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Mirrors > Home > MPE Home > Th. List > dfdec10 | Structured version Visualization version 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 12087 | . 2 ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) | |
2 | 9p1e10 12088 | . . . 4 ⊢ (9 + 1) = ;10 | |
3 | 2 | oveq1i 7155 | . . 3 ⊢ ((9 + 1) · 𝐴) = (;10 · 𝐴) |
4 | 3 | oveq1i 7155 | . 2 ⊢ (((9 + 1) · 𝐴) + 𝐵) = ((;10 · 𝐴) + 𝐵) |
5 | 1, 4 | eqtri 2841 | 1 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 9c9 11687 ;cdc 12086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-dec 12087 |
This theorem is referenced by: decnncl 12106 dec0u 12107 dec0h 12108 decnncl2 12110 declt 12114 decltc 12115 decsuc 12117 decle 12120 declti 12124 decsucc 12127 dec10p 12129 decma 12137 decmac 12138 decma2c 12139 decadd 12140 decaddc 12141 decsubi 12149 decmul1c 12151 decmul2c 12152 decmul10add 12155 5t5e25 12189 6t6e36 12194 8t6e48 12205 9t11e99 12216 3dec 13614 bpoly4 15401 3dvdsdec 15669 dec2dvds 16387 dec5dvds 16388 dec5nprm 16390 dec2nprm 16391 decsplit1 16406 decsplit 16407 4001lem1 16462 dfdec100 30473 dpfrac1 30495 dpmul10 30498 dpmul100 30500 dp3mul10 30501 dpmul1000 30502 dpmul 30516 dpmul4 30517 decpmul 39052 1t10e1p1e11 43387 3exp4mod41 43658 41prothprmlem1 43659 41prothprm 43661 tgoldbachlt 43858 |
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