Theorem subrgdvds 14653

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 14642	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
3		ringsrg 14460	. . . 4 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
4	2, 3	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ SRing)
5		reldvdsrsrg 14509	. . . 4 ⊢ (𝑆 ∈ SRing → Rel (∥r‘𝑆))
6		subrgdvds.3	. . . . 5 ⊢ 𝐸 = (∥r‘𝑆)
7	6	releqi 4860	. . . 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 14648	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
11		eqid 2238	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	11	subrgss 14640	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
13	10, 12	eqsstrrd 3285	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
14	13	sseld 3247	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑅)))
15		subrgrcl 14644	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
16		eqid 2238	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
17	1, 16	ressmulrg 13559	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑅 ∈ Ring) → (.r‘𝑅) = (.r‘𝑆))
18	15, 17	mpdan 425	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
19	18	oveqd 6105	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑧(.r‘𝑅)𝑥) = (𝑧(.r‘𝑆)𝑥))
20	19	eqeq1d 2247	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑆)𝑥) = 𝑦))
21	20	rexbidv 2551	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦))
22		ssrexv 3313	. . . . . . 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 2239	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) = (Base‘𝑆))
27	6	a1i 9	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 = (∥r‘𝑆))
28		eqidd 2239	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑆) = (.r‘𝑆))
29	26, 27, 4, 28	dvdsrd 14511	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝐸𝑦 ↔ (𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)))
30		eqidd 2239	. . . . 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 14460	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
34	15, 33	syl 14	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ SRing)
35		eqidd 2239	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑅))
36	30, 32, 34, 35	dvdsrd 14511	. . . 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 4132	. . 3 ⊢ (𝑥𝐸𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐸)
39		df-br 4132	. . 3 ⊢ (𝑥 ∥ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∥ )
40	37, 38, 39	3imtr3g 204	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (⟨𝑥, 𝑦⟩ ∈ 𝐸 → ⟨𝑥, 𝑦⟩ ∈ ∥ ))
41	9, 40	relssdv 4869	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ )

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∃wrex 2529   ⊆ wss 3220  ⟨cop 3712   class class class wbr 4131  Rel wrel 4781  ‘cfv 5380  (class class class)co 6088  Basecbs 13411   ↾_s cress 13412  ._rcmulr 13492  SRingcsrg 14375  Ringcrg 14408  ∥_rcdsr 14502  SubRingcsubrg 14635

This proof depends on 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 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-i2m1 8291  ax-0lt1 8292  ax-0id 8294  ax-rnegex 8295  ax-pre-ltirr 8298  ax-pre-lttrn 8300  ax-pre-ltadd 8302

This proof depends on definitions:  df-bi 117  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-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8369  df-mnf 8370  df-ltxr 8372  df-inn 9315  df-2 9373  df-3 9374  df-ndx 13414  df-slot 13415  df-base 13417  df-sets 13418  df-iress 13419  df-plusg 13504  df-mulr 13505  df-0g 13672  df-mgm 13736  df-sgrp 13777  df-mnd 13790  df-grp 13868  df-minusg 13869  df-subg 14033  df-cmn 14180  df-abl 14181  df-mgp 14326  df-ur 14371  df-srg 14376  df-ring 14410  df-dvdsr 14505  df-subrg 14637

This theorem is used by:  subrguss  14654