decbin0 - Metamath Proof Explorer

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Theorem decbin0 12988

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 12533	. . 3 ⊢ (2 · 2) = 4
2	1	oveq1i 7453	. 2 ⊢ ((2 · 2) · 𝐴) = (4 · 𝐴)
3		2cn 12445	. . 3 ⊢ 2 ∈ ℂ
4		decbin.1	. . . 4 ⊢ 𝐴 ∈ ℕ₀
5	4	nn0cni 12645	. . 3 ⊢ 𝐴 ∈ ℂ
6	3, 3, 5	mulassi 11347	. 2 ⊢ ((2 · 2) · 𝐴) = (2 · (2 · 𝐴))
7	2, 6	eqtr3i 2788	1 ⊢ (4 · 𝐴) = (2 · (2 · 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2147 (class class class)co 7443 · cmul 11232 2c2 12424 4c4 12426 ℕ₀cn0 12633

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pr 5395 ax-un 7775 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-mulcl 11289 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1rid 11297 ax-cnre 11300

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-ov 7446 df-om 7902 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-n0 12634

This theorem is used by: decbin2 12989