zndvds0 - Intuitionistic Logic Explorer

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Theorem zndvds0 14282

Description: Special case of zndvds 14281 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 9354	. . 3 ⊢ 0 ∈ ℤ
2		zncyg.y	. . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
3		zndvds.2	. . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌)
4	2, 3	zndvds 14281	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 0 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0)))
5	1, 4	mp3an3 1337	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0)))
6	2	zncrng 14277	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑌 ∈ CRing)
7	6	adantr 276	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝑌 ∈ CRing)
8		crngring 13640	. . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
9	3	zrhrhm 14255	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
10	7, 8, 9	3syl 17	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
11		rhmghm 13794	. . . 4 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿 ∈ (ℤ_ring GrpHom 𝑌))
12		zring0 14232	. . . . 5 ⊢ 0 = (0_g‘ℤ_ring)
13		zndvds0.3	. . . . 5 ⊢ 0 = (0_g‘𝑌)
14	12, 13	ghmid 13455	. . . 4 ⊢ (𝐿 ∈ (ℤ_ring GrpHom 𝑌) → (𝐿‘0) = 0 )
15	10, 11, 14	3syl 17	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝐿‘0) = 0 )
16	15	eqeq2d 2208	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ (𝐿‘𝐴) = 0 ))
17		simpr 110	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℤ)
18	17	zcnd 9466	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℂ)
19	18	subid1d 8343	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝐴 − 0) = 𝐴)
20	19	breq2d 4046	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝑁 ∥ (𝐴 − 0) ↔ 𝑁 ∥ 𝐴))
21	5, 16, 20	3bitr3d 218	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = 0 ↔ 𝑁 ∥ 𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2167 class class class wbr 4034 ‘cfv 5259 (class class class)co 5925 0cc0 7896 − cmin 8214 ℕ₀cn0 9266 ℤcz 9343 ∥ cdvds 11969 0_gc0g 12958 GrpHom cghm 13446 Ringcrg 13628 CRingccrg 13629 RingHom crh 13782 ℤ_ringczring 14222 ℤRHomczrh 14243 ℤ/nℤczn 14245

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-addf 8018 ax-mulf 8019

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-tp 3631 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-tpos 6312 df-recs 6372 df-frec 6458 df-er 6601 df-ec 6603 df-qs 6607 df-map 6718 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-5 9069 df-6 9070 df-7 9071 df-8 9072 df-9 9073 df-n0 9267 df-z 9344 df-dec 9475 df-uz 9619 df-rp 9746 df-fz 10101 df-fzo 10235 df-seqfrec 10557 df-cj 11024 df-abs 11181 df-dvds 11970 df-struct 12705 df-ndx 12706 df-slot 12707 df-base 12709 df-sets 12710 df-iress 12711 df-plusg 12793 df-mulr 12794 df-starv 12795 df-sca 12796 df-vsca 12797 df-ip 12798 df-tset 12799 df-ple 12800 df-ds 12802 df-unif 12803 df-0g 12960 df-topgen 12962 df-iimas 13004 df-qus 13005 df-mgm 13058 df-sgrp 13104 df-mnd 13119 df-mhm 13161 df-grp 13205 df-minusg 13206 df-sbg 13207 df-mulg 13326 df-subg 13376 df-nsg 13377 df-eqg 13378 df-ghm 13447 df-cmn 13492 df-abl 13493 df-mgp 13553 df-rng 13565 df-ur 13592 df-srg 13596 df-ring 13630 df-cring 13631 df-oppr 13700 df-dvdsr 13721 df-rhm 13784 df-subrg 13851 df-lmod 13921 df-lssm 13985 df-lsp 14019 df-sra 14067 df-rgmod 14068 df-lidl 14101 df-rsp 14102 df-2idl 14132 df-bl 14178 df-mopn 14179 df-fg 14181 df-metu 14182 df-cnfld 14189 df-zring 14223 df-zrh 14246 df-zn 14248

This theorem is referenced by: znidom 14289 znidomb 14290 znrrg 14292

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