decbin0 - Metamath Proof Explorer

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Theorem decbin0 12794

Description: Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)

**Hypothesis**
Ref	Expression
decbin.1	⊢ 𝐴 ∈ ℕ₀

**Assertion**
Ref	Expression
decbin0	⊢ (4 · 𝐴) = (2 · (2 · 𝐴))

**Proof of Theorem decbin0**
Step	Hyp	Ref	Expression
1		2t2e4 12353	. . 3 ⊢ (2 · 2) = 4
2	1	oveq1i 7398	. 2 ⊢ ((2 · 2) · 𝐴) = (4 · 𝐴)
3		2cn 12264	. . 3 ⊢ 2 ∈ ℂ
4		decbin.1	. . . 4 ⊢ 𝐴 ∈ ℕ₀
5	4	nn0cni 12461	. . 3 ⊢ 𝐴 ∈ ℂ
6	3, 3, 5	mulassi 11202	. 2 ⊢ ((2 · 2) · 𝐴) = (2 · (2 · 𝐴))
7	2, 6	eqtr3i 2761	1 ⊢ (4 · 𝐴) = (2 · (2 · 𝐴))

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2106 (class class class)co 7388 · cmul 11092 2c2 12244 4c4 12246 ℕ₀cn0 12449

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5287 ax-nul 5294 ax-pr 5415 ax-un 7703 ax-resscn 11144 ax-1cn 11145 ax-icn 11146 ax-addcl 11147 ax-mulcl 11149 ax-mulcom 11151 ax-addass 11152 ax-mulass 11153 ax-distr 11154 ax-i2m1 11155 ax-1rid 11157 ax-cnre 11160

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3375 df-rab 3429 df-v 3471 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4314 df-if 4518 df-pw 4593 df-sn 4618 df-pr 4620 df-op 4624 df-uni 4897 df-iun 4987 df-br 5137 df-opab 5199 df-mpt 5220 df-tr 5254 df-id 5562 df-eprel 5568 df-po 5576 df-so 5577 df-fr 5619 df-we 5621 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6284 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-ov 7391 df-om 7834 df-2nd 7953 df-frecs 8243 df-wrecs 8274 df-recs 8348 df-rdg 8387 df-nn 12190 df-2 12252 df-3 12253 df-4 12254 df-n0 12450

This theorem is referenced by: decbin2 12795 decexp2 16985