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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 13432 | . . 3 β’ 0 β (0[,)+β) | |
2 | digval 47472 | . . 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 1446 | . 2 β’ ((π΅ β β β§ πΎ β β€) β (πΎ(digitβπ΅)0) = ((ββ((π΅β-πΎ) Β· 0)) mod π΅)) |
4 | nncn 12217 | . . . . . . . . 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 12243 | . . . . . . . . 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 12594 | . . . . . . . . 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 14113 | . . . . . . 7 β’ ((π΅ β β β§ πΎ β β€) β (π΅β-πΎ) β β) |
11 | 10 | mul01d 11410 | . . . . . 6 β’ ((π΅ β β β§ πΎ β β€) β ((π΅β-πΎ) Β· 0) = 0) |
12 | 11 | fveq2d 6885 | . . . . 5 β’ ((π΅ β β β§ πΎ β β€) β (ββ((π΅β-πΎ) Β· 0)) = (ββ0)) |
13 | 0zd 12567 | . . . . . 6 β’ ((π΅ β β β§ πΎ β β€) β 0 β β€) | |
14 | flid 13770 | . . . . . 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 2764 | . . . 4 β’ ((π΅ β β β§ πΎ β β€) β (ββ((π΅β-πΎ) Β· 0)) = 0) |
17 | 16 | oveq1d 7416 | . . 3 β’ ((π΅ β β β§ πΎ β β€) β ((ββ((π΅β-πΎ) Β· 0)) mod π΅) = (0 mod π΅)) |
18 | nnrp 12982 | . . . . 5 β’ (π΅ β β β π΅ β β+) | |
19 | 0mod 13864 | . . . . 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 2764 | . 2 β’ ((π΅ β β β§ πΎ β β€) β ((ββ((π΅β-πΎ) Β· 0)) mod π΅) = 0) |
23 | 3, 22 | eqtrd 2764 | 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 1533 β wcel 2098 β wne 2932 βcfv 6533 (class class class)co 7401 βcc 11104 0cc0 11106 Β· cmul 11111 +βcpnf 11242 -cneg 11442 βcn 12209 β€cz 12555 β+crp 12971 [,)cico 13323 βcfl 13752 mod cmo 13831 βcexp 14024 digitcdig 47469 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-ico 13327 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-dig 47470 |
This theorem is referenced by: 0dig2pr01 47484 nn0sumshdiglem1 47495 |
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