metakunt20 - Mathbox for metakunt

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Theorem metakunt20 40144

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 2742	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑀 ↔ 𝑋 = 𝑀))
4		breq1 5077	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 < 𝐼 ↔ 𝑋 < 𝐼))
5		oveq1 7282	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 + (𝑀 − 𝐼)) = (𝑋 + (𝑀 − 𝐼)))
6		oveq1 7282	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 + (1 − 𝐼)) = (𝑋 + (1 − 𝐼)))
7	4, 5, 6	ifbieq12d 4487	. . . . . . 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 4485	. . . . . 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 482	. . . . 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 4465	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑋 = 𝑀, 𝑀, if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼)))) = 𝑀)
14	10	eqcomd 2744	. . . . . . 7 ⊢ (𝜑 → 𝑀 = 𝑋)
15	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑀 = 𝑋)
16	13, 15	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑋 = 𝑀, 𝑀, if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼)))) = 𝑋)
17	9, 16	eqtrd 2778	. . . 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 6882	. . 3 ⊢ (𝜑 → (𝐵‘𝑋) = 𝑋)
20	10	fveq2d 6778	. . . . 5 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = ({⟨𝑀, 𝑀⟩}‘𝑀))
21		metakunt20.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
22		fvsng 7052	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑀 ∈ ℕ) → ({⟨𝑀, 𝑀⟩}‘𝑀) = 𝑀)
23	21, 21, 22	syl2anc 584	. . . . 5 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑀) = 𝑀)
24	20, 23	eqtrd 2778	. . . 4 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = 𝑀)
25	24	eqcomd 2744	. . 3 ⊢ (𝜑 → 𝑀 = ({⟨𝑀, 𝑀⟩}‘𝑋))
26	19, 10, 25	3eqtrd 2782	. 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 40143	. . . . . 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 495	. . . . 5 ⊢ (𝜑 → (𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))))
33	32	simp3d 1143	. . . 4 ⊢ (𝜑 → (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
34	31	simprd 496	. . . 4 ⊢ (𝜑 → {⟨𝑀, 𝑀⟩} Fn {𝑀})
35	21	nnzd 12425	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
36		fzsn 13298	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
37	35, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
38	37	ineq2d 4146	. . . . . 6 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}))
39	38	eqcomd 2744	. . . . 5 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)))
40	27	nncnd 11989	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℂ)
41	21	nncnd 11989	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℂ)
42	40, 41	pncan3d 11335	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 + (𝑀 − 𝐼)) = 𝑀)
43	42	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (𝜑 → (1..^(𝐼 + (𝑀 − 𝐼))) = (1..^𝑀))
44		fzoval 13388	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → (1..^𝑀) = (1...(𝑀 − 1)))
45	35, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1..^𝑀) = (1...(𝑀 − 1)))
46	43, 45	eqtrd 2778	. . . . . . . . . . 11 ⊢ (𝜑 → (1..^(𝐼 + (𝑀 − 𝐼))) = (1...(𝑀 − 1)))
47	46	eqcomd 2744	. . . . . . . . . 10 ⊢ (𝜑 → (1...(𝑀 − 1)) = (1..^(𝐼 + (𝑀 − 𝐼))))
48		nnuz 12621	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
49	27, 48	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (ℤ_≥‘1))
50	27	nnzd 12425	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℤ)
51	50, 35	jca 512	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ))
52		znn0sub 12367	. . . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 − 𝐼) ∈ ℕ₀)
55		fzoun 13424	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ (ℤ_≥‘1) ∧ (𝑀 − 𝐼) ∈ ℕ₀) → (1..^(𝐼 + (𝑀 − 𝐼))) = ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))))
56	49, 54, 55	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (1..^(𝐼 + (𝑀 − 𝐼))) = ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))))
57	47, 56	eqtrd 2778	. . . . . . . . 9 ⊢ (𝜑 → (1...(𝑀 − 1)) = ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))))
58		fzoval 13388	. . . . . . . . . . 11 ⊢ (𝐼 ∈ ℤ → (1..^𝐼) = (1...(𝐼 − 1)))
59	50, 58	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1..^𝐼) = (1...(𝐼 − 1)))
60	42	oveq2d 7291	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼..^(𝐼 + (𝑀 − 𝐼))) = (𝐼..^𝑀))
61		fzoval 13388	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (𝐼..^𝑀) = (𝐼...(𝑀 − 1)))
62	35, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼..^𝑀) = (𝐼...(𝑀 − 1)))
63	60, 62	eqtrd 2778	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼..^(𝐼 + (𝑀 − 𝐼))) = (𝐼...(𝑀 − 1)))
64	59, 63	uneq12d 4098	. . . . . . . . 9 ⊢ (𝜑 → ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
65	57, 64	eqtrd 2778	. . . . . . . 8 ⊢ (𝜑 → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
66	65	ineq1d 4145	. . . . . . 7 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)))
67	66	eqcomd 2744	. . . . . 6 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∩ (𝑀...𝑀)))
68	21	nnred 11988	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
69	68	ltm1d 11907	. . . . . . 7 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
70		fzdisj 13283	. . . . . . 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 2778	. . . . 5 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = ∅)
73	39, 72	eqtrd 2778	. . . 4 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅)
74		elsng 4575	. . . . . 6 ⊢ (𝑋 ∈ (1...𝑀) → (𝑋 ∈ {𝑀} ↔ 𝑋 = 𝑀))
75	18, 74	syl 17	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ {𝑀} ↔ 𝑋 = 𝑀))
76	10, 75	mpbird 256	. . . 4 ⊢ (𝜑 → 𝑋 ∈ {𝑀})
77	33, 34, 73, 76	fvun2d 6862	. . 3 ⊢ (𝜑 → (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋) = ({⟨𝑀, 𝑀⟩}‘𝑋))
78	77	eqcomd 2744	. 2 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
79	26, 78	eqtrd 2778	1 ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∪ cun 3885 ∩ cin 3886 ∅c0 4256 ifcif 4459 {csn 4561 ⟨cop 4567 class class class wbr 5074 ↦ cmpt 5157 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 1c1 10872 + caddc 10874 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 ..^cfzo 13382

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383

This theorem is referenced by: metakunt23 40147

W3C validator