zndvds0 - Metamath Proof Explorer

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Theorem zndvds0 20803

Description: Special case of zndvds 20802 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.)

**Hypotheses**
Ref	Expression
zncyg.y	⊢ 𝑌 = (ℤ/nℤ‘𝑁)
zndvds.2	⊢ 𝐿 = (ℤRHom‘𝑌)
zndvds0.3	⊢ 0 = (0_g‘𝑌)

**Assertion**
Ref	Expression
zndvds0	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = 0 ↔ 𝑁 ∥ 𝐴))

**Proof of Theorem zndvds0**
Step	Hyp	Ref	Expression
1		0z 12376	. . 3 ⊢ 0 ∈ ℤ
2		zncyg.y	. . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
3		zndvds.2	. . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌)
4	2, 3	zndvds 20802	. . 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 20797	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑌 ∈ CRing)
7	6	adantr 482	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝑌 ∈ CRing)
8		crngring 19840	. . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
9	3	zrhrhm 20758	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
10	7, 8, 9	3syl 18	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
11		rhmghm 20014	. . . 4 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿 ∈ (ℤ_ring GrpHom 𝑌))
12		zring0 20725	. . . . 5 ⊢ 0 = (0_g‘ℤ_ring)
13		zndvds0.3	. . . . 5 ⊢ 0 = (0_g‘𝑌)
14	12, 13	ghmid 18885	. . . 4 ⊢ (𝐿 ∈ (ℤ_ring 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 486	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℤ)
18	17	zcnd 12473	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℂ)
19	18	subid1d 11367	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝐴 − 0) = 𝐴)
20	19	breq2d 5093	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝑁 ∥ (𝐴 − 0) ↔ 𝑁 ∥ 𝐴))
21	5, 16, 20	3bitr3d 309	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = 0 ↔ 𝑁 ∥ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1539 ∈ wcel 2104 class class class wbr 5081 ‘cfv 6458 (class class class)co 7307 0cc0 10917 − cmin 11251 ℕ₀cn0 12279 ℤcz 12365 ∥ cdvds 16008 0_gc0g 17195 GrpHom cghm 18876 Ringcrg 19828 CRingccrg 19829 RingHom crh 20001 ℤ_ringczring 20715 ℤRHomczrh 20746 ℤ/nℤczn 20749

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-addf 10996 ax-mulf 10997

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-tpos 8073 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-ec 8531 df-qs 8535 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-sup 9245 df-inf 9246 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-fz 13286 df-seq 13768 df-dvds 16009 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-starv 17022 df-sca 17023 df-vsca 17024 df-ip 17025 df-tset 17026 df-ple 17027 df-ds 17029 df-unif 17030 df-0g 17197 df-imas 17264 df-qus 17265 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-mhm 18475 df-grp 18625 df-minusg 18626 df-sbg 18627 df-mulg 18746 df-subg 18797 df-nsg 18798 df-eqg 18799 df-ghm 18877 df-cmn 19433 df-abl 19434 df-mgp 19766 df-ur 19783 df-ring 19830 df-cring 19831 df-oppr 19907 df-dvdsr 19928 df-rnghom 20004 df-subrg 20067 df-lmod 20170 df-lss 20239 df-lsp 20279 df-sra 20479 df-rgmod 20480 df-lidl 20481 df-rsp 20482 df-2idl 20548 df-cnfld 20643 df-zring 20716 df-zrh 20750 df-zn 20753

This theorem is referenced by: znfld 20813 znidomb 20814 znchr 20815 znrrg 20818 lgseisenlem3 26570 znfermltl 31607

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