decbin0 - Metamath Proof Explorer

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Theorem decbin0 12886

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 12431	. . 3 ⊢ (2 · 2) = 4
2	1	oveq1i 7424	. 2 ⊢ ((2 · 2) · 𝐴) = (4 · 𝐴)
3		2cn 12343	. . 3 ⊢ 2 ∈ ℂ
4		decbin.1	. . . 4 ⊢ 𝐴 ∈ ℕ₀
5	4	nn0cni 12543	. . 3 ⊢ 𝐴 ∈ ℂ
6	3, 3, 5	mulassi 11247	. 2 ⊢ ((2 · 2) · 𝐴) = (2 · (2 · 𝐴))
7	2, 6	eqtr3i 2785	1 ⊢ (4 · 𝐴) = (2 · (2 · 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2145 (class class class)co 7414 · cmul 11132 2c2 12322 4c4 12324 ℕ₀cn0 12531

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-un 7737 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-mulcl 11189 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1rid 11197 ax-cnre 11200

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 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 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7417 df-om 7864 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-n0 12532

This theorem is used by: decbin2 12887