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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 9089 | . 2 ⊢ ;1𝐴 = ((;10 · 1) + 𝐴) | |
2 | 10nn 9101 | . . . . 5 ⊢ ;10 ∈ ℕ | |
3 | 2 | nncni 8640 | . . . 4 ⊢ ;10 ∈ ℂ |
4 | 3 | mulid1i 7692 | . . 3 ⊢ (;10 · 1) = ;10 |
5 | 4 | oveq1i 5738 | . 2 ⊢ ((;10 · 1) + 𝐴) = (;10 + 𝐴) |
6 | 1, 5 | eqtr2i 2136 | 1 ⊢ (;10 + 𝐴) = ;1𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1314 (class class class)co 5728 0cc0 7547 1c1 7548 + caddc 7550 · cmul 7552 ;cdc 9086 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-mulcom 7646 ax-addass 7647 ax-mulass 7648 ax-distr 7649 ax-1rid 7652 ax-0id 7653 ax-cnre 7656 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-iota 5046 df-fv 5089 df-ov 5731 df-inn 8631 df-2 8689 df-3 8690 df-4 8691 df-5 8692 df-6 8693 df-7 8694 df-8 8695 df-9 8696 df-dec 9087 |
This theorem is referenced by: 5t3e15 9186 |
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