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Mirrors > Home > MPE Home > Th. List > zndvds0 | Structured version Visualization version GIF version |
Description: Special case of zndvds 20690 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 11986 | . . 3 ⊢ 0 ∈ ℤ | |
2 | zncyg.y | . . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
3 | zndvds.2 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
4 | 2, 3 | zndvds 20690 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 0 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0))) |
5 | 1, 4 | mp3an3 1446 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0))) |
6 | 2 | zncrng 20685 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
7 | 6 | adantr 483 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝑌 ∈ CRing) |
8 | crngring 19302 | . . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
9 | 3 | zrhrhm 20653 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
10 | 7, 8, 9 | 3syl 18 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝐿 ∈ (ℤring RingHom 𝑌)) |
11 | rhmghm 19471 | . . . 4 ⊢ (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌)) | |
12 | zring0 20621 | . . . . 5 ⊢ 0 = (0g‘ℤring) | |
13 | zndvds0.3 | . . . . 5 ⊢ 0 = (0g‘𝑌) | |
14 | 12, 13 | ghmid 18358 | . . . 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 2832 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ (𝐿‘𝐴) = 0 )) |
17 | simpr 487 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℤ) | |
18 | 17 | zcnd 12082 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℂ) |
19 | 18 | subid1d 10980 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐴 − 0) = 𝐴) |
20 | 19 | breq2d 5071 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝑁 ∥ (𝐴 − 0) ↔ 𝑁 ∥ 𝐴)) |
21 | 5, 16, 20 | 3bitr3d 311 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = 0 ↔ 𝑁 ∥ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 0cc0 10531 − cmin 10864 ℕ0cn0 11891 ℤcz 11975 ∥ cdvds 15601 0gc0g 16707 GrpHom cghm 18349 Ringcrg 19291 CRingccrg 19292 RingHom crh 19458 ℤringzring 20611 ℤRHomczrh 20641 ℤ/nℤczn 20644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-seq 13364 df-dvds 15602 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-imas 16775 df-qus 16776 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-nsg 18271 df-eqg 18272 df-ghm 18350 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-rnghom 19461 df-subrg 19527 df-lmod 19630 df-lss 19698 df-lsp 19738 df-sra 19938 df-rgmod 19939 df-lidl 19940 df-rsp 19941 df-2idl 19999 df-cnfld 20540 df-zring 20612 df-zrh 20645 df-zn 20648 |
This theorem is referenced by: znfld 20701 znidomb 20702 znchr 20703 znrrg 20706 lgseisenlem3 25947 |
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