Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > bitsval | Structured version Visualization version GIF version |
Description: Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
bitsval | ⊢ (𝑀 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑀))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bits 15774 | . . . 4 ⊢ bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚)))}) | |
2 | 1 | mptrcl 6780 | . . 3 ⊢ (𝑀 ∈ (bits‘𝑁) → 𝑁 ∈ ℤ) |
3 | bitsfval 15775 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (bits‘𝑁) = {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))}) | |
4 | 3 | eleq2d 2901 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑀 ∈ (bits‘𝑁) ↔ 𝑀 ∈ {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))})) |
5 | oveq2 7167 | . . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (2↑𝑚) = (2↑𝑀)) | |
6 | 5 | oveq2d 7175 | . . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑁 / (2↑𝑚)) = (𝑁 / (2↑𝑀))) |
7 | 6 | fveq2d 6677 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → (⌊‘(𝑁 / (2↑𝑚))) = (⌊‘(𝑁 / (2↑𝑀)))) |
8 | 7 | breq2d 5081 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (2 ∥ (⌊‘(𝑁 / (2↑𝑚))) ↔ 2 ∥ (⌊‘(𝑁 / (2↑𝑀))))) |
9 | 8 | notbid 320 | . . . . 5 ⊢ (𝑚 = 𝑀 → (¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))) ↔ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑀))))) |
10 | 9 | elrab 3683 | . . . 4 ⊢ (𝑀 ∈ {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))} ↔ (𝑀 ∈ ℕ0 ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑀))))) |
11 | 4, 10 | syl6bb 289 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑀 ∈ (bits‘𝑁) ↔ (𝑀 ∈ ℕ0 ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑀)))))) |
12 | 2, 11 | biadanii 820 | . 2 ⊢ (𝑀 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0 ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑀)))))) |
13 | 3anass 1091 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑀)))) ↔ (𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0 ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑀)))))) | |
14 | 12, 13 | bitr4i 280 | 1 ⊢ (𝑀 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑀))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 {crab 3145 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 / cdiv 11300 2c2 11695 ℕ0cn0 11900 ℤcz 11984 ⌊cfl 13163 ↑cexp 13432 ∥ cdvds 15610 bitscbits 15771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-1cn 10598 ax-addcl 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-nn 11642 df-n0 11901 df-bits 15774 |
This theorem is referenced by: bitsval2 15777 bitsss 15778 bitsfzo 15787 bitsmod 15788 bitscmp 15790 |
Copyright terms: Public domain | W3C validator |