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Mirrors > Home > ILE Home > Th. List > dec10p | GIF version |
Description: Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
dec10p | ⊢ (;10 + 𝐴) = ;1𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdec10 9417 | . 2 ⊢ ;1𝐴 = ((;10 · 1) + 𝐴) | |
2 | 10nn 9429 | . . . . 5 ⊢ ;10 ∈ ℕ | |
3 | 2 | nncni 8959 | . . . 4 ⊢ ;10 ∈ ℂ |
4 | 3 | mulid1i 7989 | . . 3 ⊢ (;10 · 1) = ;10 |
5 | 4 | oveq1i 5906 | . 2 ⊢ ((;10 · 1) + 𝐴) = (;10 + 𝐴) |
6 | 1, 5 | eqtr2i 2211 | 1 ⊢ (;10 + 𝐴) = ;1𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 (class class class)co 5896 0cc0 7841 1c1 7842 + caddc 7844 · cmul 7846 ;cdc 9414 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-sep 4136 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-1rid 7948 ax-0id 7949 ax-cnre 7952 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-iota 5196 df-fv 5243 df-ov 5899 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-5 9011 df-6 9012 df-7 9013 df-8 9014 df-9 9015 df-dec 9415 |
This theorem is referenced by: 5t3e15 9514 |
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