dec0u - Intuitionistic Logic Explorer

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Theorem dec0u 9614

Description: Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)

**Hypothesis**
Ref	Expression
dec0u.1	⊢ 𝐴 ∈ ℕ₀

**Assertion**
Ref	Expression
dec0u	⊢ (;10 · 𝐴) = ;𝐴0

**Proof of Theorem dec0u**
Step	Hyp	Ref	Expression
1		10nn0 9611	. . 3 ⊢ ;10 ∈ ℕ₀
2		dec0u.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
3	1, 2	num0u 9604	. 2 ⊢ (;10 · 𝐴) = ((;10 · 𝐴) + 0)
4		dfdec10 9597	. 2 ⊢ ;𝐴0 = ((;10 · 𝐴) + 0)
5	3, 4	eqtr4i 2253	1 ⊢ (;10 · 𝐴) = ;𝐴0

Colors of variables: wff set class

Syntax hints: = wceq 1395 ∈ wcel 2200 (class class class)co 6010 0cc0 8015 1c1 8016 + caddc 8018 · cmul 8020 ℕ₀cn0 9385 cdc 9594

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4259 ax-pr 4294 ax-setind 4630 ax-cnex 8106 ax-resscn 8107 ax-1cn 8108 ax-1re 8109 ax-icn 8110 ax-addcl 8111 ax-addrcl 8112 ax-mulcl 8113 ax-addcom 8115 ax-mulcom 8116 ax-addass 8117 ax-mulass 8118 ax-distr 8119 ax-i2m1 8120 ax-1rid 8122 ax-0id 8123 ax-rnegex 8124 ax-cnre 8126

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-br 4084 df-opab 4146 df-id 4385 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-iota 5281 df-fun 5323 df-fv 5329 df-riota 5963 df-ov 6013 df-oprab 6014 df-mpo 6015 df-sub 8335 df-inn 9127 df-2 9185 df-3 9186 df-4 9187 df-5 9188 df-6 9189 df-7 9190 df-8 9191 df-9 9192 df-n0 9386 df-dec 9595

This theorem is referenced by: decmul10add 9662 5t5e25 9696 6t6e36 9701 8t6e48 9712