metakunt20 - Mathbox for metakunt

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Theorem metakunt20 40992

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 2736	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑀 ↔ 𝑋 = 𝑀))
4		breq1 5150	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 < 𝐼 ↔ 𝑋 < 𝐼))
5		oveq1 7412	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 + (𝑀 − 𝐼)) = (𝑋 + (𝑀 − 𝐼)))
6		oveq1 7412	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 + (1 − 𝐼)) = (𝑋 + (1 − 𝐼)))
7	4, 5, 6	ifbieq12d 4555	. . . . . . 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 4553	. . . . . 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 4533	. . . . . . . 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 2738	. . . . . . 7 ⊢ (𝜑 → 𝑀 = 𝑋)
15	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑀 = 𝑋)
16	13, 15	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑋 = 𝑀, 𝑀, if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼)))) = 𝑋)
17	9, 16	eqtrd 2772	. . . 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 7002	. . 3 ⊢ (𝜑 → (𝐵‘𝑋) = 𝑋)
20	10	fveq2d 6892	. . . . 5 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = ({⟨𝑀, 𝑀⟩}‘𝑀))
21		metakunt20.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
22		fvsng 7174	. . . . . 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 2772	. . . 4 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = 𝑀)
25	24	eqcomd 2738	. . 3 ⊢ (𝜑 → 𝑀 = ({⟨𝑀, 𝑀⟩}‘𝑋))
26	19, 10, 25	3eqtrd 2776	. 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 40991	. . . . . 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 1144	. . . 4 ⊢ (𝜑 → (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
34	31	simprd 496	. . . 4 ⊢ (𝜑 → {⟨𝑀, 𝑀⟩} Fn {𝑀})
35	21	nnzd 12581	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
36		fzsn 13539	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
37	35, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
38	37	ineq2d 4211	. . . . . 6 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}))
39	38	eqcomd 2738	. . . . 5 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)))
40	27	nncnd 12224	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℂ)
41	21	nncnd 12224	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℂ)
42	40, 41	pncan3d 11570	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 + (𝑀 − 𝐼)) = 𝑀)
43	42	oveq2d 7421	. . . . . . . . . . . 12 ⊢ (𝜑 → (1..^(𝐼 + (𝑀 − 𝐼))) = (1..^𝑀))
44		fzoval 13629	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → (1..^𝑀) = (1...(𝑀 − 1)))
45	35, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1..^𝑀) = (1...(𝑀 − 1)))
46	43, 45	eqtrd 2772	. . . . . . . . . . 11 ⊢ (𝜑 → (1..^(𝐼 + (𝑀 − 𝐼))) = (1...(𝑀 − 1)))
47	46	eqcomd 2738	. . . . . . . . . 10 ⊢ (𝜑 → (1...(𝑀 − 1)) = (1..^(𝐼 + (𝑀 − 𝐼))))
48		nnuz 12861	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
49	27, 48	eleqtrdi 2843	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (ℤ_≥‘1))
50	27	nnzd 12581	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℤ)
51	50, 35	jca 512	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ))
52		znn0sub 12605	. . . . . . . . . . . . 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 13665	. . . . . . . . . . 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 2772	. . . . . . . . 9 ⊢ (𝜑 → (1...(𝑀 − 1)) = ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))))
58		fzoval 13629	. . . . . . . . . . 11 ⊢ (𝐼 ∈ ℤ → (1..^𝐼) = (1...(𝐼 − 1)))
59	50, 58	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1..^𝐼) = (1...(𝐼 − 1)))
60	42	oveq2d 7421	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼..^(𝐼 + (𝑀 − 𝐼))) = (𝐼..^𝑀))
61		fzoval 13629	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (𝐼..^𝑀) = (𝐼...(𝑀 − 1)))
62	35, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼..^𝑀) = (𝐼...(𝑀 − 1)))
63	60, 62	eqtrd 2772	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼..^(𝐼 + (𝑀 − 𝐼))) = (𝐼...(𝑀 − 1)))
64	59, 63	uneq12d 4163	. . . . . . . . 9 ⊢ (𝜑 → ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
65	57, 64	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
66	65	ineq1d 4210	. . . . . . 7 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)))
67	66	eqcomd 2738	. . . . . 6 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∩ (𝑀...𝑀)))
68	21	nnred 12223	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
69	68	ltm1d 12142	. . . . . . 7 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
70		fzdisj 13524	. . . . . . 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 2772	. . . . 5 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = ∅)
73	39, 72	eqtrd 2772	. . . 4 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅)
74		elsng 4641	. . . . . 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 6982	. . 3 ⊢ (𝜑 → (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋) = ({⟨𝑀, 𝑀⟩}‘𝑋))
78	77	eqcomd 2738	. 2 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
79	26, 78	eqtrd 2772	1 ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∪ cun 3945 ∩ cin 3946 ∅c0 4321 ifcif 4527 {csn 4627 ⟨cop 4633 class class class wbr 5147 ↦ cmpt 5230 Fn wfn 6535 ‘cfv 6540 (class class class)co 7405 1c1 11107 + caddc 11109 < clt 11244 ≤ cle 11245 − cmin 11440 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 ..^cfzo 13623

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624

This theorem is referenced by: metakunt23 40995

W3C validator