bitsinvp1 - Metamath Proof Explorer

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Theorem bitsinvp1 16329

Description: Recursive definition of the inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.)

**Hypothesis**
Ref	Expression
bitsinv.k	⊢ 𝐾 = ^◡(bits ↾ ℕ₀)

**Assertion**
Ref	Expression
bitsinvp1	⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐴, (2↑𝑁), 0)))

Proof of Theorem bitsinvp1 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzonel 13586	. . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
2	1	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ¬ 𝑁 ∈ (0..^𝑁))
3		disjsn 4672	. . . . . 6 ⊢ (((0..^𝑁) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0..^𝑁))
4	2, 3	sylibr 233	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑁) ∩ {𝑁}) = ∅)
5	4	ineq2d 4172	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ ((0..^𝑁) ∩ {𝑁})) = (𝐴 ∩ ∅))
6		inindi 4186	. . . 4 ⊢ (𝐴 ∩ ((0..^𝑁) ∩ {𝑁})) = ((𝐴 ∩ (0..^𝑁)) ∩ (𝐴 ∩ {𝑁}))
7		in0 4351	. . . 4 ⊢ (𝐴 ∩ ∅) = ∅
8	5, 6, 7	3eqtr3g 2799	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 ∩ (0..^𝑁)) ∩ (𝐴 ∩ {𝑁})) = ∅)
9		simpr 485	. . . . . . 7 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
10		nn0uz 12805	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtrdi 2848	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
12		fzosplitsn 13680	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
13	11, 12	syl 17	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
14	13	ineq2d 4172	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) = (𝐴 ∩ ((0..^𝑁) ∪ {𝑁})))
15		indi 4233	. . . 4 ⊢ (𝐴 ∩ ((0..^𝑁) ∪ {𝑁})) = ((𝐴 ∩ (0..^𝑁)) ∪ (𝐴 ∩ {𝑁}))
16	14, 15	eqtrdi 2792	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) = ((𝐴 ∩ (0..^𝑁)) ∪ (𝐴 ∩ {𝑁})))
17		fzofi 13879	. . . . 5 ⊢ (0..^(𝑁 + 1)) ∈ Fin
18	17	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑁 + 1)) ∈ Fin)
19		inss2 4189	. . . 4 ⊢ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ (0..^(𝑁 + 1))
20		ssfi 9117	. . . 4 ⊢ (((0..^(𝑁 + 1)) ∈ Fin ∧ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ (0..^(𝑁 + 1))) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin)
21	18, 19, 20	sylancl 586	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin)
22		2nn 12226	. . . . . 6 ⊢ 2 ∈ ℕ
23	22	a1i 11	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → 2 ∈ ℕ)
24		inss1 4188	. . . . . . 7 ⊢ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ 𝐴
25		simpl 483	. . . . . . 7 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ⊆ ℕ₀)
26	24, 25	sstrid 3955	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ ℕ₀)
27	26	sselda 3944	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → 𝑘 ∈ ℕ₀)
28	23, 27	nnexpcld 14148	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → (2↑𝑘) ∈ ℕ)
29	28	nncnd 12169	. . 3 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → (2↑𝑘) ∈ ℂ)
30	8, 16, 21, 29	fsumsplit 15626	. 2 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → Σ𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))(2↑𝑘) = (Σ𝑘 ∈ (𝐴 ∩ (0..^𝑁))(2↑𝑘) + Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘)))
31		elfpw 9298	. . . 4 ⊢ ((𝐴 ∩ (0..^(𝑁 + 1))) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^(𝑁 + 1))) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin))
32	26, 21, 31	sylanbrc 583	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ (𝒫 ℕ₀ ∩ Fin))
33		bitsinv.k	. . . 4 ⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
34	33	bitsinv 16328	. . 3 ⊢ ((𝐴 ∩ (0..^(𝑁 + 1))) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = Σ𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))(2↑𝑘))
35	32, 34	syl 17	. 2 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = Σ𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))(2↑𝑘))
36		inss1 4188	. . . . . 6 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
37	36, 25	sstrid 3955	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀)
38		fzofi 13879	. . . . . . 7 ⊢ (0..^𝑁) ∈ Fin
39	38	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^𝑁) ∈ Fin)
40		inss2 4189	. . . . . 6 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
41		ssfi 9117	. . . . . 6 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
42	39, 40, 41	sylancl 586	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
43		elfpw 9298	. . . . 5 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^𝑁)) ∈ Fin))
44	37, 42, 43	sylanbrc 583	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
45	33	bitsinv 16328	. . . 4 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘(𝐴 ∩ (0..^𝑁))) = Σ𝑘 ∈ (𝐴 ∩ (0..^𝑁))(2↑𝑘))
46	44, 45	syl 17	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾‘(𝐴 ∩ (0..^𝑁))) = Σ𝑘 ∈ (𝐴 ∩ (0..^𝑁))(2↑𝑘))
47		snssi 4768	. . . . . . . 8 ⊢ (𝑁 ∈ 𝐴 → {𝑁} ⊆ 𝐴)
48	47	adantl 482	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → {𝑁} ⊆ 𝐴)
49		sseqin2 4175	. . . . . . 7 ⊢ ({𝑁} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑁}) = {𝑁})
50	48, 49	sylib 217	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (𝐴 ∩ {𝑁}) = {𝑁})
51	50	sumeq1d 15586	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘) = Σ𝑘 ∈ {𝑁} (2↑𝑘))
52		simpr 485	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ 𝐴)
53	22	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 2 ∈ ℕ)
54		simplr 767	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ ℕ₀)
55	53, 54	nnexpcld 14148	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℕ)
56	55	nncnd 12169	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℂ)
57		oveq2 7365	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (2↑𝑘) = (2↑𝑁))
58	57	sumsn 15631	. . . . . 6 ⊢ ((𝑁 ∈ 𝐴 ∧ (2↑𝑁) ∈ ℂ) → Σ𝑘 ∈ {𝑁} (2↑𝑘) = (2↑𝑁))
59	52, 56, 58	syl2anc 584	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ {𝑁} (2↑𝑘) = (2↑𝑁))
60	51, 59	eqtr2d 2777	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
61		simpr 485	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → ¬ 𝑁 ∈ 𝐴)
62		disjsn 4672	. . . . . . 7 ⊢ ((𝐴 ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ 𝐴)
63	61, 62	sylibr 233	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → (𝐴 ∩ {𝑁}) = ∅)
64	63	sumeq1d 15586	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘) = Σ𝑘 ∈ ∅ (2↑𝑘))
65		sum0 15606	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (2↑𝑘) = 0
66	64, 65	eqtr2di 2793	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → 0 = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
67	60, 66	ifeqda 4522	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
68	46, 67	oveq12d 7375	. 2 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐴, (2↑𝑁), 0)) = (Σ𝑘 ∈ (𝐴 ∩ (0..^𝑁))(2↑𝑘) + Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘)))
69	30, 35, 68	3eqtr4d 2786	1 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐴, (2↑𝑁), 0)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∪ cun 3908 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 ifcif 4486 𝒫 cpw 4560 {csn 4586 ^◡ccnv 5632 ↾ cres 5635 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℂcc 11049 0cc0 11051 1c1 11052 + caddc 11054 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤ_≥cuz 12763 ..^cfzo 13567 ↑cexp 13967 Σcsu 15570 bitscbits 16299

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-dvds 16137 df-bits 16302

This theorem is referenced by: sadcaddlem 16337 sadadd2lem 16339

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