bitsinvp1 - Metamath Proof Explorer

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Theorem bitsinvp1 16532

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 13721	. . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
2	1	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ¬ 𝑁 ∈ (0..^𝑁))
3		disjsn 4682	. . . . . 6 ⊢ (((0..^𝑁) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0..^𝑁))
4	2, 3	sylibr 237	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑁) ∩ {𝑁}) = ∅)
5	4	ineq2d 4176	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ ((0..^𝑁) ∩ {𝑁})) = (𝐴 ∩ ∅))
6		inindi 4190	. . . 4 ⊢ (𝐴 ∩ ((0..^𝑁) ∩ {𝑁})) = ((𝐴 ∩ (0..^𝑁)) ∩ (𝐴 ∩ {𝑁}))
7		in0 4355	. . . 4 ⊢ (𝐴 ∩ ∅) = ∅
8	5, 6, 7	3eqtr3g 2824	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 ∩ (0..^𝑁)) ∩ (𝐴 ∩ {𝑁})) = ∅)
9		simpr 490	. . . . . . 7 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
10		nn0uz 12918	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtrdi 2876	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
12		fzosplitsn 13824	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
13	11, 12	syl 18	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
14	13	ineq2d 4176	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) = (𝐴 ∩ ((0..^𝑁) ∪ {𝑁})))
15		indi 4240	. . . 4 ⊢ (𝐴 ∩ ((0..^𝑁) ∪ {𝑁})) = ((𝐴 ∩ (0..^𝑁)) ∪ (𝐴 ∩ {𝑁}))
16	14, 15	eqtrdi 2817	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) = ((𝐴 ∩ (0..^𝑁)) ∪ (𝐴 ∩ {𝑁})))
17		fzofi 14030	. . . . 5 ⊢ (0..^(𝑁 + 1)) ∈ Fin
18	17	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑁 + 1)) ∈ Fin)
19		inss2 4193	. . . 4 ⊢ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ (0..^(𝑁 + 1))
20		ssfi 9167	. . . 4 ⊢ (((0..^(𝑁 + 1)) ∈ Fin ∧ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ (0..^(𝑁 + 1))) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin)
21	18, 19, 20	sylancl 598	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin)
22		2nn 12332	. . . . . 6 ⊢ 2 ∈ ℕ
23	22	a1i 11	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → 2 ∈ ℕ)
24		inss1 4192	. . . . . . 7 ⊢ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ 𝐴
25		simpl 488	. . . . . . 7 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ⊆ ℕ₀)
26	24, 25	sstrid 3951	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ ℕ₀)
27	26	sselda 3940	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → 𝑘 ∈ ℕ₀)
28	23, 27	nnexpcld 14301	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → (2↑𝑘) ∈ ℕ)
29	28	nncnd 12267	. . 3 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → (2↑𝑘) ∈ ℂ)
30	8, 16, 21, 29	fsumsplit 15818	. 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 9321	. . . 4 ⊢ ((𝐴 ∩ (0..^(𝑁 + 1))) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^(𝑁 + 1))) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin))
32	26, 21, 31	sylanbrc 595	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ (𝒫 ℕ₀ ∩ Fin))
33		bitsinv.k	. . . 4 ⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
34	33	bitsinv 16531	. . 3 ⊢ ((𝐴 ∩ (0..^(𝑁 + 1))) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = Σ𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))(2↑𝑘))
35	32, 34	syl 18	. 2 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = Σ𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))(2↑𝑘))
36		inss1 4192	. . . . . 6 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
37	36, 25	sstrid 3951	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀)
38		fzofi 14030	. . . . . . 7 ⊢ (0..^𝑁) ∈ Fin
39	38	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^𝑁) ∈ Fin)
40		inss2 4193	. . . . . 6 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
41		ssfi 9167	. . . . . 6 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
42	39, 40, 41	sylancl 598	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
43		elfpw 9321	. . . . 5 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^𝑁)) ∈ Fin))
44	37, 42, 43	sylanbrc 595	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
45	33	bitsinv 16531	. . . 4 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘(𝐴 ∩ (0..^𝑁))) = Σ𝑘 ∈ (𝐴 ∩ (0..^𝑁))(2↑𝑘))
46	44, 45	syl 18	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾‘(𝐴 ∩ (0..^𝑁))) = Σ𝑘 ∈ (𝐴 ∩ (0..^𝑁))(2↑𝑘))
47		snssi 4756	. . . . . . . 8 ⊢ (𝑁 ∈ 𝐴 → {𝑁} ⊆ 𝐴)
48	47	adantl 487	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → {𝑁} ⊆ 𝐴)
49		sseqin2 4179	. . . . . . 7 ⊢ ({𝑁} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑁}) = {𝑁})
50	48, 49	sylib 221	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (𝐴 ∩ {𝑁}) = {𝑁})
51	50	sumeq1d 15777	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘) = Σ𝑘 ∈ {𝑁} (2↑𝑘))
52		simpr 490	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ 𝐴)
53	22	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 2 ∈ ℕ)
54		simplr 781	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ ℕ₀)
55	53, 54	nnexpcld 14301	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℕ)
56	55	nncnd 12267	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℂ)
57		oveq2 7431	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (2↑𝑘) = (2↑𝑁))
58	57	sumsn 15823	. . . . . 6 ⊢ ((𝑁 ∈ 𝐴 ∧ (2↑𝑁) ∈ ℂ) → Σ𝑘 ∈ {𝑁} (2↑𝑘) = (2↑𝑁))
59	52, 56, 58	syl2anc 596	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ {𝑁} (2↑𝑘) = (2↑𝑁))
60	51, 59	eqtr2d 2802	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
61		simpr 490	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → ¬ 𝑁 ∈ 𝐴)
62		disjsn 4682	. . . . . . 7 ⊢ ((𝐴 ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ 𝐴)
63	61, 62	sylibr 237	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → (𝐴 ∩ {𝑁}) = ∅)
64	63	sumeq1d 15777	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘) = Σ𝑘 ∈ ∅ (2↑𝑘))
65		sum0 15798	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (2↑𝑘) = 0
66	64, 65	eqtr2di 2818	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → 0 = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
67	60, 66	ifeqda 4529	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
68	46, 67	oveq12d 7441	. 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 2811	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∪ cun 3906 ∩ cin 3907 ⊆ wss 3908 ∅c0 4289 ifcif 4492 𝒫 cpw 4567 {csn 4594 ^◡ccnv 5665 ↾ cres 5668 ‘cfv 6543 (class class class)co 7423 Fincfn 8952 ℂcc 11116 0cc0 11118 1c1 11119 + caddc 11121 ℕcn 12251 2c2 12313 ℕ₀cn0 12522 ℤ_≥cuz 12880 ..^cfzo 13701 ↑cexp 14117 Σcsu 15763 bitscbits 16502

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-disj 5082 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8703 df-map 8835 df-pm 8836 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-xnn0 12596 df-z 12610 df-uz 12881 df-rp 13035 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-sum 15764 df-dvds 16336 df-bits 16505

This theorem is used by: sadcaddlem 16540 sadadd2lem 16542

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