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Theorem subrgpropd 20392
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 (𝜑𝐵 = (Base‘𝐾))
subrgpropd.2 (𝜑𝐵 = (Base‘𝐿))
subrgpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
subrgpropd.4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
Assertion
Ref Expression
subrgpropd (𝜑 → (SubRing‘𝐾) = (SubRing‘𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝐿,𝑦

Proof of Theorem subrgpropd
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 subrgpropd.1 . . . . . 6 (𝜑𝐵 = (Base‘𝐾))
2 subrgpropd.2 . . . . . 6 (𝜑𝐵 = (Base‘𝐿))
3 subrgpropd.3 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
4 subrgpropd.4 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
51, 2, 3, 4ringpropd 20095 . . . . 5 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
61ineq2d 4211 . . . . . . 7 (𝜑 → (𝑠𝐵) = (𝑠 ∩ (Base‘𝐾)))
7 eqid 2732 . . . . . . . . 9 (𝐾s 𝑠) = (𝐾s 𝑠)
8 eqid 2732 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
97, 8ressbas 17175 . . . . . . . 8 (𝑠 ∈ V → (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝑠)))
109elv 3480 . . . . . . 7 (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝑠))
116, 10eqtrdi 2788 . . . . . 6 (𝜑 → (𝑠𝐵) = (Base‘(𝐾s 𝑠)))
122ineq2d 4211 . . . . . . 7 (𝜑 → (𝑠𝐵) = (𝑠 ∩ (Base‘𝐿)))
13 eqid 2732 . . . . . . . . 9 (𝐿s 𝑠) = (𝐿s 𝑠)
14 eqid 2732 . . . . . . . . 9 (Base‘𝐿) = (Base‘𝐿)
1513, 14ressbas 17175 . . . . . . . 8 (𝑠 ∈ V → (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿s 𝑠)))
1615elv 3480 . . . . . . 7 (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿s 𝑠))
1712, 16eqtrdi 2788 . . . . . 6 (𝜑 → (𝑠𝐵) = (Base‘(𝐿s 𝑠)))
18 elinel2 4195 . . . . . . . 8 (𝑥 ∈ (𝑠𝐵) → 𝑥𝐵)
19 elinel2 4195 . . . . . . . 8 (𝑦 ∈ (𝑠𝐵) → 𝑦𝐵)
2018, 19anim12i 613 . . . . . . 7 ((𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵)) → (𝑥𝐵𝑦𝐵))
21 eqid 2732 . . . . . . . . . . 11 (+g𝐾) = (+g𝐾)
227, 21ressplusg 17231 . . . . . . . . . 10 (𝑠 ∈ V → (+g𝐾) = (+g‘(𝐾s 𝑠)))
2322elv 3480 . . . . . . . . 9 (+g𝐾) = (+g‘(𝐾s 𝑠))
2423oveqi 7418 . . . . . . . 8 (𝑥(+g𝐾)𝑦) = (𝑥(+g‘(𝐾s 𝑠))𝑦)
25 eqid 2732 . . . . . . . . . . 11 (+g𝐿) = (+g𝐿)
2613, 25ressplusg 17231 . . . . . . . . . 10 (𝑠 ∈ V → (+g𝐿) = (+g‘(𝐿s 𝑠)))
2726elv 3480 . . . . . . . . 9 (+g𝐿) = (+g‘(𝐿s 𝑠))
2827oveqi 7418 . . . . . . . 8 (𝑥(+g𝐿)𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦)
293, 24, 283eqtr3g 2795 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g‘(𝐾s 𝑠))𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦))
3020, 29sylan2 593 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵))) → (𝑥(+g‘(𝐾s 𝑠))𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦))
31 eqid 2732 . . . . . . . . . . 11 (.r𝐾) = (.r𝐾)
327, 31ressmulr 17248 . . . . . . . . . 10 (𝑠 ∈ V → (.r𝐾) = (.r‘(𝐾s 𝑠)))
3332elv 3480 . . . . . . . . 9 (.r𝐾) = (.r‘(𝐾s 𝑠))
3433oveqi 7418 . . . . . . . 8 (𝑥(.r𝐾)𝑦) = (𝑥(.r‘(𝐾s 𝑠))𝑦)
35 eqid 2732 . . . . . . . . . . 11 (.r𝐿) = (.r𝐿)
3613, 35ressmulr 17248 . . . . . . . . . 10 (𝑠 ∈ V → (.r𝐿) = (.r‘(𝐿s 𝑠)))
3736elv 3480 . . . . . . . . 9 (.r𝐿) = (.r‘(𝐿s 𝑠))
3837oveqi 7418 . . . . . . . 8 (𝑥(.r𝐿)𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦)
394, 34, 383eqtr3g 2795 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r‘(𝐾s 𝑠))𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦))
4020, 39sylan2 593 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵))) → (𝑥(.r‘(𝐾s 𝑠))𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦))
4111, 17, 30, 40ringpropd 20095 . . . . 5 (𝜑 → ((𝐾s 𝑠) ∈ Ring ↔ (𝐿s 𝑠) ∈ Ring))
425, 41anbi12d 631 . . . 4 (𝜑 → ((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ↔ (𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring)))
431, 2eqtr3d 2774 . . . . . 6 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4443sseq2d 4013 . . . . 5 (𝜑 → (𝑠 ⊆ (Base‘𝐾) ↔ 𝑠 ⊆ (Base‘𝐿)))
451, 2, 4rngidpropd 20221 . . . . . 6 (𝜑 → (1r𝐾) = (1r𝐿))
4645eleq1d 2818 . . . . 5 (𝜑 → ((1r𝐾) ∈ 𝑠 ↔ (1r𝐿) ∈ 𝑠))
4744, 46anbi12d 631 . . . 4 (𝜑 → ((𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠)))
4842, 47anbi12d 631 . . 3 (𝜑 → (((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠)) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠))))
49 eqid 2732 . . . 4 (1r𝐾) = (1r𝐾)
508, 49issubrg 20355 . . 3 (𝑠 ∈ (SubRing‘𝐾) ↔ ((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠)))
51 eqid 2732 . . . 4 (1r𝐿) = (1r𝐿)
5214, 51issubrg 20355 . . 3 (𝑠 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠)))
5348, 50, 523bitr4g 313 . 2 (𝜑 → (𝑠 ∈ (SubRing‘𝐾) ↔ 𝑠 ∈ (SubRing‘𝐿)))
5453eqrdv 2730 1 (𝜑 → (SubRing‘𝐾) = (SubRing‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cin 3946  wss 3947  cfv 6540  (class class class)co 7405  Basecbs 17140  s cress 17169  +gcplusg 17193  .rcmulr 17194  1rcur 19998  Ringcrg 20049  SubRingcsubrg 20351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-mgp 19982  df-ur 19999  df-ring 20051  df-subrg 20353
This theorem is referenced by:  ply1subrg  21712  subrgply1  21746  srasubrg  32662
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