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Mirrors > Home > MPE Home > Th. List > zndvds0 | Structured version Visualization version GIF version |
Description: Special case of zndvds 20956 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
zncyg.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
zndvds.2 | ⊢ 𝐿 = (ℤRHom‘𝑌) |
zndvds0.3 | ⊢ 0 = (0g‘𝑌) |
Ref | Expression |
---|---|
zndvds0 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = 0 ↔ 𝑁 ∥ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 12510 | . . 3 ⊢ 0 ∈ ℤ | |
2 | zncyg.y | . . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
3 | zndvds.2 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
4 | 2, 3 | zndvds 20956 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 0 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0))) |
5 | 1, 4 | mp3an3 1450 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0))) |
6 | 2 | zncrng 20951 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
7 | 6 | adantr 481 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝑌 ∈ CRing) |
8 | crngring 19976 | . . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
9 | 3 | zrhrhm 20912 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
10 | 7, 8, 9 | 3syl 18 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝐿 ∈ (ℤring RingHom 𝑌)) |
11 | rhmghm 20157 | . . . 4 ⊢ (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌)) | |
12 | zring0 20879 | . . . . 5 ⊢ 0 = (0g‘ℤring) | |
13 | zndvds0.3 | . . . . 5 ⊢ 0 = (0g‘𝑌) | |
14 | 12, 13 | ghmid 19014 | . . . 4 ⊢ (𝐿 ∈ (ℤring GrpHom 𝑌) → (𝐿‘0) = 0 ) |
15 | 10, 11, 14 | 3syl 18 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐿‘0) = 0 ) |
16 | 15 | eqeq2d 2747 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ (𝐿‘𝐴) = 0 )) |
17 | simpr 485 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℤ) | |
18 | 17 | zcnd 12608 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℂ) |
19 | 18 | subid1d 11501 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐴 − 0) = 𝐴) |
20 | 19 | breq2d 5117 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝑁 ∥ (𝐴 − 0) ↔ 𝑁 ∥ 𝐴)) |
21 | 5, 16, 20 | 3bitr3d 308 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = 0 ↔ 𝑁 ∥ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 0cc0 11051 − cmin 11385 ℕ0cn0 12413 ℤcz 12499 ∥ cdvds 16136 0gc0g 17321 GrpHom cghm 19005 Ringcrg 19964 CRingccrg 19965 RingHom crh 20143 ℤringczring 20869 ℤRHomczrh 20900 ℤ/nℤczn 20903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-seq 13907 df-dvds 16137 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-0g 17323 df-imas 17390 df-qus 17391 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-nsg 18926 df-eqg 18927 df-ghm 19006 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-rnghom 20146 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-sra 20633 df-rgmod 20634 df-lidl 20635 df-rsp 20636 df-2idl 20702 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-zn 20907 |
This theorem is referenced by: znfld 20967 znidomb 20968 znchr 20969 znrrg 20972 lgseisenlem3 26725 znfermltl 32155 |
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