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Theorem subsubrg 13304
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 13285 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ Ring)
21adantr 276 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝑅 ∈ Ring)
3 eqid 2177 . . . . . . . . 9 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
43subrgss 13281 . . . . . . . 8 (𝐡 ∈ (SubRingβ€˜π‘†) β†’ 𝐡 βŠ† (Baseβ€˜π‘†))
54adantl 277 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 βŠ† (Baseβ€˜π‘†))
6 subsubrg.s . . . . . . . . 9 𝑆 = (𝑅 β†Ύs 𝐴)
76subrgbas 13289 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 = (Baseβ€˜π‘†))
87adantr 276 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐴 = (Baseβ€˜π‘†))
95, 8sseqtrrd 3194 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 βŠ† 𝐴)
106oveq1i 5882 . . . . . . 7 (𝑆 β†Ύs 𝐡) = ((𝑅 β†Ύs 𝐴) β†Ύs 𝐡)
11 ressabsg 12527 . . . . . . . . 9 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴 ∧ 𝑅 ∈ Ring) β†’ ((𝑅 β†Ύs 𝐴) β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
12113expa 1203 . . . . . . . 8 (((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴) ∧ 𝑅 ∈ Ring) β†’ ((𝑅 β†Ύs 𝐴) β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
131, 12mpidan 423 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴) β†’ ((𝑅 β†Ύs 𝐴) β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
1410, 13eqtrid 2222 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴) β†’ (𝑆 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
159, 14syldan 282 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝑆 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
16 eqid 2177 . . . . . . 7 (𝑆 β†Ύs 𝐡) = (𝑆 β†Ύs 𝐡)
1716subrgring 13283 . . . . . 6 (𝐡 ∈ (SubRingβ€˜π‘†) β†’ (𝑆 β†Ύs 𝐡) ∈ Ring)
1817adantl 277 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝑆 β†Ύs 𝐡) ∈ Ring)
1915, 18eqeltrrd 2255 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝑅 β†Ύs 𝐡) ∈ Ring)
20 eqid 2177 . . . . . . . 8 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2120subrgss 13281 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
2221adantr 276 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
239, 22sstrd 3165 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 βŠ† (Baseβ€˜π‘…))
24 eqid 2177 . . . . . . . 8 (1rβ€˜π‘…) = (1rβ€˜π‘…)
256, 24subrg1 13290 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
2625adantr 276 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
27 eqid 2177 . . . . . . . 8 (1rβ€˜π‘†) = (1rβ€˜π‘†)
2827subrg1cl 13288 . . . . . . 7 (𝐡 ∈ (SubRingβ€˜π‘†) β†’ (1rβ€˜π‘†) ∈ 𝐡)
2928adantl 277 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (1rβ€˜π‘†) ∈ 𝐡)
3026, 29eqeltrd 2254 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (1rβ€˜π‘…) ∈ 𝐡)
3123, 30jca 306 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝐡 βŠ† (Baseβ€˜π‘…) ∧ (1rβ€˜π‘…) ∈ 𝐡))
3220, 24issubrg 13280 . . . 4 (𝐡 ∈ (SubRingβ€˜π‘…) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐡) ∈ Ring) ∧ (𝐡 βŠ† (Baseβ€˜π‘…) ∧ (1rβ€˜π‘…) ∈ 𝐡)))
332, 19, 31, 32syl21anbrc 1182 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ 𝐡 ∈ (SubRingβ€˜π‘…))
3433, 9jca 306 . 2 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 ∈ (SubRingβ€˜π‘†)) β†’ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴))
356subrgring 13283 . . . 4 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ Ring)
3635adantr 276 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝑆 ∈ Ring)
3714adantrl 478 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝑆 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡))
38 eqid 2177 . . . . . 6 (𝑅 β†Ύs 𝐡) = (𝑅 β†Ύs 𝐡)
3938subrgring 13283 . . . . 5 (𝐡 ∈ (SubRingβ€˜π‘…) β†’ (𝑅 β†Ύs 𝐡) ∈ Ring)
4039ad2antrl 490 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝑅 β†Ύs 𝐡) ∈ Ring)
4137, 40eqeltrd 2254 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝑆 β†Ύs 𝐡) ∈ Ring)
42 simprr 531 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐡 βŠ† 𝐴)
437adantr 276 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐴 = (Baseβ€˜π‘†))
4442, 43sseqtrd 3193 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐡 βŠ† (Baseβ€˜π‘†))
4525adantr 276 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
4624subrg1cl 13288 . . . . . 6 (𝐡 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘…) ∈ 𝐡)
4746ad2antrl 490 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (1rβ€˜π‘…) ∈ 𝐡)
4845, 47eqeltrrd 2255 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (1rβ€˜π‘†) ∈ 𝐡)
4944, 48jca 306 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ (𝐡 βŠ† (Baseβ€˜π‘†) ∧ (1rβ€˜π‘†) ∈ 𝐡))
503, 27issubrg 13280 . . 3 (𝐡 ∈ (SubRingβ€˜π‘†) ↔ ((𝑆 ∈ Ring ∧ (𝑆 β†Ύs 𝐡) ∈ Ring) ∧ (𝐡 βŠ† (Baseβ€˜π‘†) ∧ (1rβ€˜π‘†) ∈ 𝐡)))
5136, 41, 49, 50syl21anbrc 1182 . 2 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)) β†’ 𝐡 ∈ (SubRingβ€˜π‘†))
5234, 51impbida 596 1 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝐡 ∈ (SubRingβ€˜π‘†) ↔ (𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐡 βŠ† 𝐴)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   βŠ† wss 3129  β€˜cfv 5215  (class class class)co 5872  Basecbs 12454   β†Ύs cress 12455  1rcur 13073  Ringcrg 13110  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-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-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-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-subg 12961  df-mgp 13062  df-ur 13074  df-ring 13112  df-subrg 13278
This theorem is referenced by:  subsubrg2  13305
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