![]() |
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 | β’ (π β (bitsβπ) β (π β β€ β§ π β β0 β§ Β¬ 2 β₯ (ββ(π / (2βπ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bits 16404 | . . . 4 β’ bits = (π β β€ β¦ {π β β0 β£ Β¬ 2 β₯ (ββ(π / (2βπ)))}) | |
2 | 1 | mptrcl 7019 | . . 3 β’ (π β (bitsβπ) β π β β€) |
3 | bitsfval 16405 | . . . . 5 β’ (π β β€ β (bitsβπ) = {π β β0 β£ Β¬ 2 β₯ (ββ(π / (2βπ)))}) | |
4 | 3 | eleq2d 2815 | . . . 4 β’ (π β β€ β (π β (bitsβπ) β π β {π β β0 β£ Β¬ 2 β₯ (ββ(π / (2βπ)))})) |
5 | oveq2 7434 | . . . . . . . . 9 β’ (π = π β (2βπ) = (2βπ)) | |
6 | 5 | oveq2d 7442 | . . . . . . . 8 β’ (π = π β (π / (2βπ)) = (π / (2βπ))) |
7 | 6 | fveq2d 6906 | . . . . . . 7 β’ (π = π β (ββ(π / (2βπ))) = (ββ(π / (2βπ)))) |
8 | 7 | breq2d 5164 | . . . . . 6 β’ (π = π β (2 β₯ (ββ(π / (2βπ))) β 2 β₯ (ββ(π / (2βπ))))) |
9 | 8 | notbid 317 | . . . . 5 β’ (π = π β (Β¬ 2 β₯ (ββ(π / (2βπ))) β Β¬ 2 β₯ (ββ(π / (2βπ))))) |
10 | 9 | elrab 3684 | . . . 4 β’ (π β {π β β0 β£ Β¬ 2 β₯ (ββ(π / (2βπ)))} β (π β β0 β§ Β¬ 2 β₯ (ββ(π / (2βπ))))) |
11 | 4, 10 | bitrdi 286 | . . 3 β’ (π β β€ β (π β (bitsβπ) β (π β β0 β§ Β¬ 2 β₯ (ββ(π / (2βπ)))))) |
12 | 2, 11 | biadanii 820 | . 2 β’ (π β (bitsβπ) β (π β β€ β§ (π β β0 β§ Β¬ 2 β₯ (ββ(π / (2βπ)))))) |
13 | 3anass 1092 | . 2 β’ ((π β β€ β§ π β β0 β§ Β¬ 2 β₯ (ββ(π / (2βπ)))) β (π β β€ β§ (π β β0 β§ Β¬ 2 β₯ (ββ(π / (2βπ)))))) | |
14 | 12, 13 | bitr4i 277 | 1 β’ (π β (bitsβπ) β (π β β€ β§ π β β0 β§ Β¬ 2 β₯ (ββ(π / (2βπ))))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 {crab 3430 class class class wbr 5152 βcfv 6553 (class class class)co 7426 / cdiv 11909 2c2 12305 β0cn0 12510 β€cz 12596 βcfl 13795 βcexp 14066 β₯ cdvds 16238 bitscbits 16401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-1cn 11204 ax-addcl 11206 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-nn 12251 df-n0 12511 df-bits 16404 |
This theorem is referenced by: bitsval2 16407 bitsss 16408 bitsfzo 16417 bitsmod 16418 bitscmp 16420 |
Copyright terms: Public domain | W3C validator |