bitsinvp1 - Metamath Proof Explorer

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Theorem bitsinvp1 15619

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 12890	. . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
2	1	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ¬ 𝑁 ∈ (0..^𝑁))
3		disjsn 4548	. . . . . 6 ⊢ (((0..^𝑁) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0..^𝑁))
4	2, 3	sylibr 235	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑁) ∩ {𝑁}) = ∅)
5	4	ineq2d 4104	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ ((0..^𝑁) ∩ {𝑁})) = (𝐴 ∩ ∅))
6		inindi 4118	. . . 4 ⊢ (𝐴 ∩ ((0..^𝑁) ∩ {𝑁})) = ((𝐴 ∩ (0..^𝑁)) ∩ (𝐴 ∩ {𝑁}))
7		in0 4259	. . . 4 ⊢ (𝐴 ∩ ∅) = ∅
8	5, 6, 7	3eqtr3g 2852	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 ∩ (0..^𝑁)) ∩ (𝐴 ∩ {𝑁})) = ∅)
9		simpr 485	. . . . . . 7 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
10		nn0uz 12118	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	syl6eleq 2891	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
12		fzosplitsn 12983	. . . . . 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 4104	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) = (𝐴 ∩ ((0..^𝑁) ∪ {𝑁})))
15		indi 4165	. . . 4 ⊢ (𝐴 ∩ ((0..^𝑁) ∪ {𝑁})) = ((𝐴 ∩ (0..^𝑁)) ∪ (𝐴 ∩ {𝑁}))
16	14, 15	syl6eq 2845	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) = ((𝐴 ∩ (0..^𝑁)) ∪ (𝐴 ∩ {𝑁})))
17		fzofi 13180	. . . . 5 ⊢ (0..^(𝑁 + 1)) ∈ Fin
18	17	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑁 + 1)) ∈ Fin)
19		inss2 4121	. . . 4 ⊢ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ (0..^(𝑁 + 1))
20		ssfi 8574	. . . 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 11547	. . . . . 6 ⊢ 2 ∈ ℕ
23	22	a1i 11	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → 2 ∈ ℕ)
24		inss1 4120	. . . . . . 7 ⊢ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ 𝐴
25		simpl 483	. . . . . . 7 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ⊆ ℕ₀)
26	24, 25	syl5ss 3895	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ ℕ₀)
27	26	sselda 3884	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → 𝑘 ∈ ℕ₀)
28	23, 27	nnexpcld 13444	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → (2↑𝑘) ∈ ℕ)
29	28	nncnd 11491	. . 3 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → (2↑𝑘) ∈ ℂ)
30	8, 16, 21, 29	fsumsplit 14918	. 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 8662	. . . 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 15618	. . 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 4120	. . . . . 6 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
37	36, 25	syl5ss 3895	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀)
38		fzofi 13180	. . . . . . 7 ⊢ (0..^𝑁) ∈ Fin
39	38	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^𝑁) ∈ Fin)
40		inss2 4121	. . . . . 6 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
41		ssfi 8574	. . . . . 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 8662	. . . . 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 15618	. . . 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 4642	. . . . . . . 8 ⊢ (𝑁 ∈ 𝐴 → {𝑁} ⊆ 𝐴)
48	47	adantl 482	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → {𝑁} ⊆ 𝐴)
49		sseqin2 4107	. . . . . . 7 ⊢ ({𝑁} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑁}) = {𝑁})
50	48, 49	sylib 219	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (𝐴 ∩ {𝑁}) = {𝑁})
51	50	sumeq1d 14879	. . . . 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 765	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ ℕ₀)
55	53, 54	nnexpcld 13444	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℕ)
56	55	nncnd 11491	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℂ)
57		oveq2 7015	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (2↑𝑘) = (2↑𝑁))
58	57	sumsn 14922	. . . . . 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 2830	. . . 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 4548	. . . . . . 7 ⊢ ((𝐴 ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ 𝐴)
63	61, 62	sylibr 235	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → (𝐴 ∩ {𝑁}) = ∅)
64	63	sumeq1d 14879	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘) = Σ𝑘 ∈ ∅ (2↑𝑘))
65		sum0 14899	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (2↑𝑘) = 0
66	64, 65	syl6req 2846	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → 0 = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
67	60, 66	ifeqda 4410	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
68	46, 67	oveq12d 7025	. 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 2839	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 1520 ∈ wcel 2079 ∪ cun 3852 ∩ cin 3853 ⊆ wss 3854 ∅c0 4206 ifcif 4375 𝒫 cpw 4447 {csn 4466 ^◡ccnv 5434 ↾ cres 5437 ‘cfv 6217 (class class class)co 7007 Fincfn 8347 ℂcc 10370 0cc0 10372 1c1 10373 + caddc 10375 ℕcn 11475 2c2 11529 ℕ₀cn0 11734 ℤ_≥cuz 12082 ..^cfzo 12872 ↑cexp 13267 Σcsu 14864 bitscbits 15589

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-inf2 8939 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-disj 4925 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-2o 7945 df-oadd 7948 df-er 8130 df-map 8249 df-pm 8250 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-sup 8742 df-inf 8743 df-oi 8810 df-dju 9165 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-n0 11735 df-xnn0 11805 df-z 11819 df-uz 12083 df-rp 12229 df-fz 12732 df-fzo 12873 df-fl 13000 df-mod 13076 df-seq 13208 df-exp 13268 df-hash 13529 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-clim 14667 df-sum 14865 df-dvds 15429 df-bits 15592

This theorem is referenced by: sadcaddlem 15627 sadadd2lem 15629

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