Theorem subrgdvds 13294

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 13283	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
3		ringsrg 13155	. . . 4 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
4	2, 3	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ SRing)
5		reldvdsrsrg 13192	. . . 4 ⊢ (𝑆 ∈ SRing → Rel (∥r‘𝑆))
6		subrgdvds.3	. . . . 5 ⊢ 𝐸 = (∥r‘𝑆)
7	6	releqi 4708	. . . 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 13289	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
11		eqid 2177	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	11	subrgss 13281	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
13	10, 12	eqsstrrd 3192	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
14	13	sseld 3154	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑅)))
15		subrgrcl 13285	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
16		eqid 2177	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
17	1, 16	ressmulrg 12595	. . . . . . . . . 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 5889	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑧(.r‘𝑅)𝑥) = (𝑧(.r‘𝑆)𝑥))
20	19	eqeq1d 2186	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑆)𝑥) = 𝑦))
21	20	rexbidv 2478	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦))
22		ssrexv 3220	. . . . . . 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 2178	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) = (Base‘𝑆))
27	6	a1i 9	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 = (∥r‘𝑆))
28		eqidd 2178	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑆) = (.r‘𝑆))
29	26, 27, 4, 28	dvdsrd 13194	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝐸𝑦 ↔ (𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)))
30		eqidd 2178	. . . . 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 13155	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
34	15, 33	syl 14	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ SRing)
35		eqidd 2178	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑅))
36	30, 32, 34, 35	dvdsrd 13194	. . . 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 4003	. . 3 ⊢ (𝑥𝐸𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐸)
39		df-br 4003	. . 3 ⊢ (𝑥 ∥ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∥ )
40	37, 38, 39	3imtr3g 204	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (⟨𝑥, 𝑦⟩ ∈ 𝐸 → ⟨𝑥, 𝑦⟩ ∈ ∥ ))
41	9, 40	relssdv 4717	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  ∃wrex 2456   ⊆ wss 3129  ⟨cop 3595   class class class wbr 4002  Rel wrel 4630  ‘cfv 5215  (class class class)co 5872  Basecbs 12454   ↾_s cress 12455  ._rcmulr 12529  SRingcsrg 13077  Ringcrg 13110  ∥_rcdsr 13186  SubRingcsubrg 13276

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-dvdsr 13189  df-subrg 13278

This theorem is referenced by:  subrguss  13295