dec0u - Intuitionistic Logic Explorer

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Theorem dec0u 9403

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 9400	. . 3 ⊢ ;10 ∈ ℕ₀
2		dec0u.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
3	1, 2	num0u 9393	. 2 ⊢ (;10 · 𝐴) = ((;10 · 𝐴) + 0)
4		dfdec10 9386	. 2 ⊢ ;𝐴0 = ((;10 · 𝐴) + 0)
5	3, 4	eqtr4i 2201	1 ⊢ (;10 · 𝐴) = ;𝐴0

Colors of variables: wff set class

Syntax hints: = wceq 1353 ∈ wcel 2148 (class class class)co 5874 0cc0 7810 1c1 7811 + caddc 7813 · cmul 7815 ℕ₀cn0 9175 cdc 9383

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-sub 8129 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-5 8980 df-6 8981 df-7 8982 df-8 8983 df-9 8984 df-n0 9176 df-dec 9384

This theorem is referenced by: decmul10add 9451 5t5e25 9485 6t6e36 9490 8t6e48 9501