MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subsubrg Structured version   Visualization version   GIF version
Theorem subsubrg 20488
Description: A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypothesis
Ref Expression
subsubrg.s 𝑆 = (𝑅 β†Ύs 𝐴)
Assertion
Ref Expression
subsubrg (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝐡 ∈ (SubRingβ€˜π‘†) ↔ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)))
Proof of Theorem subsubrg
StepHypRef Expression
1 subrgrcl 20466 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ Ring)
21adantr 479 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝑅 ∈ Ring)
3 eqid 2730 . . . . . . . . 9 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
43subrgss 20462 . . . . . . . 8 (𝐡 ∈ (SubRingβ€˜π‘†) β†’ 𝐡 βŠ† (Baseβ€˜π‘†))
54adantl 480 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 βŠ† (Baseβ€˜π‘†))
6 subsubrg.s . . . . . . . . 9 𝑆 = (𝑅 β†Ύs 𝐴)
76subrgbas 20471 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 = (Baseβ€˜π‘†))
87adantr 479 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐴 = (Baseβ€˜π‘†))
95, 8sseqtrrd 4022 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 βŠ† 𝐴)
106oveq1i 7421 . . . . . . 7 (𝑆 β†Ύs 𝐡) = ((𝑅 β†Ύs 𝐴) β†Ύs 𝐡)
11 ressabs 17198 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴) β†’ ((𝑅 β†Ύs 𝐴) β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
1210, 11eqtrid 2782 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴) β†’ (𝑆 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
139, 12syldan 589 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝑆 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
14 eqid 2730 . . . . . . 7 (𝑆 β†Ύs 𝐡) = (𝑆 β†Ύs 𝐡)
1514subrgring 20464 . . . . . 6 (𝐡 ∈ (SubRingβ€˜π‘†) β†’ (𝑆 β†Ύs 𝐡) ∈ Ring)
1615adantl 480 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝑆 β†Ύs 𝐡) ∈ Ring)
1713, 16eqeltrrd 2832 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝑅 β†Ύs 𝐡) ∈ Ring)
18 eqid 2730 . . . . . . . 8 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
1918subrgss 20462 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
2019adantr 479 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
219, 20sstrd 3991 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 βŠ† (Baseβ€˜π‘…))
22 eqid 2730 . . . . . . . 8 (1rβ€˜π‘…) = (1rβ€˜π‘…)
236, 22subrg1 20472 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
2423adantr 479 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
25 eqid 2730 . . . . . . . 8 (1rβ€˜π‘†) = (1rβ€˜π‘†)
2625subrg1cl 20470 . . . . . . 7 (𝐡 ∈ (SubRingβ€˜π‘†) β†’ (1rβ€˜π‘†) ∈ 𝐡)
2726adantl 480 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (1rβ€˜π‘†) ∈ 𝐡)
2824, 27eqeltrd 2831 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (1rβ€˜π‘…) ∈ 𝐡)
2921, 28jca 510 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝐡 βŠ† (Baseβ€˜π‘…) ∧ (1rβ€˜π‘…) ∈ 𝐡))
3018, 22issubrg 20461 . . . 4 (𝐡 ∈ (SubRingβ€˜π‘…) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐡) ∈ Ring) ∧ (𝐡 βŠ† (Baseβ€˜π‘…) ∧ (1rβ€˜π‘…) ∈ 𝐡)))
312, 17, 29, 30syl21anbrc 1342 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 ∈ (SubRingβ€˜π‘…))
3231, 9jca 510 . 2 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴))
336subrgring 20464 . . . 4 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ Ring)
3433adantr 479 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝑆 ∈ Ring)
3512adantrl 712 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝑆 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
36 eqid 2730 . . . . . 6 (𝑅 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡)
3736subrgring 20464 . . . . 5 (𝐡 ∈ (SubRingβ€˜π‘…) β†’ (𝑅 β†Ύs 𝐡) ∈ Ring)
3837ad2antrl 724 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝑅 β†Ύs 𝐡) ∈ Ring)
3935, 38eqeltrd 2831 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝑆 β†Ύs 𝐡) ∈ Ring)
40 simprr 769 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐡 βŠ† 𝐴)
417adantr 479 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐴 = (Baseβ€˜π‘†))
4240, 41sseqtrd 4021 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐡 βŠ† (Baseβ€˜π‘†))
4323adantr 479 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
4422subrg1cl 20470 . . . . . 6 (𝐡 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘…) ∈ 𝐡)
4544ad2antrl 724 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (1rβ€˜π‘…) ∈ 𝐡)
4643, 45eqeltrrd 2832 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (1rβ€˜π‘†) ∈ 𝐡)
4742, 46jca 510 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝐡 βŠ† (Baseβ€˜π‘†) ∧ (1rβ€˜π‘†) ∈ 𝐡))
483, 25issubrg 20461 . . 3 (𝐡 ∈ (SubRingβ€˜π‘†) ↔ ((𝑆 ∈ Ring ∧ (𝑆 β†Ύs 𝐡) ∈ Ring) ∧ (𝐡 βŠ† (Baseβ€˜π‘†) ∧ (1rβ€˜π‘†) ∈ 𝐡)))
4934, 39, 47, 48syl21anbrc 1342 . 2 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐡 ∈ (SubRingβ€˜π‘†))
5032, 49impbida 797 1 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝐡 ∈ (SubRingβ€˜π‘†) ↔ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   βŠ† wss 3947  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148   β†Ύs cress 17177  1rcur 20075  Ringcrg 20127  SubRingcsubrg 20457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-subg 19039  df-mgp 20029  df-ur 20076  df-ring 20129  df-subrg 20459
This theorem is referenced by:  subsubrg2  20489  zringunit  21237  rzgrp  21395  subrgmpl  21806  mplbas2  21816  mplind  21850  lsssra  32963  fedgmullem1  33002  fedgmullem2  33003  fedgmul  33004  fldexttr  33025  algextdeglem2  33063  algextdeglem4  33065
  Copyright terms: Public domain W3C validator