ 1 = (0 + 1)
95	93, 94	eqtr4di 2786	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((𝑀 − 𝐼) + 1) = 1)
96	95	oveq1d 7435	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (((𝑀 − 𝐼) + 1)...(𝑀 − 1)) = (1...(𝑀 − 1)))
97	89, 96	uneq12d 4163	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) = ((1...0) ∪ (1...(𝑀 − 1))))
98	97	eqcomd 2734	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
99	82, 98	eqtrd 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
100	42, 45	zsubcld 12702	. . . . . . . . . . 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 11246	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℝ)
104	101, 102, 103, 60	lesubd 11849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ (𝑀 − 𝐼))
105	103, 101, 102, 47	lesub2dd 11862	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 𝐼) ≤ (𝑀 − 1))
106	41, 43, 100, 104, 105	elfzd 13525	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 𝐼) ∈ (1...(𝑀 − 1)))
107		fzsplit 13560	. . . . . . . . . 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 812	. . . . . . . 8 ⊢ (𝜑 → (1...(𝑀 − 1)) = ((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))))
110	109	uneq1d 4161	. . . . . . 7 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
111	21, 110	eqtrd 2768	. . . . . 6 ⊢ (𝜑 → (1...𝑀) = (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
112	111	eqcomd 2734	. . . . 5 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (1...𝑀))
113	72, 112	eqtrd 2768	. . . 4 ⊢ (𝜑 → (((1...(𝑀 − 𝐼)) ∪ (((𝑀 − 𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (1...𝑀))
114	70, 113	eqtrd 2768	. . 3 ⊢ (𝜑 → (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀}) = (1...𝑀))
115	114	eqcomd 2734	. 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 1534 ∈ wcel 2099 ≠ wne 2937 ∪ cun 3945 ∩ cin 3946 ∅c0 4323 {csn 4629 class class class wbr 5148 (class class class)co 7420 ℝcr 11138 0cc0 11139 1c1 11140 + caddc 11142 < clt 11279 ≤ cle 11280 − cmin 11475 ℕcn 12243 ℤcz 12589 ...cfz 13517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-fz 13518

This theorem is referenced by: metakunt25 41681

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