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Mirrors > Home > MPE Home > Th. List > decbin2 | Structured version Visualization version GIF version |
Description: Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
decbin.1 | ⊢ 𝐴 ∈ ℕ0 |
Ref | Expression |
---|---|
decbin2 | ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2t1e2 11788 | . . 3 ⊢ (2 · 1) = 2 | |
2 | 1 | oveq2i 7156 | . 2 ⊢ ((2 · (2 · 𝐴)) + (2 · 1)) = ((2 · (2 · 𝐴)) + 2) |
3 | 2cn 11700 | . . 3 ⊢ 2 ∈ ℂ | |
4 | decbin.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
5 | 4 | nn0cni 11897 | . . . 4 ⊢ 𝐴 ∈ ℂ |
6 | 3, 5 | mulcli 10636 | . . 3 ⊢ (2 · 𝐴) ∈ ℂ |
7 | ax-1cn 10583 | . . 3 ⊢ 1 ∈ ℂ | |
8 | 3, 6, 7 | adddii 10641 | . 2 ⊢ (2 · ((2 · 𝐴) + 1)) = ((2 · (2 · 𝐴)) + (2 · 1)) |
9 | 4 | decbin0 12226 | . . 3 ⊢ (4 · 𝐴) = (2 · (2 · 𝐴)) |
10 | 9 | oveq1i 7155 | . 2 ⊢ ((4 · 𝐴) + 2) = ((2 · (2 · 𝐴)) + 2) |
11 | 2, 8, 10 | 3eqtr4ri 2852 | 1 ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 1c1 10526 + caddc 10528 · cmul 10530 2c2 11680 4c4 11682 ℕ0cn0 11885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-mulcl 10587 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1rid 10595 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 |
This theorem is referenced by: decbin3 12228 |
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