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Mirrors > Home > ILE Home > Th. List > dfdec10 | GIF version |
Description: Version of the definition of the "decimal constructor" using ;10 instead of the symbol 10. Of course, this statement cannot be used as definition, because it uses the "decimal constructor". (Contributed by AV, 1-Aug-2021.) |
Ref | Expression |
---|---|
dfdec10 | ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dec 9035 | . 2 ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) | |
2 | 9p1e10 9036 | . . . 4 ⊢ (9 + 1) = ;10 | |
3 | 2 | oveq1i 5716 | . . 3 ⊢ ((9 + 1) · 𝐴) = (;10 · 𝐴) |
4 | 3 | oveq1i 5716 | . 2 ⊢ (((9 + 1) · 𝐴) + 𝐵) = ((;10 · 𝐴) + 𝐵) |
5 | 1, 4 | eqtri 2120 | 1 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1299 (class class class)co 5706 0cc0 7500 1c1 7501 + caddc 7503 · cmul 7505 9c9 8636 ;cdc 9034 |
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 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-1rid 7602 ax-0id 7603 ax-cnre 7606 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-iota 5024 df-fv 5067 df-ov 5709 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-5 8640 df-6 8641 df-7 8642 df-8 8643 df-9 8644 df-dec 9035 |
This theorem is referenced by: decnncl 9053 dec0u 9054 dec0h 9055 decnncl2 9057 declt 9061 decltc 9062 decsuc 9064 decle 9067 declti 9071 decsucc 9074 dec10p 9076 decma 9084 decmac 9085 decma2c 9086 decadd 9087 decaddc 9088 decsubi 9096 decmul1 9097 decmul1c 9098 decmul2c 9099 decmul10add 9102 5t5e25 9136 6t6e36 9141 8t6e48 9152 9t11e99 9163 3dec 10302 3dvdsdec 11357 |
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