metakunt20 - Mathbox for metakunt

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Theorem metakunt20 42206

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 2739	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑀 ↔ 𝑋 = 𝑀))
4		breq1 5151	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 < 𝐼 ↔ 𝑋 < 𝐼))
5		oveq1 7438	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 + (𝑀 − 𝐼)) = (𝑋 + (𝑀 − 𝐼)))
6		oveq1 7438	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 + (1 − 𝐼)) = (𝑋 + (1 − 𝐼)))
7	4, 5, 6	ifbieq12d 4559	. . . . . . 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 4557	. . . . . 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 481	. . . . 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 4537	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑋 = 𝑀, 𝑀, if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼)))) = 𝑀)
14	10	eqcomd 2741	. . . . . . 7 ⊢ (𝜑 → 𝑀 = 𝑋)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑀 = 𝑋)
16	13, 15	eqtrd 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑋 = 𝑀, 𝑀, if(𝑋 < 𝐼, (𝑋 + (𝑀 − 𝐼)), (𝑋 + (1 − 𝐼)))) = 𝑋)
17	9, 16	eqtrd 2775	. . . 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 7023	. . 3 ⊢ (𝜑 → (𝐵‘𝑋) = 𝑋)
20	10	fveq2d 6911	. . . . 5 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = ({⟨𝑀, 𝑀⟩}‘𝑀))
21		metakunt20.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
22		fvsng 7200	. . . . . 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 2775	. . . 4 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = 𝑀)
25	24	eqcomd 2741	. . 3 ⊢ (𝜑 → 𝑀 = ({⟨𝑀, 𝑀⟩}‘𝑋))
26	19, 10, 25	3eqtrd 2779	. 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 42205	. . . . . 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 494	. . . . 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 495	. . . 4 ⊢ (𝜑 → {⟨𝑀, 𝑀⟩} Fn {𝑀})
35	21	nnzd 12638	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
36		fzsn 13603	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
37	35, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
38	37	ineq2d 4228	. . . . . 6 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}))
39	38	eqcomd 2741	. . . . 5 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)))
40	27	nncnd 12280	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℂ)
41	21	nncnd 12280	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℂ)
42	40, 41	pncan3d 11621	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 + (𝑀 − 𝐼)) = 𝑀)
43	42	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝜑 → (1..^(𝐼 + (𝑀 − 𝐼))) = (1..^𝑀))
44		fzoval 13697	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → (1..^𝑀) = (1...(𝑀 − 1)))
45	35, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1..^𝑀) = (1...(𝑀 − 1)))
46	43, 45	eqtrd 2775	. . . . . . . . . . 11 ⊢ (𝜑 → (1..^(𝐼 + (𝑀 − 𝐼))) = (1...(𝑀 − 1)))
47	46	eqcomd 2741	. . . . . . . . . 10 ⊢ (𝜑 → (1...(𝑀 − 1)) = (1..^(𝐼 + (𝑀 − 𝐼))))
48		nnuz 12919	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
49	27, 48	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (ℤ_≥‘1))
50	27	nnzd 12638	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℤ)
51	50, 35	jca 511	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ))
52		znn0sub 12662	. . . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 − 𝐼) ∈ ℕ₀)
55		fzoun 13733	. . . . . . . . . . 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 2775	. . . . . . . . 9 ⊢ (𝜑 → (1...(𝑀 − 1)) = ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))))
58		fzoval 13697	. . . . . . . . . . 11 ⊢ (𝐼 ∈ ℤ → (1..^𝐼) = (1...(𝐼 − 1)))
59	50, 58	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1..^𝐼) = (1...(𝐼 − 1)))
60	42	oveq2d 7447	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼..^(𝐼 + (𝑀 − 𝐼))) = (𝐼..^𝑀))
61		fzoval 13697	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (𝐼..^𝑀) = (𝐼...(𝑀 − 1)))
62	35, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼..^𝑀) = (𝐼...(𝑀 − 1)))
63	60, 62	eqtrd 2775	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼..^(𝐼 + (𝑀 − 𝐼))) = (𝐼...(𝑀 − 1)))
64	59, 63	uneq12d 4179	. . . . . . . . 9 ⊢ (𝜑 → ((1..^𝐼) ∪ (𝐼..^(𝐼 + (𝑀 − 𝐼)))) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
65	57, 64	eqtrd 2775	. . . . . . . 8 ⊢ (𝜑 → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
66	65	ineq1d 4227	. . . . . . 7 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)))
67	66	eqcomd 2741	. . . . . 6 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∩ (𝑀...𝑀)))
68	21	nnred 12279	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
69	68	ltm1d 12198	. . . . . . 7 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
70		fzdisj 13588	. . . . . . 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 2775	. . . . 5 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ (𝑀...𝑀)) = ∅)
73	39, 72	eqtrd 2775	. . . 4 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅)
74		elsng 4645	. . . . . 6 ⊢ (𝑋 ∈ (1...𝑀) → (𝑋 ∈ {𝑀} ↔ 𝑋 = 𝑀))
75	18, 74	syl 17	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ {𝑀} ↔ 𝑋 = 𝑀))
76	10, 75	mpbird 257	. . . 4 ⊢ (𝜑 → 𝑋 ∈ {𝑀})
77	33, 34, 73, 76	fvun2d 7003	. . 3 ⊢ (𝜑 → (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋) = ({⟨𝑀, 𝑀⟩}‘𝑋))
78	77	eqcomd 2741	. 2 ⊢ (𝜑 → ({⟨𝑀, 𝑀⟩}‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
79	26, 78	eqtrd 2775	1 ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 ifcif 4531 {csn 4631 ⟨cop 4637 class class class wbr 5148 ↦ cmpt 5231 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 − cmin 11490 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ...cfz 13544 ..^cfzo 13691

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692

This theorem is referenced by: metakunt23 42209

W3C validator