Theorem subrgdvds 13734

Description: If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
subrgdvds.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)
subrgdvds.2	⊢ ∥ = (∥r‘𝑅)
subrgdvds.3	⊢ 𝐸 = (∥r‘𝑆)

Assertion

Ref	Expression
subrgdvds	⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ )

Proof of Theorem subrgdvds

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgdvds.1	. . . . 5 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
2	1	subrgring 13723	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
3		ringsrg 13546	. . . 4 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
4	2, 3	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ SRing)
5		reldvdsrsrg 13591	. . . 4 ⊢ (𝑆 ∈ SRing → Rel (∥r‘𝑆))
6		subrgdvds.3	. . . . 5 ⊢ 𝐸 = (∥r‘𝑆)
7	6	releqi 4743	. . . 4 ⊢ (Rel 𝐸 ↔ Rel (∥r‘𝑆))
8	5, 7	sylibr 134	. . 3 ⊢ (𝑆 ∈ SRing → Rel 𝐸)
9	4, 8	syl 14	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → Rel 𝐸)
10	1	subrgbas 13729	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
11		eqid 2193	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	11	subrgss 13721	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
13	10, 12	eqsstrrd 3217	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
14	13	sseld 3179	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑅)))
15		subrgrcl 13725	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
16		eqid 2193	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
17	1, 16	ressmulrg 12765	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑅 ∈ Ring) → (.r‘𝑅) = (.r‘𝑆))
18	15, 17	mpdan 421	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
19	18	oveqd 5936	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑧(.r‘𝑅)𝑥) = (𝑧(.r‘𝑆)𝑥))
20	19	eqeq1d 2202	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑆)𝑥) = 𝑦))
21	20	rexbidv 2495	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦))
22		ssrexv 3245	. . . . . . 7 ⊢ ((Base‘𝑆) ⊆ (Base‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 → ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
23	13, 22	syl 14	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 → ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
24	21, 23	sylbird 170	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦 → ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
25	14, 24	anim12d 335	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦) → (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)))
26		eqidd 2194	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) = (Base‘𝑆))
27	6	a1i 9	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 = (∥r‘𝑆))
28		eqidd 2194	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑆) = (.r‘𝑆))
29	26, 27, 4, 28	dvdsrd 13593	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝐸𝑦 ↔ (𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)))
30		eqidd 2194	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑅) = (Base‘𝑅))
31		subrgdvds.2	. . . . . 6 ⊢ ∥ = (∥r‘𝑅)
32	31	a1i 9	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∥ = (∥r‘𝑅))
33		ringsrg 13546	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
34	15, 33	syl 14	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ SRing)
35		eqidd 2194	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑅))
36	30, 32, 34, 35	dvdsrd 13593	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∥ 𝑦 ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)))
37	25, 29, 36	3imtr4d 203	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝐸𝑦 → 𝑥 ∥ 𝑦))
38		df-br 4031	. . 3 ⊢ (𝑥𝐸𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐸)
39		df-br 4031	. . 3 ⊢ (𝑥 ∥ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∥ )
40	37, 38, 39	3imtr3g 204	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (⟨𝑥, 𝑦⟩ ∈ 𝐸 → ⟨𝑥, 𝑦⟩ ∈ ∥ ))
41	9, 40	relssdv 4752	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∃wrex 2473   ⊆ wss 3154  ⟨cop 3622   class class class wbr 4030  Rel wrel 4665  ‘cfv 5255  (class class class)co 5919  Basecbs 12621   ↾_s cress 12622  ._rcmulr 12699  SRingcsrg 13462  Ringcrg 13495  ∥_rcdsr 13585  SubRingcsubrg 13716

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-subg 13243  df-cmn 13359  df-abl 13360  df-mgp 13420  df-ur 13459  df-srg 13463  df-ring 13497  df-dvdsr 13588  df-subrg 13718

This theorem is referenced by:  subrguss  13735