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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 12427 | . . 3 ⊢ (2 · 1) = 2 | |
2 | 1 | oveq2i 7442 | . 2 ⊢ ((2 · (2 · 𝐴)) + (2 · 1)) = ((2 · (2 · 𝐴)) + 2) |
3 | 2cn 12339 | . . 3 ⊢ 2 ∈ ℂ | |
4 | decbin.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
5 | 4 | nn0cni 12536 | . . . 4 ⊢ 𝐴 ∈ ℂ |
6 | 3, 5 | mulcli 11266 | . . 3 ⊢ (2 · 𝐴) ∈ ℂ |
7 | ax-1cn 11211 | . . 3 ⊢ 1 ∈ ℂ | |
8 | 3, 6, 7 | adddii 11271 | . 2 ⊢ (2 · ((2 · 𝐴) + 1)) = ((2 · (2 · 𝐴)) + (2 · 1)) |
9 | 4 | decbin0 12871 | . . 3 ⊢ (4 · 𝐴) = (2 · (2 · 𝐴)) |
10 | 9 | oveq1i 7441 | . 2 ⊢ ((4 · 𝐴) + 2) = ((2 · (2 · 𝐴)) + 2) |
11 | 2, 8, 10 | 3eqtr4ri 2774 | 1 ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 (class class class)co 7431 1c1 11154 + caddc 11156 · cmul 11158 2c2 12319 4c4 12321 ℕ0cn0 12524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1rid 11223 ax-cnre 11226 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 |
This theorem is referenced by: decbin3 12873 |
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