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Theorem subsubrg 20489
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 20467 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ Ring)
21adantr 480 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝑅 ∈ Ring)
3 eqid 2731 . . . . . . . . 9 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
43subrgss 20463 . . . . . . . 8 (𝐡 ∈ (SubRingβ€˜π‘†) β†’ 𝐡 βŠ† (Baseβ€˜π‘†))
54adantl 481 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 βŠ† (Baseβ€˜π‘†))
6 subsubrg.s . . . . . . . . 9 𝑆 = (𝑅 β†Ύs 𝐴)
76subrgbas 20472 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 = (Baseβ€˜π‘†))
87adantr 480 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐴 = (Baseβ€˜π‘†))
95, 8sseqtrrd 4023 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 βŠ† 𝐴)
106oveq1i 7422 . . . . . . 7 (𝑆 β†Ύs 𝐡) = ((𝑅 β†Ύs 𝐴) β†Ύs 𝐡)
11 ressabs 17199 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴) β†’ ((𝑅 β†Ύs 𝐴) β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
1210, 11eqtrid 2783 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴) β†’ (𝑆 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
139, 12syldan 590 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝑆 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
14 eqid 2731 . . . . . . 7 (𝑆 β†Ύs 𝐡) = (𝑆 β†Ύs 𝐡)
1514subrgring 20465 . . . . . 6 (𝐡 ∈ (SubRingβ€˜π‘†) β†’ (𝑆 β†Ύs 𝐡) ∈ Ring)
1615adantl 481 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝑆 β†Ύs 𝐡) ∈ Ring)
1713, 16eqeltrrd 2833 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝑅 β†Ύs 𝐡) ∈ Ring)
18 eqid 2731 . . . . . . . 8 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
1918subrgss 20463 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
2019adantr 480 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
219, 20sstrd 3992 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 βŠ† (Baseβ€˜π‘…))
22 eqid 2731 . . . . . . . 8 (1rβ€˜π‘…) = (1rβ€˜π‘…)
236, 22subrg1 20473 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
2423adantr 480 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
25 eqid 2731 . . . . . . . 8 (1rβ€˜π‘†) = (1rβ€˜π‘†)
2625subrg1cl 20471 . . . . . . 7 (𝐡 ∈ (SubRingβ€˜π‘†) β†’ (1rβ€˜π‘†) ∈ 𝐡)
2726adantl 481 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (1rβ€˜π‘†) ∈ 𝐡)
2824, 27eqeltrd 2832 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (1rβ€˜π‘…) ∈ 𝐡)
2921, 28jca 511 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝐡 βŠ† (Baseβ€˜π‘…) ∧ (1rβ€˜π‘…) ∈ 𝐡))
3018, 22issubrg 20462 . . . 4 (𝐡 ∈ (SubRingβ€˜π‘…) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐡) ∈ Ring) ∧ (𝐡 βŠ† (Baseβ€˜π‘…) ∧ (1rβ€˜π‘…) ∈ 𝐡)))
312, 17, 29, 30syl21anbrc 1343 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 ∈ (SubRingβ€˜π‘…))
3231, 9jca 511 . 2 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴))
336subrgring 20465 . . . 4 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ Ring)
3433adantr 480 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝑆 ∈ Ring)
3512adantrl 713 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝑆 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
36 eqid 2731 . . . . . 6 (𝑅 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡)
3736subrgring 20465 . . . . 5 (𝐡 ∈ (SubRingβ€˜π‘…) β†’ (𝑅 β†Ύs 𝐡) ∈ Ring)
3837ad2antrl 725 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝑅 β†Ύs 𝐡) ∈ Ring)
3935, 38eqeltrd 2832 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝑆 β†Ύs 𝐡) ∈ Ring)
40 simprr 770 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐡 βŠ† 𝐴)
417adantr 480 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐴 = (Baseβ€˜π‘†))
4240, 41sseqtrd 4022 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐡 βŠ† (Baseβ€˜π‘†))
4323adantr 480 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
4422subrg1cl 20471 . . . . . 6 (𝐡 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘…) ∈ 𝐡)
4544ad2antrl 725 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (1rβ€˜π‘…) ∈ 𝐡)
4643, 45eqeltrrd 2833 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (1rβ€˜π‘†) ∈ 𝐡)
4742, 46jca 511 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝐡 βŠ† (Baseβ€˜π‘†) ∧ (1rβ€˜π‘†) ∈ 𝐡))
483, 25issubrg 20462 . . 3 (𝐡 ∈ (SubRingβ€˜π‘†) ↔ ((𝑆 ∈ Ring ∧ (𝑆 β†Ύs 𝐡) ∈ Ring) ∧ (𝐡 βŠ† (Baseβ€˜π‘†) ∧ (1rβ€˜π‘†) ∈ 𝐡)))
4934, 39, 47, 48syl21anbrc 1343 . 2 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐡 ∈ (SubRingβ€˜π‘†))
5032, 49impbida 798 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 395   = wceq 1540   ∈ wcel 2105   βŠ† wss 3948  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149   β†Ύs cress 17178  1rcur 20076  Ringcrg 20128  SubRingcsubrg 20458
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-0g 17392  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-subg 19040  df-mgp 20030  df-ur 20077  df-ring 20130  df-subrg 20460
This theorem is referenced by:  subsubrg2  20490  zringunit  21238  rzgrp  21396  subrgmpl  21807  mplbas2  21817  mplind  21851  lsssra  32964  fedgmullem1  33003  fedgmullem2  33004  fedgmul  33005  fldexttr  33026  algextdeglem2  33064  algextdeglem4  33066
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