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Mirrors  >  Home  >  ILE Home  >  Th. List  >  subrgpropd GIF version
Theorem subrgpropd 13374
Description: If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypotheses
Ref Expression
subrgpropd.1 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
subrgpropd.2 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))
subrgpropd.3 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
subrgpropd.4 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))
Assertion
Ref Expression
subrgpropd (πœ‘ β†’ (SubRingβ€˜πΎ) = (SubRingβ€˜πΏ))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐾,𝑦   πœ‘,π‘₯,𝑦   π‘₯,𝐿,𝑦
Proof of Theorem subrgpropd
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 subrgrcl 13352 . . . 4 (𝑠 ∈ (SubRingβ€˜πΎ) β†’ 𝐾 ∈ Ring)
21a1i 9 . . 3 (πœ‘ β†’ (𝑠 ∈ (SubRingβ€˜πΎ) β†’ 𝐾 ∈ Ring))
3 subrgrcl 13352 . . . 4 (𝑠 ∈ (SubRingβ€˜πΏ) β†’ 𝐿 ∈ Ring)
4 subrgpropd.1 . . . . 5 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
5 subrgpropd.2 . . . . 5 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))
6 subrgpropd.3 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
7 subrgpropd.4 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))
84, 5, 6, 7ringpropd 13222 . . . 4 (πœ‘ β†’ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
93, 8imbitrrid 156 . . 3 (πœ‘ β†’ (𝑠 ∈ (SubRingβ€˜πΏ) β†’ 𝐾 ∈ Ring))
108adantr 276 . . . . . . 7 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
114ineq2d 3338 . . . . . . . . . 10 (πœ‘ β†’ (𝑠 ∩ 𝐡) = (𝑠 ∩ (Baseβ€˜πΎ)))
1211adantr 276 . . . . . . . . 9 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (𝑠 ∩ 𝐡) = (𝑠 ∩ (Baseβ€˜πΎ)))
13 eqidd 2178 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ (𝐾 β†Ύs 𝑠) = (𝐾 β†Ύs 𝑠))
14 eqidd 2178 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΎ))
15 simplr 528 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ 𝐾 ∈ Ring)
16 simpr 110 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ 𝑠 ∈ V)
1713, 14, 15, 16ressbasd 12529 . . . . . . . . . 10 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ (𝑠 ∩ (Baseβ€˜πΎ)) = (Baseβ€˜(𝐾 β†Ύs 𝑠)))
1817elvd 2744 . . . . . . . . 9 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (𝑠 ∩ (Baseβ€˜πΎ)) = (Baseβ€˜(𝐾 β†Ύs 𝑠)))
1912, 18eqtrd 2210 . . . . . . . 8 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (𝑠 ∩ 𝐡) = (Baseβ€˜(𝐾 β†Ύs 𝑠)))
205ineq2d 3338 . . . . . . . . . 10 (πœ‘ β†’ (𝑠 ∩ 𝐡) = (𝑠 ∩ (Baseβ€˜πΏ)))
2120adantr 276 . . . . . . . . 9 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (𝑠 ∩ 𝐡) = (𝑠 ∩ (Baseβ€˜πΏ)))
22 eqidd 2178 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ (𝐿 β†Ύs 𝑠) = (𝐿 β†Ύs 𝑠))
23 eqidd 2178 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ (Baseβ€˜πΏ) = (Baseβ€˜πΏ))
248biimpa 296 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ 𝐿 ∈ Ring)
2524adantr 276 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ 𝐿 ∈ Ring)
2622, 23, 25, 16ressbasd 12529 . . . . . . . . . 10 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ (𝑠 ∩ (Baseβ€˜πΏ)) = (Baseβ€˜(𝐿 β†Ύs 𝑠)))
2726elvd 2744 . . . . . . . . 9 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (𝑠 ∩ (Baseβ€˜πΏ)) = (Baseβ€˜(𝐿 β†Ύs 𝑠)))
2821, 27eqtrd 2210 . . . . . . . 8 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (𝑠 ∩ 𝐡) = (Baseβ€˜(𝐿 β†Ύs 𝑠)))
29 elinel2 3324 . . . . . . . . . 10 (π‘₯ ∈ (𝑠 ∩ 𝐡) β†’ π‘₯ ∈ 𝐡)
30 elinel2 3324 . . . . . . . . . 10 (𝑦 ∈ (𝑠 ∩ 𝐡) β†’ 𝑦 ∈ 𝐡)
3129, 30anim12i 338 . . . . . . . . 9 ((π‘₯ ∈ (𝑠 ∩ 𝐡) ∧ 𝑦 ∈ (𝑠 ∩ 𝐡)) β†’ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡))
326adantlr 477 . . . . . . . . . 10 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
33 eqidd 2178 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ (+gβ€˜πΎ) = (+gβ€˜πΎ))
3413, 33, 16, 15ressplusgd 12589 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ (+gβ€˜πΎ) = (+gβ€˜(𝐾 β†Ύs 𝑠)))
3534elvd 2744 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (+gβ€˜πΎ) = (+gβ€˜(𝐾 β†Ύs 𝑠)))
3635oveqdr 5905 . . . . . . . . . 10 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜(𝐾 β†Ύs 𝑠))𝑦))
37 eqidd 2178 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ (+gβ€˜πΏ) = (+gβ€˜πΏ))
3822, 37, 16, 25ressplusgd 12589 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) β†’ (+gβ€˜πΏ) = (+gβ€˜(𝐿 β†Ύs 𝑠)))
3938elvd 2744 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (+gβ€˜πΏ) = (+gβ€˜(𝐿 β†Ύs 𝑠)))
4039oveqdr 5905 . . . . . . . . . 10 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΏ)𝑦) = (π‘₯(+gβ€˜(𝐿 β†Ύs 𝑠))𝑦))
4132, 36, 403eqtr3d 2218 . . . . . . . . 9 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜(𝐾 β†Ύs 𝑠))𝑦) = (π‘₯(+gβ€˜(𝐿 β†Ύs 𝑠))𝑦))
4231, 41sylan2 286 . . . . . . . 8 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ (π‘₯ ∈ (𝑠 ∩ 𝐡) ∧ 𝑦 ∈ (𝑠 ∩ 𝐡))) β†’ (π‘₯(+gβ€˜(𝐾 β†Ύs 𝑠))𝑦) = (π‘₯(+gβ€˜(𝐿 β†Ύs 𝑠))𝑦))
437adantlr 477 . . . . . . . . . 10 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))
44 vex 2742 . . . . . . . . . . . . 13 𝑠 ∈ V
45 eqid 2177 . . . . . . . . . . . . . 14 (𝐾 β†Ύs 𝑠) = (𝐾 β†Ύs 𝑠)
46 eqid 2177 . . . . . . . . . . . . . 14 (.rβ€˜πΎ) = (.rβ€˜πΎ)
4745, 46ressmulrg 12605 . . . . . . . . . . . . 13 ((𝑠 ∈ V ∧ 𝐾 ∈ Ring) β†’ (.rβ€˜πΎ) = (.rβ€˜(𝐾 β†Ύs 𝑠)))
4844, 47mpan 424 . . . . . . . . . . . 12 (𝐾 ∈ Ring β†’ (.rβ€˜πΎ) = (.rβ€˜(𝐾 β†Ύs 𝑠)))
4948adantl 277 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (.rβ€˜πΎ) = (.rβ€˜(𝐾 β†Ύs 𝑠)))
5049oveqdr 5905 . . . . . . . . . 10 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜(𝐾 β†Ύs 𝑠))𝑦))
51 eqid 2177 . . . . . . . . . . . . 13 (𝐿 β†Ύs 𝑠) = (𝐿 β†Ύs 𝑠)
52 eqid 2177 . . . . . . . . . . . . 13 (.rβ€˜πΏ) = (.rβ€˜πΏ)
5351, 52ressmulrg 12605 . . . . . . . . . . . 12 ((𝑠 ∈ V ∧ 𝐿 ∈ Ring) β†’ (.rβ€˜πΏ) = (.rβ€˜(𝐿 β†Ύs 𝑠)))
5444, 24, 53sylancr 414 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (.rβ€˜πΏ) = (.rβ€˜(𝐿 β†Ύs 𝑠)))
5554oveqdr 5905 . . . . . . . . . 10 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΏ)𝑦) = (π‘₯(.rβ€˜(𝐿 β†Ύs 𝑠))𝑦))
5643, 50, 553eqtr3d 2218 . . . . . . . . 9 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜(𝐾 β†Ύs 𝑠))𝑦) = (π‘₯(.rβ€˜(𝐿 β†Ύs 𝑠))𝑦))
5731, 56sylan2 286 . . . . . . . 8 (((πœ‘ ∧ 𝐾 ∈ Ring) ∧ (π‘₯ ∈ (𝑠 ∩ 𝐡) ∧ 𝑦 ∈ (𝑠 ∩ 𝐡))) β†’ (π‘₯(.rβ€˜(𝐾 β†Ύs 𝑠))𝑦) = (π‘₯(.rβ€˜(𝐿 β†Ύs 𝑠))𝑦))
5819, 28, 42, 57ringpropd 13222 . . . . . . 7 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ ((𝐾 β†Ύs 𝑠) ∈ Ring ↔ (𝐿 β†Ύs 𝑠) ∈ Ring))
5910, 58anbi12d 473 . . . . . 6 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ ((𝐾 ∈ Ring ∧ (𝐾 β†Ύs 𝑠) ∈ Ring) ↔ (𝐿 ∈ Ring ∧ (𝐿 β†Ύs 𝑠) ∈ Ring)))
604, 5eqtr3d 2212 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΏ))
6160sseq2d 3187 . . . . . . . 8 (πœ‘ β†’ (𝑠 βŠ† (Baseβ€˜πΎ) ↔ 𝑠 βŠ† (Baseβ€˜πΏ)))
6261adantr 276 . . . . . . 7 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (𝑠 βŠ† (Baseβ€˜πΎ) ↔ 𝑠 βŠ† (Baseβ€˜πΏ)))
634adantr 276 . . . . . . . . 9 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ 𝐡 = (Baseβ€˜πΎ))
645adantr 276 . . . . . . . . 9 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ 𝐡 = (Baseβ€˜πΏ))
65 simpr 110 . . . . . . . . 9 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ 𝐾 ∈ Ring)
6663, 64, 43, 65, 24rngidpropdg 13320 . . . . . . . 8 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (1rβ€˜πΎ) = (1rβ€˜πΏ))
6766eleq1d 2246 . . . . . . 7 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ ((1rβ€˜πΎ) ∈ 𝑠 ↔ (1rβ€˜πΏ) ∈ 𝑠))
6862, 67anbi12d 473 . . . . . 6 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ ((𝑠 βŠ† (Baseβ€˜πΎ) ∧ (1rβ€˜πΎ) ∈ 𝑠) ↔ (𝑠 βŠ† (Baseβ€˜πΏ) ∧ (1rβ€˜πΏ) ∈ 𝑠)))
6959, 68anbi12d 473 . . . . 5 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (((𝐾 ∈ Ring ∧ (𝐾 β†Ύs 𝑠) ∈ Ring) ∧ (𝑠 βŠ† (Baseβ€˜πΎ) ∧ (1rβ€˜πΎ) ∈ 𝑠)) ↔ ((𝐿 ∈ Ring ∧ (𝐿 β†Ύs 𝑠) ∈ Ring) ∧ (𝑠 βŠ† (Baseβ€˜πΏ) ∧ (1rβ€˜πΏ) ∈ 𝑠))))
70 eqid 2177 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
71 eqid 2177 . . . . . 6 (1rβ€˜πΎ) = (1rβ€˜πΎ)
7270, 71issubrg 13347 . . . . 5 (𝑠 ∈ (SubRingβ€˜πΎ) ↔ ((𝐾 ∈ Ring ∧ (𝐾 β†Ύs 𝑠) ∈ Ring) ∧ (𝑠 βŠ† (Baseβ€˜πΎ) ∧ (1rβ€˜πΎ) ∈ 𝑠)))
73 eqid 2177 . . . . . 6 (Baseβ€˜πΏ) = (Baseβ€˜πΏ)
74 eqid 2177 . . . . . 6 (1rβ€˜πΏ) = (1rβ€˜πΏ)
7573, 74issubrg 13347 . . . . 5 (𝑠 ∈ (SubRingβ€˜πΏ) ↔ ((𝐿 ∈ Ring ∧ (𝐿 β†Ύs 𝑠) ∈ Ring) ∧ (𝑠 βŠ† (Baseβ€˜πΏ) ∧ (1rβ€˜πΏ) ∈ 𝑠)))
7669, 72, 753bitr4g 223 . . . 4 ((πœ‘ ∧ 𝐾 ∈ Ring) β†’ (𝑠 ∈ (SubRingβ€˜πΎ) ↔ 𝑠 ∈ (SubRingβ€˜πΏ)))
7776ex 115 . . 3 (πœ‘ β†’ (𝐾 ∈ Ring β†’ (𝑠 ∈ (SubRingβ€˜πΎ) ↔ 𝑠 ∈ (SubRingβ€˜πΏ))))
782, 9, 77pm5.21ndd 705 . 2 (πœ‘ β†’ (𝑠 ∈ (SubRingβ€˜πΎ) ↔ 𝑠 ∈ (SubRingβ€˜πΏ)))
7978eqrdv 2175 1 (πœ‘ β†’ (SubRingβ€˜πΎ) = (SubRingβ€˜πΏ))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ∩ cin 3130   βŠ† wss 3131  β€˜cfv 5218  (class class class)co 5877  Basecbs 12464   β†Ύs cress 12465  +gcplusg 12538  .rcmulr 12539  1rcur 13147  Ringcrg 13184  SubRingcsubrg 13343
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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-ltadd 7929
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-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-mgp 13136  df-ur 13148  df-ring 13186  df-subrg 13345
This theorem is referenced by: (None)
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