Theorem bitsfval 12505

Description: Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)

Assertion

Ref	Expression
bitsfval	⊢ (𝑁 ∈ ℤ → (bits‘𝑁) = {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))})

Distinct variable group:   𝑚,𝑁

Proof of Theorem bitsfval

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvoveq1 6041	. . . . 5 ⊢ (𝑛 = 𝑁 → (⌊‘(𝑛 / (2↑𝑚))) = (⌊‘(𝑁 / (2↑𝑚))))
2	1	breq2d 4100	. . . 4 ⊢ (𝑛 = 𝑁 → (2 ∥ (⌊‘(𝑛 / (2↑𝑚))) ↔ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
3	2	notbid 673	. . 3 ⊢ (𝑛 = 𝑁 → (¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚))) ↔ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
4	3	rabbidv 2791	. 2 ⊢ (𝑛 = 𝑁 → {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚)))} = {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))})
5		df-bits 12504	. 2 ⊢ bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚)))})
6		nn0ex 9408	. . 3 ⊢ ℕ0 ∈ V
7	6	rabex 4234	. 2 ⊢ {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))} ∈ V
8	4, 5, 7	fvmpt 5723	1 ⊢ (𝑁 ∈ ℤ → (bits‘𝑁) = {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1397   ∈ wcel 2202  {crab 2514   class class class wbr 4088  ‘cfv 5326  (class class class)co 6018   / cdiv 8852  2c2 9194  ℕ₀cn0 9402  ℤcz 9479  ⌊cfl 10529  ↑cexp 10801   ∥ cdvds 12350  bitscbits 12503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-i2m1 8137

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6021  df-inn 9144  df-n0 9403  df-bits 12504

This theorem is referenced by:  bitsval  12506