zndvds0 - Metamath Proof Explorer

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Theorem zndvds0 21582

Description: Special case of zndvds 21581 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 12576	. . 3 ⊢ 0 ∈ ℤ
2		zncyg.y	. . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
3		zndvds.2	. . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌)
4	2, 3	zndvds 21581	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 0 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0)))
5	1, 4	mp3an3 1470	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0)))
6	2	zncrng 21576	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑌 ∈ CRing)
7	6	adantr 484	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝑌 ∈ CRing)
8		crngring 20274	. . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
9	3	zrhrhm 21543	. . . . 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 20511	. . . 4 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿 ∈ (ℤ_ring GrpHom 𝑌))
12		zring0 21490	. . . . 5 ⊢ 0 = (0_g‘ℤ_ring)
13		zndvds0.3	. . . . 5 ⊢ 0 = (0_g‘𝑌)
14	12, 13	ghmid 19245	. . . 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 2772	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ (𝐿‘𝐴) = 0 ))
17		simpr 488	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℤ)
18	17	zcnd 12675	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℂ)
19	18	subid1d 11528	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝐴 − 0) = 𝐴)
20	19	breq2d 5111	. 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 399 = wceq 1559 ∈ wcel 2141 class class class wbr 5099 ‘cfv 6517 (class class class)co 7392 0cc0 11070 − cmin 11411 ℕ₀cn0 12478 ℤcz 12565 ∥ cdvds 16269 0_gc0g 17451 GrpHom cghm 19236 Ringcrg 20262 CRingccrg 20263 RingHom crh 20497 ℤ_ringczring 21478 ℤRHomczrh 21531 ℤ/nℤczn 21534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-addf 11149 ax-mulf 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-tpos 8201 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-ec 8675 df-qs 8679 df-map 8805 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-sup 9385 df-inf 9386 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-fz 13510 df-seq 14012 df-dvds 16270 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-0g 17453 df-imas 17521 df-qus 17522 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-mhm 18800 df-grp 18961 df-minusg 18962 df-sbg 18963 df-mulg 19093 df-subg 19148 df-nsg 19149 df-eqg 19150 df-ghm 19237 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-cring 20265 df-oppr 20365 df-dvdsr 20385 df-rhm 20500 df-subrng 20575 df-subrg 20599 df-lmod 20909 df-lss 20979 df-lsp 21019 df-sra 21220 df-rgmod 21221 df-lidl 21258 df-rsp 21259 df-2idl 21300 df-cnfld 21405 df-zring 21479 df-zrh 21535 df-zn 21538

This theorem is referenced by: znfld 21592 znidomb 21593 znchr 21594 znrrg 21597 lgseisenlem3 27418 znfermltl 33513

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