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Mirrors > Home > MPE Home > Th. List > zndvds0 | Structured version Visualization version GIF version |
Description: Special case of zndvds 19717 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 11265 | . . 3 ⊢ 0 ∈ ℤ | |
2 | zncyg.y | . . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
3 | zndvds.2 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
4 | 2, 3 | zndvds 19717 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 0 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0))) |
5 | 1, 4 | mp3an3 1405 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0))) |
6 | 2 | zncrng 19712 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
7 | 6 | adantr 480 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝑌 ∈ CRing) |
8 | crngring 18381 | . . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
9 | 3 | zrhrhm 19679 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
10 | 7, 8, 9 | 3syl 18 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝐿 ∈ (ℤring RingHom 𝑌)) |
11 | rhmghm 18548 | . . . 4 ⊢ (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌)) | |
12 | zring0 19647 | . . . . 5 ⊢ 0 = (0g‘ℤring) | |
13 | zndvds0.3 | . . . . 5 ⊢ 0 = (0g‘𝑌) | |
14 | 12, 13 | ghmid 17489 | . . . 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 2620 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ (𝐿‘𝐴) = 0 )) |
17 | simpr 476 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℤ) | |
18 | 17 | zcnd 11359 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℂ) |
19 | 18 | subid1d 10260 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐴 − 0) = 𝐴) |
20 | 19 | breq2d 4595 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝑁 ∥ (𝐴 − 0) ↔ 𝑁 ∥ 𝐴)) |
21 | 5, 16, 20 | 3bitr3d 297 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = 0 ↔ 𝑁 ∥ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 − cmin 10145 ℕ0cn0 11169 ℤcz 11254 ∥ cdvds 14821 0gc0g 15923 GrpHom cghm 17480 Ringcrg 18370 CRingccrg 18371 RingHom crh 18535 ℤringzring 19637 ℤRHomczrh 19667 ℤ/nℤczn 19670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-seq 12664 df-dvds 14822 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-zn 19674 |
This theorem is referenced by: znfld 19728 znidomb 19729 znchr 19730 znrrg 19733 lgseisenlem3 24902 |
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