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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 16307 | . . . 4 β’ bits = (π β β€ β¦ {π β β0 β£ Β¬ 2 β₯ (ββ(π / (2βπ)))}) | |
2 | 1 | mptrcl 6958 | . . 3 β’ (π β (bitsβπ) β π β β€) |
3 | bitsfval 16308 | . . . . 5 β’ (π β β€ β (bitsβπ) = {π β β0 β£ Β¬ 2 β₯ (ββ(π / (2βπ)))}) | |
4 | 3 | eleq2d 2820 | . . . 4 β’ (π β β€ β (π β (bitsβπ) β π β {π β β0 β£ Β¬ 2 β₯ (ββ(π / (2βπ)))})) |
5 | oveq2 7366 | . . . . . . . . 9 β’ (π = π β (2βπ) = (2βπ)) | |
6 | 5 | oveq2d 7374 | . . . . . . . 8 β’ (π = π β (π / (2βπ)) = (π / (2βπ))) |
7 | 6 | fveq2d 6847 | . . . . . . 7 β’ (π = π β (ββ(π / (2βπ))) = (ββ(π / (2βπ)))) |
8 | 7 | breq2d 5118 | . . . . . 6 β’ (π = π β (2 β₯ (ββ(π / (2βπ))) β 2 β₯ (ββ(π / (2βπ))))) |
9 | 8 | notbid 318 | . . . . 5 β’ (π = π β (Β¬ 2 β₯ (ββ(π / (2βπ))) β Β¬ 2 β₯ (ββ(π / (2βπ))))) |
10 | 9 | elrab 3646 | . . . 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 287 | . . 3 β’ (π β β€ β (π β (bitsβπ) β (π β β0 β§ Β¬ 2 β₯ (ββ(π / (2βπ)))))) |
12 | 2, 11 | biadanii 821 | . 2 β’ (π β (bitsβπ) β (π β β€ β§ (π β β0 β§ Β¬ 2 β₯ (ββ(π / (2βπ)))))) |
13 | 3anass 1096 | . 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 278 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 {crab 3406 class class class wbr 5106 βcfv 6497 (class class class)co 7358 / cdiv 11817 2c2 12213 β0cn0 12418 β€cz 12504 βcfl 13701 βcexp 13973 β₯ cdvds 16141 bitscbits 16304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-1cn 11114 ax-addcl 11116 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-nn 12159 df-n0 12419 df-bits 16307 |
This theorem is referenced by: bitsval2 16310 bitsss 16311 bitsfzo 16320 bitsmod 16321 bitscmp 16323 |
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