decbin0 - Metamath Proof Explorer

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Theorem decbin0 12239

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 11802	. . 3 ⊢ (2 · 2) = 4
2	1	oveq1i 7166	. 2 ⊢ ((2 · 2) · 𝐴) = (4 · 𝐴)
3		2cn 11713	. . 3 ⊢ 2 ∈ ℂ
4		decbin.1	. . . 4 ⊢ 𝐴 ∈ ℕ₀
5	4	nn0cni 11910	. . 3 ⊢ 𝐴 ∈ ℂ
6	3, 3, 5	mulassi 10652	. 2 ⊢ ((2 · 2) · 𝐴) = (2 · (2 · 𝐴))
7	2, 6	eqtr3i 2846	1 ⊢ (4 · 𝐴) = (2 · (2 · 𝐴))

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 · cmul 10542 2c2 11693 4c4 11695 ℕ₀cn0 11898

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-mulcl 10599 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1rid 10607 ax-cnre 10610

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-n0 11899

This theorem is referenced by: decbin2 12240 decexp2 16411