metakunt20 - Mathbox for metakunt

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Theorem metakunt20 39651

Description: Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)

**Hypotheses**
Ref	Expression
metakunt20.1	⊢ (𝜑 → 𝑀 ∈ ℕ)
metakunt20.2	⊢ (𝜑 → 𝐼 ∈ ℕ)
metakunt20.3	⊢ (𝜑 → 𝐼 ≤ 𝑀)
metakunt20.4	⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼)))))
metakunt20.5	⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼)))
metakunt20.6	⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))
metakunt20.7	⊢ (𝜑 → 𝑋 ∈ (1...𝑀))
metakunt20.8	⊢ (𝜑 → 𝑋 = 𝑀)

**Assertion**
Ref	Expression
metakunt20	⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))

Distinct variable groups: 𝑥,𝐼 𝑥,𝑀 𝑥,𝑋 𝜑,𝑥 Allowed substitution hints: 𝐵(𝑥) 𝐶(𝑥) 𝐷(𝑥)

**Proof of Theorem metakunt20**
Step	Hyp	Ref	Expression
1		metakunt20.4	. . . . 5 ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼)))))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))))
3		eqeq1 2763	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑀 ↔ 𝑋 = 𝑀))
4		breq1 5028	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 < 𝐼 ↔ 𝑋 < 𝐼))
5		oveq1 7150	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 + (𝑀 − 𝐼)) = (𝑋 + (𝑀 − 𝐼)))
6		oveq1 7150	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 + (1 − 𝐼)) = (𝑋 + (1 − 𝐼)))
7	4, 5, 6	ifbieq12d 4441	. . . . . . 7 ⊢ (𝑥 = 𝑋 → if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))) = if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼))))
8	3, 7	ifbieq2d 4439	. . . . . 6 ⊢ (𝑥 = 𝑋 → if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼)))) = if(𝑋 = 𝑀, 𝑀, if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼)))))
9	8	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼)))) = if(𝑋 = 𝑀, 𝑀, if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼)))))
10		metakunt20.8	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = 𝑀)
11		iftrue 4419	. . . . . . . 8 ⊢ (𝑋 = 𝑀 → if(𝑋 = 𝑀, 𝑀, if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼)))) = 𝑀)
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → if(𝑋 = 𝑀, 𝑀, if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼)))) = 𝑀)
13	12	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑋 = 𝑀, 𝑀, if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼)))) = 𝑀)
14	10	eqcomd 2765	. . . . . . 7 ⊢ (𝜑 → 𝑀 = 𝑋)
15	14	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑀 = 𝑋)
16	13, 15	eqtrd 2794	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑋 = 𝑀, 𝑀, if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼)))) = 𝑋)
17	9, 16	eqtrd 2794	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼)))) = 𝑋)
18		metakunt20.7	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀))
19	2, 17, 18, 18	fvmptd 6759	. . 3 ⊢ (𝜑 → (𝐵‘𝑋) = 𝑋)
20	10	fveq2d 6655	. . . . 5 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = ({⟨𝑀, 𝑀⟩}‘𝑀))
21		metakunt20.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
22		fvsng 6926	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑀 ∈ ℕ) → ({⟨𝑀, 𝑀⟩}‘𝑀) = 𝑀)
23	21, 21, 22	syl2anc 588	. . . . 5 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑀) = 𝑀)
24	20, 23	eqtrd 2794	. . . 4 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = 𝑀)
25	24	eqcomd 2765	. . 3 ⊢ (𝜑 → 𝑀 = ({⟨𝑀, 𝑀⟩}‘𝑋))
26	19, 10, 25	3eqtrd 2798	. 2 ⊢ (𝜑 → (𝐵‘𝑋) = ({⟨𝑀, 𝑀⟩}‘𝑋))
27		metakunt20.2	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℕ)
28		metakunt20.3	. . . . . . 7 ⊢ (𝜑 → 𝐼 ≤ 𝑀)
29		metakunt20.5	. . . . . . 7 ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼)))
30		metakunt20.6	. . . . . . 7 ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))
31	21, 27, 28, 1, 29, 30	metakunt19 39650	. . . . . 6 ⊢ (𝜑 → ((𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))) ∧ {⟨𝑀, 𝑀⟩} Fn {𝑀}))
32	31	simpld 499	. . . . 5 ⊢ (𝜑 → (𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))))
33	32	simp3d 1142	. . . 4 ⊢ (𝜑 → (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
34	31	simprd 500	. . . 4 ⊢ (𝜑 → {⟨𝑀, 𝑀⟩} Fn {𝑀})
35	21	nnzd 12110	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
36		fzsn 12983	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
37	35, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
38	37	ineq2d 4113	. . . . . 6 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}))
39	38	eqcomd 2765	. . . . 5 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)))
40	27	nncnd 11675	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℂ)
41	21	nncnd 11675	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℂ)
42	40, 41	pncan3d 11023	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 + (𝑀 − 𝐼)) = 𝑀)
43	42	oveq2d 7159	. . . . . . . . . . . 12 ⊢ (𝜑 → (1..^(𝐼 + (𝑀 − 𝐼))) = (1..^𝑀))
44		fzoval 13073	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → (1..^𝑀) = (1...(𝑀 − 1)))
45	35, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1..^𝑀) = (1...(𝑀 − 1)))
46	43, 45	eqtrd 2794	. . . . . . . . . . 11 ⊢ (𝜑 → (1..^(𝐼 + (𝑀 − 𝐼))) = (1...(𝑀 − 1)))
47	46	eqcomd 2765	. . . . . . . . . 10 ⊢ (𝜑 → (1...(𝑀 − 1)) = (1..^(𝐼 + (𝑀 − 𝐼))))
48		nnuz 12306	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
49	27, 48	eleqtrdi 2861	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (ℤ_≥‘1))
50	27	nnzd 12110	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℤ)
51	50, 35	jca 516	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ))
52		znn0sub 12053	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐼 ≤ 𝑀 ↔ (𝑀 − 𝐼) ∈ ℕ₀))
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 ≤ 𝑀 ↔ (𝑀 − 𝐼) ∈ ℕ₀))
54	28, 53	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 − 𝐼) ∈ ℕ₀)
55		fzoun 13108	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ (ℤ_≥‘1) ∧ (𝑀 − 𝐼) ∈ ℕ₀) → (1..^(𝐼 + (𝑀 − 𝐼))) = ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))))
56	49, 54, 55	syl2anc 588	. . . . . . . . . 10 ⊢ (𝜑 → (1..^(𝐼 + (𝑀 − 𝐼))) = ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))))
57	47, 56	eqtrd 2794	. . . . . . . . 9 ⊢ (𝜑 → (1...(𝑀 − 1)) = ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))))
58		fzoval 13073	. . . . . . . . . . 11 ⊢ (𝐼 ∈ ℤ → (1..^𝐼) = (1...(𝐼 − 1)))
59	50, 58	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1..^𝐼) = (1...(𝐼 − 1)))
60	42	oveq2d 7159	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼..^(𝐼 + (𝑀 − 𝐼))) = (𝐼..^𝑀))
61		fzoval 13073	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (𝐼..^𝑀) = (𝐼...(𝑀 − 1)))
62	35, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼..^𝑀) = (𝐼...(𝑀 − 1)))
63	60, 62	eqtrd 2794	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼..^(𝐼 + (𝑀 − 𝐼))) = (𝐼...(𝑀 − 1)))
64	59, 63	uneq12d 4065	. . . . . . . . 9 ⊢ (𝜑 → ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
65	57, 64	eqtrd 2794	. . . . . . . 8 ⊢ (𝜑 → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
66	65	ineq1d 4112	. . . . . . 7 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)))
67	66	eqcomd 2765	. . . . . 6 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∩ (𝑀...𝑀)))
68	21	nnred 11674	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
69	68	ltm1d 11595	. . . . . . 7 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
70		fzdisj 12968	. . . . . . 7 ⊢ ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑀)) = ∅)
71	69, 70	syl 17	. . . . . 6 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑀)) = ∅)
72	67, 71	eqtrd 2794	. . . . 5 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = ∅)
73	39, 72	eqtrd 2794	. . . 4 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅)
74		elsng 4529	. . . . . 6 ⊢ (𝑋 ∈ (1...𝑀) → (𝑋 ∈ {𝑀} ↔ 𝑋 = 𝑀))
75	18, 74	syl 17	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ {𝑀} ↔ 𝑋 = 𝑀))
76	10, 75	mpbird 260	. . . 4 ⊢ (𝜑 → 𝑋 ∈ {𝑀})
77	33, 34, 73, 76	fvun2d 6739	. . 3 ⊢ (𝜑 → (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋) = ({⟨𝑀, 𝑀⟩}‘𝑋))
78	77	eqcomd 2765	. 2 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
79	26, 78	eqtrd 2794	1 ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ∪ cun 3852 ∩ cin 3853 ∅c0 4221 ifcif 4413 {csn 4515 ⟨cop 4521 class class class wbr 5025 ↦ cmpt 5105 Fn wfn 6323 ‘cfv 6328 (class class class)co 7143 1c1 10561 + caddc 10563 < clt 10698 ≤ cle 10699 − cmin 10893 ℕcn 11659 ℕ₀cn0 11919 ℤcz 12005 ℤ_≥cuz 12267 ...cfz 12924 ..^cfzo 13067

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-n0 11920 df-z 12006 df-uz 12268 df-rp 12416 df-fz 12925 df-fzo 13068

This theorem is referenced by: metakunt23 39654

W3C validator