Theorem subrgdvds 20511

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.3	. . . 4 ⊢ 𝐸 = (∥r‘𝑆)
2	1	reldvdsr 20288	. . 3 ⊢ Rel 𝐸
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → Rel 𝐸)
4		subrgdvds.1	. . . . . . . 8 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
5	4	subrgbas 20506	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
6		eqid 2733	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
7	6	subrgss 20497	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
8	5, 7	eqsstrrd 3967	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
9	8	sseld 3930	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑅)))
10		eqid 2733	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
11	4, 10	ressmulr 17221	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
12	11	oveqd 7372	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑧(.r‘𝑅)𝑥) = (𝑧(.r‘𝑆)𝑥))
13	12	eqeq1d 2735	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑆)𝑥) = 𝑦))
14	13	rexbidv 3158	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦))
15		ssrexv 4001	. . . . . . 7 ⊢ ((Base‘𝑆) ⊆ (Base‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 → ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
16	8, 15	syl 17	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 → ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
17	14, 16	sylbird 260	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦 → ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
18	9, 17	anim12d 609	. . . 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)))
19		eqid 2733	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
20		eqid 2733	. . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆)
21	19, 1, 20	dvdsr 20290	. . . 4 ⊢ (𝑥𝐸𝑦 ↔ (𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦))
22		subrgdvds.2	. . . . 5 ⊢ ∥ = (∥r‘𝑅)
23	6, 22, 10	dvdsr 20290	. . . 4 ⊢ (𝑥 ∥ 𝑦 ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
24	18, 21, 23	3imtr4g 296	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝐸𝑦 → 𝑥 ∥ 𝑦))
25		df-br 5096	. . 3 ⊢ (𝑥𝐸𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐸)
26		df-br 5096	. . 3 ⊢ (𝑥 ∥ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∥ )
27	24, 25, 26	3imtr3g 295	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (⟨𝑥, 𝑦⟩ ∈ 𝐸 → ⟨𝑥, 𝑦⟩ ∈ ∥ ))
28	3, 27	relssdv 5734	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058   ⊆ wss 3899  ⟨cop 4583   class class class wbr 5095  Rel wrel 5626  ‘cfv 6489  (class class class)co 7355  Basecbs 17130   ↾_s cress 17151  ._rcmulr 17172  ∥_rcdsr 20282  SubRingcsubrg 20494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-er 8631  df-en 8879  df-dom 8880  df-sdom 8881  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-nn 12136  df-2 12198  df-3 12199  df-sets 17085  df-slot 17103  df-ndx 17115  df-base 17131  df-ress 17152  df-mulr 17185  df-subg 19046  df-ring 20163  df-dvdsr 20285  df-subrg 20495

This theorem is referenced by:  subrguss  20512