![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dp20h | Structured version Visualization version GIF version |
Description: Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
dp20h.1 | ⊢ 𝐴 ∈ ℝ+ |
Ref | Expression |
---|---|
dp20h | ⊢ _0𝐴 = (𝐴 / ;10) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dp2 30203 | . 2 ⊢ _0𝐴 = (0 + (𝐴 / ;10)) | |
2 | dp20h.1 | . . . . 5 ⊢ 𝐴 ∈ ℝ+ | |
3 | rpcn 12238 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 𝐴 ∈ ℂ |
5 | 10nn0 11954 | . . . . 5 ⊢ ;10 ∈ ℕ0 | |
6 | 5 | nn0cni 11746 | . . . 4 ⊢ ;10 ∈ ℂ |
7 | 0re 10478 | . . . . 5 ⊢ 0 ∈ ℝ | |
8 | 10pos 11953 | . . . . 5 ⊢ 0 < ;10 | |
9 | 7, 8 | gtneii 10588 | . . . 4 ⊢ ;10 ≠ 0 |
10 | 4, 6, 9 | divcli 11219 | . . 3 ⊢ (𝐴 / ;10) ∈ ℂ |
11 | 10 | addid2i 10664 | . 2 ⊢ (0 + (𝐴 / ;10)) = (𝐴 / ;10) |
12 | 1, 11 | eqtri 2817 | 1 ⊢ _0𝐴 = (𝐴 / ;10) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1520 ∈ wcel 2079 (class class class)co 7007 ℂcc 10370 0cc0 10372 1c1 10373 + caddc 10375 / cdiv 11134 ;cdc 11936 ℝ+crp 12228 _cdp2 30202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-dec 11937 df-rp 12229 df-dp2 30203 |
This theorem is referenced by: dp0h 30233 dpexpp1 30239 |
Copyright terms: Public domain | W3C validator |