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Theorem bitsinvp1 16418

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 13628	. . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
2	1	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ¬ 𝑁 ∈ (0..^𝑁))
3		disjsn 4655	. . . . . 6 ⊢ (((0..^𝑁) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0..^𝑁))
4	2, 3	sylibr 234	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑁) ∩ {𝑁}) = ∅)
5	4	ineq2d 4160	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ ((0..^𝑁) ∩ {𝑁})) = (𝐴 ∩ ∅))
6		inindi 4175	. . . 4 ⊢ (𝐴 ∩ ((0..^𝑁) ∩ {𝑁})) = ((𝐴 ∩ (0..^𝑁)) ∩ (𝐴 ∩ {𝑁}))
7		in0 4335	. . . 4 ⊢ (𝐴 ∩ ∅) = ∅
8	5, 6, 7	3eqtr3g 2794	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 ∩ (0..^𝑁)) ∩ (𝐴 ∩ {𝑁})) = ∅)
9		simpr 484	. . . . . . 7 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
10		nn0uz 12826	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtrdi 2846	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
12		fzosplitsn 13731	. . . . . 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 4160	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) = (𝐴 ∩ ((0..^𝑁) ∪ {𝑁})))
15		indi 4224	. . . 4 ⊢ (𝐴 ∩ ((0..^𝑁) ∪ {𝑁})) = ((𝐴 ∩ (0..^𝑁)) ∪ (𝐴 ∩ {𝑁}))
16	14, 15	eqtrdi 2787	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) = ((𝐴 ∩ (0..^𝑁)) ∪ (𝐴 ∩ {𝑁})))
17		fzofi 13936	. . . . 5 ⊢ (0..^(𝑁 + 1)) ∈ Fin
18	17	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑁 + 1)) ∈ Fin)
19		inss2 4178	. . . 4 ⊢ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ (0..^(𝑁 + 1))
20		ssfi 9107	. . . 4 ⊢ (((0..^(𝑁 + 1)) ∈ Fin ∧ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ (0..^(𝑁 + 1))) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin)
21	18, 19, 20	sylancl 587	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin)
22		2nn 12254	. . . . . 6 ⊢ 2 ∈ ℕ
23	22	a1i 11	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → 2 ∈ ℕ)
24		inss1 4177	. . . . . . 7 ⊢ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ 𝐴
25		simpl 482	. . . . . . 7 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ⊆ ℕ₀)
26	24, 25	sstrid 3933	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ ℕ₀)
27	26	sselda 3921	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → 𝑘 ∈ ℕ₀)
28	23, 27	nnexpcld 14207	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → (2↑𝑘) ∈ ℕ)
29	28	nncnd 12190	. . 3 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → (2↑𝑘) ∈ ℂ)
30	8, 16, 21, 29	fsumsplit 15703	. 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 9264	. . . 4 ⊢ ((𝐴 ∩ (0..^(𝑁 + 1))) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^(𝑁 + 1))) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin))
32	26, 21, 31	sylanbrc 584	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ (𝒫 ℕ₀ ∩ Fin))
33		bitsinv.k	. . . 4 ⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
34	33	bitsinv 16417	. . 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 4177	. . . . . 6 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
37	36, 25	sstrid 3933	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀)
38		fzofi 13936	. . . . . . 7 ⊢ (0..^𝑁) ∈ Fin
39	38	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^𝑁) ∈ Fin)
40		inss2 4178	. . . . . 6 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
41		ssfi 9107	. . . . . 6 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
42	39, 40, 41	sylancl 587	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
43		elfpw 9264	. . . . 5 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^𝑁)) ∈ Fin))
44	37, 42, 43	sylanbrc 584	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
45	33	bitsinv 16417	. . . 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 4729	. . . . . . . 8 ⊢ (𝑁 ∈ 𝐴 → {𝑁} ⊆ 𝐴)
48	47	adantl 481	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → {𝑁} ⊆ 𝐴)
49		sseqin2 4163	. . . . . . 7 ⊢ ({𝑁} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑁}) = {𝑁})
50	48, 49	sylib 218	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (𝐴 ∩ {𝑁}) = {𝑁})
51	50	sumeq1d 15662	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘) = Σ𝑘 ∈ {𝑁} (2↑𝑘))
52		simpr 484	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ 𝐴)
53	22	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 2 ∈ ℕ)
54		simplr 769	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ ℕ₀)
55	53, 54	nnexpcld 14207	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℕ)
56	55	nncnd 12190	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℂ)
57		oveq2 7375	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (2↑𝑘) = (2↑𝑁))
58	57	sumsn 15708	. . . . . 6 ⊢ ((𝑁 ∈ 𝐴 ∧ (2↑𝑁) ∈ ℂ) → Σ𝑘 ∈ {𝑁} (2↑𝑘) = (2↑𝑁))
59	52, 56, 58	syl2anc 585	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ {𝑁} (2↑𝑘) = (2↑𝑁))
60	51, 59	eqtr2d 2772	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
61		simpr 484	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → ¬ 𝑁 ∈ 𝐴)
62		disjsn 4655	. . . . . . 7 ⊢ ((𝐴 ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ 𝐴)
63	61, 62	sylibr 234	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → (𝐴 ∩ {𝑁}) = ∅)
64	63	sumeq1d 15662	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘) = Σ𝑘 ∈ ∅ (2↑𝑘))
65		sum0 15683	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (2↑𝑘) = 0
66	64, 65	eqtr2di 2788	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → 0 = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
67	60, 66	ifeqda 4503	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
68	46, 67	oveq12d 7385	. 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 2781	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 395 = wceq 1542 ∈ wcel 2114 ∪ cun 3887 ∩ cin 3888 ⊆ wss 3889 ∅c0 4273 ifcif 4466 𝒫 cpw 4541 {csn 4567 ^◡ccnv 5630 ↾ cres 5633 ‘cfv 6498 (class class class)co 7367 Fincfn 8893 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 ℕcn 12174 2c2 12236 ℕ₀cn0 12437 ℤ_≥cuz 12788 ..^cfzo 13608 ↑cexp 14023 Σcsu 15648 bitscbits 16388

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-xnn0 12511 df-z 12525 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-dvds 16222 df-bits 16391

This theorem is referenced by: sadcaddlem 16426 sadadd2lem 16428

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