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Theorem bitsinvp1 15171

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 12483	. . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
2	1	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ¬ 𝑁 ∈ (0..^𝑁))
3		disjsn 4246	. . . . . 6 ⊢ (((0..^𝑁) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0..^𝑁))
4	2, 3	sylibr 224	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑁) ∩ {𝑁}) = ∅)
5	4	ineq2d 3814	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ ((0..^𝑁) ∩ {𝑁})) = (𝐴 ∩ ∅))
6		inindi 3830	. . . 4 ⊢ (𝐴 ∩ ((0..^𝑁) ∩ {𝑁})) = ((𝐴 ∩ (0..^𝑁)) ∩ (𝐴 ∩ {𝑁}))
7		in0 3968	. . . 4 ⊢ (𝐴 ∩ ∅) = ∅
8	5, 6, 7	3eqtr3g 2679	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 ∩ (0..^𝑁)) ∩ (𝐴 ∩ {𝑁})) = ∅)
9		simpr 477	. . . . . . 7 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
10		nn0uz 11722	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	syl6eleq 2711	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
12		fzosplitsn 12576	. . . . . 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 3814	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) = (𝐴 ∩ ((0..^𝑁) ∪ {𝑁})))
15		indi 3873	. . . 4 ⊢ (𝐴 ∩ ((0..^𝑁) ∪ {𝑁})) = ((𝐴 ∩ (0..^𝑁)) ∪ (𝐴 ∩ {𝑁}))
16	14, 15	syl6eq 2672	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) = ((𝐴 ∩ (0..^𝑁)) ∪ (𝐴 ∩ {𝑁})))
17		fzofi 12773	. . . . 5 ⊢ (0..^(𝑁 + 1)) ∈ Fin
18	17	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑁 + 1)) ∈ Fin)
19		inss2 3834	. . . 4 ⊢ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ (0..^(𝑁 + 1))
20		ssfi 8180	. . . 4 ⊢ (((0..^(𝑁 + 1)) ∈ Fin ∧ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ (0..^(𝑁 + 1))) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin)
21	18, 19, 20	sylancl 694	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin)
22		2nn 11185	. . . . . 6 ⊢ 2 ∈ ℕ
23	22	a1i 11	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → 2 ∈ ℕ)
24		inss1 3833	. . . . . . 7 ⊢ (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ 𝐴
25		simpl 473	. . . . . . 7 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ⊆ ℕ₀)
26	24, 25	syl5ss 3614	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ⊆ ℕ₀)
27	26	sselda 3603	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → 𝑘 ∈ ℕ₀)
28	23, 27	nnexpcld 13030	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → (2↑𝑘) ∈ ℕ)
29	28	nncnd 11036	. . 3 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (𝐴 ∩ (0..^(𝑁 + 1)))) → (2↑𝑘) ∈ ℂ)
30	8, 16, 21, 29	fsumsplit 14471	. 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 8268	. . . 4 ⊢ ((𝐴 ∩ (0..^(𝑁 + 1))) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^(𝑁 + 1))) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^(𝑁 + 1))) ∈ Fin))
32	26, 21, 31	sylanbrc 698	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^(𝑁 + 1))) ∈ (𝒫 ℕ₀ ∩ Fin))
33		bitsinv.k	. . . 4 ⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
34	33	bitsinv 15170	. . 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 3833	. . . . . 6 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
37	36, 25	syl5ss 3614	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀)
38		fzofi 12773	. . . . . . 7 ⊢ (0..^𝑁) ∈ Fin
39	38	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^𝑁) ∈ Fin)
40		inss2 3834	. . . . . 6 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
41		ssfi 8180	. . . . . 6 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
42	39, 40, 41	sylancl 694	. . . . 5 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
43		elfpw 8268	. . . . 5 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^𝑁)) ∈ Fin))
44	37, 42, 43	sylanbrc 698	. . . 4 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
45	33	bitsinv 15170	. . . 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 4339	. . . . . . . 8 ⊢ (𝑁 ∈ 𝐴 → {𝑁} ⊆ 𝐴)
48	47	adantl 482	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → {𝑁} ⊆ 𝐴)
49		sseqin2 3817	. . . . . . 7 ⊢ ({𝑁} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑁}) = {𝑁})
50	48, 49	sylib 208	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (𝐴 ∩ {𝑁}) = {𝑁})
51	50	sumeq1d 14431	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘) = Σ𝑘 ∈ {𝑁} (2↑𝑘))
52		simpr 477	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ 𝐴)
53	22	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 2 ∈ ℕ)
54		simplr 792	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ ℕ₀)
55	53, 54	nnexpcld 13030	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℕ)
56	55	nncnd 11036	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℂ)
57		oveq2 6658	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (2↑𝑘) = (2↑𝑁))
58	57	sumsn 14475	. . . . . 6 ⊢ ((𝑁 ∈ 𝐴 ∧ (2↑𝑁) ∈ ℂ) → Σ𝑘 ∈ {𝑁} (2↑𝑘) = (2↑𝑁))
59	52, 56, 58	syl2anc 693	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ {𝑁} (2↑𝑘) = (2↑𝑁))
60	51, 59	eqtr2d 2657	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
61		simpr 477	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → ¬ 𝑁 ∈ 𝐴)
62		disjsn 4246	. . . . . . 7 ⊢ ((𝐴 ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ 𝐴)
63	61, 62	sylibr 224	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → (𝐴 ∩ {𝑁}) = ∅)
64	63	sumeq1d 14431	. . . . 5 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘) = Σ𝑘 ∈ ∅ (2↑𝑘))
65		sum0 14452	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (2↑𝑘) = 0
66	64, 65	syl6req 2673	. . . 4 ⊢ (((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ ¬ 𝑁 ∈ 𝐴) → 0 = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
67	60, 66	ifeqda 4121	. . 3 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = Σ𝑘 ∈ (𝐴 ∩ {𝑁})(2↑𝑘))
68	46, 67	oveq12d 6668	. 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 2666	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 384 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 ifcif 4086 𝒫 cpw 4158 {csn 4177 ^◡ccnv 5113 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤ_≥cuz 11687 ..^cfzo 12465 ↑cexp 12860 Σcsu 14416 bitscbits 15141

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-bits 15144

This theorem is referenced by: sadcaddlem 15179 sadadd2lem 15181

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