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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 12679 | . 2 ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) | |
2 | 9p1e10 12680 | . . . 4 ⊢ (9 + 1) = ;10 | |
3 | 2 | oveq1i 7414 | . . 3 ⊢ ((9 + 1) · 𝐴) = (;10 · 𝐴) |
4 | 3 | oveq1i 7414 | . 2 ⊢ (((9 + 1) · 𝐴) + 𝐵) = ((;10 · 𝐴) + 𝐵) |
5 | 1, 4 | eqtri 2754 | 1 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7404 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 9c9 12275 ;cdc 12678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-ltxr 11254 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-dec 12679 |
This theorem is referenced by: decnncl 12698 dec0u 12699 dec0h 12700 decnncl2 12702 declt 12706 decltc 12707 decsuc 12709 decle 12712 declti 12716 decsucc 12719 dec10p 12721 decma 12729 decmac 12730 decma2c 12731 decadd 12732 decaddc 12733 decsubi 12741 decmul1c 12743 decmul2c 12744 decmul10add 12747 5t5e25 12781 6t6e36 12786 8t6e48 12797 9t11e99 12808 3dec 14228 bpoly4 16006 3dvdsdec 16279 dec2dvds 17002 dec5dvds 17003 dec5nprm 17005 dec2nprm 17006 decsplit1 17021 decsplit 17022 4001lem1 17080 dfdec100 32538 dpfrac1 32560 dpmul10 32563 dpmul100 32565 dp3mul10 32566 dpmul1000 32567 dpmul 32581 dpmul4 32582 decpmul 41740 1t10e1p1e11 46572 3exp4mod41 46838 41prothprmlem1 46839 41prothprm 46841 tgoldbachlt 47038 |
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