decbin0 - Metamath Proof Explorer

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Theorem decbin0 12796

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 12355	. . 3 ⊢ (2 · 2) = 4
2	1	oveq1i 7400	. 2 ⊢ ((2 · 2) · 𝐴) = (4 · 𝐴)
3		2cn 12266	. . 3 ⊢ 2 ∈ ℂ
4		decbin.1	. . . 4 ⊢ 𝐴 ∈ ℕ₀
5	4	nn0cni 12463	. . 3 ⊢ 𝐴 ∈ ℂ
6	3, 3, 5	mulassi 11204	. 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 7390 · cmul 11094 2c2 12246 4c4 12248 ℕ₀cn0 12451

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 5289 ax-nul 5296 ax-pr 5417 ax-un 7705 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-mulcl 11151 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1rid 11159 ax-cnre 11162

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 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-ov 7393 df-om 7836 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-n0 12452

This theorem is referenced by: decbin2 12797 decexp2 16987