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Mirrors > Home > MPE Home > Th. List > Mathboxes > dig0 | Structured version Visualization version GIF version |
Description: All digits of 0 are 0. (Contributed by AV, 24-May-2020.) |
Ref | Expression |
---|---|
dig0 | β’ ((π΅ β β β§ πΎ β β€) β (πΎ(digitβπ΅)0) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0e0icopnf 13459 | . . 3 β’ 0 β (0[,)+β) | |
2 | digval 47594 | . . 3 β’ ((π΅ β β β§ πΎ β β€ β§ 0 β (0[,)+β)) β (πΎ(digitβπ΅)0) = ((ββ((π΅β-πΎ) Β· 0)) mod π΅)) | |
3 | 1, 2 | mp3an3 1447 | . 2 β’ ((π΅ β β β§ πΎ β β€) β (πΎ(digitβπ΅)0) = ((ββ((π΅β-πΎ) Β· 0)) mod π΅)) |
4 | nncn 12242 | . . . . . . . . 9 β’ (π΅ β β β π΅ β β) | |
5 | 4 | adantr 480 | . . . . . . . 8 β’ ((π΅ β β β§ πΎ β β€) β π΅ β β) |
6 | nnne0 12268 | . . . . . . . . 9 β’ (π΅ β β β π΅ β 0) | |
7 | 6 | adantr 480 | . . . . . . . 8 β’ ((π΅ β β β§ πΎ β β€) β π΅ β 0) |
8 | znegcl 12619 | . . . . . . . . 9 β’ (πΎ β β€ β -πΎ β β€) | |
9 | 8 | adantl 481 | . . . . . . . 8 β’ ((π΅ β β β§ πΎ β β€) β -πΎ β β€) |
10 | 5, 7, 9 | expclzd 14139 | . . . . . . 7 β’ ((π΅ β β β§ πΎ β β€) β (π΅β-πΎ) β β) |
11 | 10 | mul01d 11435 | . . . . . 6 β’ ((π΅ β β β§ πΎ β β€) β ((π΅β-πΎ) Β· 0) = 0) |
12 | 11 | fveq2d 6895 | . . . . 5 β’ ((π΅ β β β§ πΎ β β€) β (ββ((π΅β-πΎ) Β· 0)) = (ββ0)) |
13 | 0zd 12592 | . . . . . 6 β’ ((π΅ β β β§ πΎ β β€) β 0 β β€) | |
14 | flid 13797 | . . . . . 6 β’ (0 β β€ β (ββ0) = 0) | |
15 | 13, 14 | syl 17 | . . . . 5 β’ ((π΅ β β β§ πΎ β β€) β (ββ0) = 0) |
16 | 12, 15 | eqtrd 2767 | . . . 4 β’ ((π΅ β β β§ πΎ β β€) β (ββ((π΅β-πΎ) Β· 0)) = 0) |
17 | 16 | oveq1d 7429 | . . 3 β’ ((π΅ β β β§ πΎ β β€) β ((ββ((π΅β-πΎ) Β· 0)) mod π΅) = (0 mod π΅)) |
18 | nnrp 13009 | . . . . 5 β’ (π΅ β β β π΅ β β+) | |
19 | 0mod 13891 | . . . . 5 β’ (π΅ β β+ β (0 mod π΅) = 0) | |
20 | 18, 19 | syl 17 | . . . 4 β’ (π΅ β β β (0 mod π΅) = 0) |
21 | 20 | adantr 480 | . . 3 β’ ((π΅ β β β§ πΎ β β€) β (0 mod π΅) = 0) |
22 | 17, 21 | eqtrd 2767 | . 2 β’ ((π΅ β β β§ πΎ β β€) β ((ββ((π΅β-πΎ) Β· 0)) mod π΅) = 0) |
23 | 3, 22 | eqtrd 2767 | 1 β’ ((π΅ β β β§ πΎ β β€) β (πΎ(digitβπ΅)0) = 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2935 βcfv 6542 (class class class)co 7414 βcc 11128 0cc0 11130 Β· cmul 11135 +βcpnf 11267 -cneg 11467 βcn 12234 β€cz 12580 β+crp 12998 [,)cico 13350 βcfl 13779 mod cmo 13858 βcexp 14050 digitcdig 47591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-ico 13354 df-fl 13781 df-mod 13859 df-seq 13991 df-exp 14051 df-dig 47592 |
This theorem is referenced by: 0dig2pr01 47606 nn0sumshdiglem1 47617 |
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