zndvds - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > zndvds		Structured version Visualization version GIF version

Theorem zndvds 21768

Description: Express equality of equivalence classes in ℤ / 𝑛ℤ in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)

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

**Assertion**
Ref	Expression
zndvds	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ 𝑁 ∥ (𝐴 − 𝐵)))

Proof of Theorem zndvds Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2769	. 2 ⊢ ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ (𝐿‘𝐵) = (𝐿‘𝐴))
2		eqid 2762	. . . . . 6 ⊢ (RSpan‘ℤ_ring) = (RSpan‘ℤ_ring)
3		eqid 2762	. . . . . 6 ⊢ (ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) = (ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))
4		zncyg.y	. . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
5		zndvds.2	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌)
6	2, 3, 4, 5	znzrhval 21765	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐵) = [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
7	6	3adant2 1149	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐵) = [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
8	2, 3, 4, 5	znzrhval 21765	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝐿‘𝐴) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
9	8	3adant3 1150	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐴) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
10	7, 9	eqeq12d 2778	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐵) = (𝐿‘𝐴) ↔ [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))))
11		zringring 21668	. . . . . 6 ⊢ ℤ_ring ∈ Ring
12		nn0z 12643	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
13	12	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝑁 ∈ ℤ)
14	13	snssd 4750	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑁} ⊆ ℤ)
15		zringbas 21672	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
16		eqid 2762	. . . . . . . 8 ⊢ (LIdeal‘ℤ_ring) = (LIdeal‘ℤ_ring)
17	2, 15, 16	rspcl 21433	. . . . . . 7 ⊢ ((ℤ_ring ∈ Ring ∧ {𝑁} ⊆ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring))
18	11, 14, 17	sylancr 599	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring))
19	16	lidlsubg 21417	. . . . . 6 ⊢ ((ℤ_ring ∈ Ring ∧ ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring)) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (SubGrp‘ℤ_ring))
20	11, 18, 19	sylancr 599	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (SubGrp‘ℤ_ring))
21	15, 3	eqger 19309	. . . . 5 ⊢ (((RSpan‘ℤ_ring)‘{𝑁}) ∈ (SubGrp‘ℤ_ring) → (ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) Er ℤ)
22	20, 21	syl 18	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) Er ℤ)
23		simp3 1156	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
24	22, 23	erth 8755	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵(ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))𝐴 ↔ [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))))
25		zringabl 21670	. . . . 5 ⊢ ℤ_ring ∈ Abel
26	15, 16	lidlss 21405	. . . . . 6 ⊢ (((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring) → ((RSpan‘ℤ_ring)‘{𝑁}) ⊆ ℤ)
27	18, 26	syl 18	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ⊆ ℤ)
28		eqid 2762	. . . . . 6 ⊢ (-_g‘ℤ_ring) = (-_g‘ℤ_ring)
29	15, 28, 3	eqgabl 19967	. . . . 5 ⊢ ((ℤ_ring ∈ Abel ∧ ((RSpan‘ℤ_ring)‘{𝑁}) ⊆ ℤ) → (𝐵(ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))𝐴 ↔ (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}))))
30	25, 27, 29	sylancr 599	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵(ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))𝐴 ↔ (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}))))
31		simp2 1155	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
32	23, 31	jca 521	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ))
33	32	biantrurd 542	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}))))
34		df-3an 1105	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁})) ↔ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁})))
35	33, 34	bitr4di 292	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}))))
36		zsubrg 21639	. . . . . . . . 9 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
37		subrgsubg 20745	. . . . . . . . 9 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
38	36, 37	mp1i 14	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ℤ ∈ (SubGrp‘ℂ_fld))
39		cnfldsub 21619	. . . . . . . . 9 ⊢ − = (-_g‘ℂ_fld)
40		df-zring 21666	. . . . . . . . 9 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
41	39, 40, 28	subgsub 19268	. . . . . . . 8 ⊢ ((ℤ ∈ (SubGrp‘ℂ_fld) ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) = (𝐴(-_g‘ℤ_ring)𝐵))
42	38, 41	syld3an1 1437	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) = (𝐴(-_g‘ℤ_ring)𝐵))
43	42	eqcomd 2768	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴(-_g‘ℤ_ring)𝐵) = (𝐴 − 𝐵))
44		dvdsrzring 21680	. . . . . . . 8 ⊢ ∥ = (∥_r‘ℤ_ring)
45	15, 2, 44	rspsn 21570	. . . . . . 7 ⊢ ((ℤ_ring ∈ Ring ∧ 𝑁 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥})
46	11, 13, 45	sylancr 599	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥})
47	43, 46	eleq12d 2856	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ (𝐴 − 𝐵) ∈ {𝑥 ∣ 𝑁 ∥ 𝑥}))
48		ovex 7450	. . . . . 6 ⊢ (𝐴 − 𝐵) ∈ V
49		breq2 5111	. . . . . 6 ⊢ (𝑥 = (𝐴 − 𝐵) → (𝑁 ∥ 𝑥 ↔ 𝑁 ∥ (𝐴 − 𝐵)))
50	48, 49	elab 3636	. . . . 5 ⊢ ((𝐴 − 𝐵) ∈ {𝑥 ∣ 𝑁 ∥ 𝑥} ↔ 𝑁 ∥ (𝐴 − 𝐵))
51	47, 50	bitrdi 290	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ 𝑁 ∥ (𝐴 − 𝐵)))
52	30, 35, 51	3bitr2d 310	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵(ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))𝐴 ↔ 𝑁 ∥ (𝐴 − 𝐵)))
53	10, 24, 52	3bitr2d 310	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐵) = (𝐿‘𝐴) ↔ 𝑁 ∥ (𝐴 − 𝐵)))
54	1, 53	bitrid 286	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ 𝑁 ∥ (𝐴 − 𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 {cab 2740 ⊆ wss 3902 {csn 4587 class class class wbr 5107 ‘cfv 6537 (class class class)co 7417 Er wer 8697 [cec 8698 − cmin 11469 ℕ₀cn0 12532 ℤcz 12619 ∥ cdvds 16348 -_gcsg 19065 SubGrpcsubg 19249 ~_QG cqg 19251 Abelcabl 19914 Ringcrg 20378 SubRingcsubrg 20737 LIdealclidl 21399 RSpancrsp 21400 ℂ_fldccnfld 21591 ℤ_ringczring 21665 ℤRHomczrh 21718 ℤ/nℤczn 21721

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-addf 11207 ax-mulf 11208

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-ec 8702 df-qs 8706 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-sup 9416 df-inf 9417 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-fz 13566 df-seq 14070 df-dvds 16349 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-starv 17363 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-0g 17532 df-imas 17600 df-qus 17601 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-mhm 18897 df-grp 19066 df-minusg 19067 df-sbg 19068 df-mulg 19197 df-subg 19252 df-nsg 19253 df-eqg 19254 df-ghm 19347 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-ring 20380 df-cring 20381 df-oppr 20484 df-dvdsr 20504 df-rhm 20619 df-subrng 20714 df-subrg 20738 df-lmod 21052 df-lss 21122 df-lsp 21162 df-sra 21363 df-rgmod 21364 df-lidl 21401 df-rsp 21402 df-2idl 21458 df-cnfld 21592 df-zring 21666 df-zrh 21722 df-zn 21725

This theorem is used by: zndvds0 21769 znf1o 21770 znunit 21782 cygznlem1 21785 lgsqrlem1 27590 lgsqrlem2 27591 lgsqrlem4 27593 lgsdchrval 27598 lgseisenlem3 27621 lgseisenlem4 27622 dchrisumlem1 27733 dirith 27773 hashscontpow1 42995 aks6d1c2 43004

W3C validator