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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 9331 | . 2 ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) | |
2 | 9p1e10 9332 | . . . 4 ⊢ (9 + 1) = ;10 | |
3 | 2 | oveq1i 5860 | . . 3 ⊢ ((9 + 1) · 𝐴) = (;10 · 𝐴) |
4 | 3 | oveq1i 5860 | . 2 ⊢ (((9 + 1) · 𝐴) + 𝐵) = ((;10 · 𝐴) + 𝐵) |
5 | 1, 4 | eqtri 2191 | 1 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 (class class class)co 5850 0cc0 7761 1c1 7762 + caddc 7764 · cmul 7766 9c9 8923 ;cdc 9330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4105 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-1rid 7868 ax-0id 7869 ax-cnre 7872 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5853 df-inn 8866 df-2 8924 df-3 8925 df-4 8926 df-5 8927 df-6 8928 df-7 8929 df-8 8930 df-9 8931 df-dec 9331 |
This theorem is referenced by: decnncl 9349 dec0u 9350 dec0h 9351 decnncl2 9353 declt 9357 decltc 9358 decsuc 9360 decle 9363 declti 9367 decsucc 9370 dec10p 9372 decma 9380 decmac 9381 decma2c 9382 decadd 9383 decaddc 9384 decsubi 9392 decmul1 9393 decmul1c 9394 decmul2c 9395 decmul10add 9398 5t5e25 9432 6t6e36 9437 8t6e48 9448 9t11e99 9459 3dec 10635 3dvdsdec 11811 |
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