zndvds0 - Intuitionistic Logic Explorer

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Theorem zndvds0 14968

Description: Special case of zndvds 14967 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 9638	. . 3 ⊢ 0 ∈ ℤ
2		zncyg.y	. . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
3		zndvds.2	. . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌)
4	2, 3	zndvds 14967	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 0 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0)))
5	1, 4	mp3an3 1367	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘0) ↔ 𝑁 ∥ (𝐴 − 0)))
6	2	zncrng 14963	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑌 ∈ CRing)
7	6	adantr 276	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝑌 ∈ CRing)
8		crngring 14295	. . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
9	3	zrhrhm 14941	. . . . 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 14452	. . . 4 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿 ∈ (ℤ_ring GrpHom 𝑌))
12		zring0 14918	. . . . 5 ⊢ 0 = (0_g‘ℤ_ring)
13		zndvds0.3	. . . . 5 ⊢ 0 = (0_g‘𝑌)
14	12, 13	ghmid 14035	. . . 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 2250	. 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 9752	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℂ)
19	18	subid1d 8620	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝐴 − 0) = 𝐴)
20	19	breq2d 4140	. 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 1402 ∈ wcel 2209 class class class wbr 4128 ‘cfv 5375 (class class class)co 6079 0cc0 8173 − cmin 8491 ℕ₀cn0 9546 ℤcz 9627 ∥ cdvds 12537 0_gc0g 13593 GrpHom cghm 14026 Ringcrg 14283 CRingccrg 14284 RingHom crh 14440 ℤ_ringczring 14908 ℤRHomczrh 14929 ℤ/nℤczn 14931

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-addf 8295 ax-mulf 8296

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-tp 3716 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-tpos 6510 df-recs 6570 df-frec 6656 df-er 6801 df-ec 6803 df-qs 6807 df-map 6918 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-5 9349 df-6 9350 df-7 9351 df-8 9352 df-9 9353 df-n0 9547 df-z 9628 df-dec 9761 df-uz 9905 df-rp 10038 df-fz 10395 df-fzo 10533 df-seqfrec 10868 df-cj 11590 df-abs 11748 df-dvds 12538 df-struct 13337 df-ndx 13338 df-slot 13339 df-base 13341 df-sets 13342 df-iress 13343 df-plusg 13427 df-mulr 13428 df-starv 13429 df-sca 13430 df-vsca 13431 df-ip 13432 df-tset 13433 df-ple 13434 df-ds 13436 df-unif 13437 df-0g 13595 df-topgen 13597 df-iimas 13607 df-qus 13608 df-mgm 13659 df-sgrp 13700 df-mnd 13713 df-mhm 13749 df-grp 13791 df-minusg 13792 df-sbg 13793 df-mulg 13906 df-subg 13956 df-nsg 13957 df-eqg 13958 df-ghm 14027 df-cmn 14072 df-abl 14073 df-mgp 14201 df-rng 14215 df-ur 14246 df-srg 14251 df-ring 14285 df-cring 14286 df-oppr 14356 df-dvdsr 14378 df-rhm 14442 df-subrg 14510 df-lmod 14608 df-lssm 14673 df-lsp 14707 df-sra 14755 df-rgmod 14756 df-lidl 14789 df-rsp 14790 df-2idl 14820 df-bl 14866 df-mopn 14867 df-fg 14869 df-metu 14870 df-cnfld 14877 df-zring 14909 df-zrh 14932 df-zn 14934

This theorem is referenced by: znidom 14975 znidomb 14976 znrrg 14978

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