MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subsubrg Structured version   Visualization version   GIF version

Theorem subsubrg 20615
Description: A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypothesis
Ref Expression
subsubrg.s 𝑆 = (𝑅s 𝐴)
Assertion
Ref Expression
subsubrg (𝐴 ∈ (SubRing‘𝑅) → (𝐵 ∈ (SubRing‘𝑆) ↔ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)))

Proof of Theorem subsubrg
StepHypRef Expression
1 subrgrcl 20593 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
21adantr 480 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝑅 ∈ Ring)
3 eqid 2735 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
43subrgss 20589 . . . . . . . 8 (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ⊆ (Base‘𝑆))
54adantl 481 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵 ⊆ (Base‘𝑆))
6 subsubrg.s . . . . . . . . 9 𝑆 = (𝑅s 𝐴)
76subrgbas 20598 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
87adantr 480 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐴 = (Base‘𝑆))
95, 8sseqtrrd 4037 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵𝐴)
106oveq1i 7441 . . . . . . 7 (𝑆s 𝐵) = ((𝑅s 𝐴) ↾s 𝐵)
11 ressabs 17295 . . . . . . 7 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴) → ((𝑅s 𝐴) ↾s 𝐵) = (𝑅s 𝐵))
1210, 11eqtrid 2787 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴) → (𝑆s 𝐵) = (𝑅s 𝐵))
139, 12syldan 591 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝑆s 𝐵) = (𝑅s 𝐵))
14 eqid 2735 . . . . . . 7 (𝑆s 𝐵) = (𝑆s 𝐵)
1514subrgring 20591 . . . . . 6 (𝐵 ∈ (SubRing‘𝑆) → (𝑆s 𝐵) ∈ Ring)
1615adantl 481 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝑆s 𝐵) ∈ Ring)
1713, 16eqeltrrd 2840 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝑅s 𝐵) ∈ Ring)
18 eqid 2735 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
1918subrgss 20589 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
2019adantr 480 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐴 ⊆ (Base‘𝑅))
219, 20sstrd 4006 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵 ⊆ (Base‘𝑅))
22 eqid 2735 . . . . . . . 8 (1r𝑅) = (1r𝑅)
236, 22subrg1 20599 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
2423adantr 480 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (1r𝑅) = (1r𝑆))
25 eqid 2735 . . . . . . . 8 (1r𝑆) = (1r𝑆)
2625subrg1cl 20597 . . . . . . 7 (𝐵 ∈ (SubRing‘𝑆) → (1r𝑆) ∈ 𝐵)
2726adantl 481 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (1r𝑆) ∈ 𝐵)
2824, 27eqeltrd 2839 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (1r𝑅) ∈ 𝐵)
2921, 28jca 511 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝐵 ⊆ (Base‘𝑅) ∧ (1r𝑅) ∈ 𝐵))
3018, 22issubrg 20588 . . . 4 (𝐵 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ (Base‘𝑅) ∧ (1r𝑅) ∈ 𝐵)))
312, 17, 29, 30syl21anbrc 1343 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵 ∈ (SubRing‘𝑅))
3231, 9jca 511 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴))
336subrgring 20591 . . . 4 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
3433adantr 480 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → 𝑆 ∈ Ring)
3512adantrl 716 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → (𝑆s 𝐵) = (𝑅s 𝐵))
36 eqid 2735 . . . . . 6 (𝑅s 𝐵) = (𝑅s 𝐵)
3736subrgring 20591 . . . . 5 (𝐵 ∈ (SubRing‘𝑅) → (𝑅s 𝐵) ∈ Ring)
3837ad2antrl 728 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → (𝑅s 𝐵) ∈ Ring)
3935, 38eqeltrd 2839 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → (𝑆s 𝐵) ∈ Ring)
40 simprr 773 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → 𝐵𝐴)
417adantr 480 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → 𝐴 = (Base‘𝑆))
4240, 41sseqtrd 4036 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → 𝐵 ⊆ (Base‘𝑆))
4323adantr 480 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → (1r𝑅) = (1r𝑆))
4422subrg1cl 20597 . . . . . 6 (𝐵 ∈ (SubRing‘𝑅) → (1r𝑅) ∈ 𝐵)
4544ad2antrl 728 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → (1r𝑅) ∈ 𝐵)
4643, 45eqeltrrd 2840 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → (1r𝑆) ∈ 𝐵)
4742, 46jca 511 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → (𝐵 ⊆ (Base‘𝑆) ∧ (1r𝑆) ∈ 𝐵))
483, 25issubrg 20588 . . 3 (𝐵 ∈ (SubRing‘𝑆) ↔ ((𝑆 ∈ Ring ∧ (𝑆s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ (Base‘𝑆) ∧ (1r𝑆) ∈ 𝐵)))
4934, 39, 47, 48syl21anbrc 1343 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)) → 𝐵 ∈ (SubRing‘𝑆))
5032, 49impbida 801 1 (𝐴 ∈ (SubRing‘𝑅) → (𝐵 ∈ (SubRing‘𝑆) ↔ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  1rcur 20199  Ringcrg 20251  SubRingcsubrg 20586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-subg 19154  df-mgp 20153  df-ur 20200  df-ring 20253  df-subrg 20587
This theorem is referenced by:  subsubrg2  20616  zringunit  21495  rzgrp  21659  subrgmpl  22068  mplbas2  22078  mplind  22112  lsssra  33618  fedgmullem1  33657  fedgmullem2  33658  fedgmul  33659  fldexttr  33686  algextdeglem2  33724  algextdeglem4  33726
  Copyright terms: Public domain W3C validator