Theorem subrgdvds 14369

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 14358	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
3		ringsrg 14180	. . . 4 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
4	2, 3	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ SRing)
5		reldvdsrsrg 14226	. . . 4 ⊢ (𝑆 ∈ SRing → Rel (∥r‘𝑆))
6		subrgdvds.3	. . . . 5 ⊢ 𝐸 = (∥r‘𝑆)
7	6	releqi 4832	. . . 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 14364	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
11		eqid 2232	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	11	subrgss 14356	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
13	10, 12	eqsstrrd 3274	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
14	13	sseld 3236	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑅)))
15		subrgrcl 14360	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
16		eqid 2232	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
17	1, 16	ressmulrg 13347	. . . . . . . . . 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 6066	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑧(.r‘𝑅)𝑥) = (𝑧(.r‘𝑆)𝑥))
20	19	eqeq1d 2241	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑆)𝑥) = 𝑦))
21	20	rexbidv 2543	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦))
22		ssrexv 3302	. . . . . . 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 2233	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) = (Base‘𝑆))
27	6	a1i 9	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 = (∥r‘𝑆))
28		eqidd 2233	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑆) = (.r‘𝑆))
29	26, 27, 4, 28	dvdsrd 14228	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝐸𝑦 ↔ (𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)))
30		eqidd 2233	. . . . 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 14180	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
34	15, 33	syl 14	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ SRing)
35		eqidd 2233	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑅))
36	30, 32, 34, 35	dvdsrd 14228	. . . 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 4109	. . 3 ⊢ (𝑥𝐸𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐸)
39		df-br 4109	. . 3 ⊢ (𝑥 ∥ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∥ )
40	37, 38, 39	3imtr3g 204	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (⟨𝑥, 𝑦⟩ ∈ 𝐸 → ⟨𝑥, 𝑦⟩ ∈ ∥ ))
41	9, 40	relssdv 4841	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ∃wrex 2521   ⊆ wss 3210  ⟨cop 3691   class class class wbr 4108  Rel wrel 4753  ‘cfv 5351  (class class class)co 6049  Basecbs 13201   ↾_s cress 13202  ._rcmulr 13280  SRingcsrg 14096  Ringcrg 14129  ∥_rcdsr 14219  SubRingcsubrg 14351

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-pre-ltirr 8235  ax-pre-lttrn 8237  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8306  df-mnf 8307  df-ltxr 8309  df-inn 9234  df-2 9292  df-3 9293  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-iress 13209  df-plusg 13292  df-mulr 13293  df-0g 13460  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-grp 13705  df-minusg 13706  df-subg 13876  df-cmn 13992  df-abl 13993  df-mgp 14054  df-ur 14093  df-srg 14097  df-ring 14131  df-dvdsr 14222  df-subrg 14353

This theorem is referenced by:  subrguss  14370