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Theorem subrgpropd 19492
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 19254 . . . . 5 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
61ineq2d 4192 . . . . . . 7 (𝜑 → (𝑠𝐵) = (𝑠 ∩ (Base‘𝐾)))
7 eqid 2825 . . . . . . . . 9 (𝐾s 𝑠) = (𝐾s 𝑠)
8 eqid 2825 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
97, 8ressbas 16546 . . . . . . . 8 (𝑠 ∈ V → (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝑠)))
109elv 3504 . . . . . . 7 (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝑠))
116, 10syl6eq 2876 . . . . . 6 (𝜑 → (𝑠𝐵) = (Base‘(𝐾s 𝑠)))
122ineq2d 4192 . . . . . . 7 (𝜑 → (𝑠𝐵) = (𝑠 ∩ (Base‘𝐿)))
13 eqid 2825 . . . . . . . . 9 (𝐿s 𝑠) = (𝐿s 𝑠)
14 eqid 2825 . . . . . . . . 9 (Base‘𝐿) = (Base‘𝐿)
1513, 14ressbas 16546 . . . . . . . 8 (𝑠 ∈ V → (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿s 𝑠)))
1615elv 3504 . . . . . . 7 (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿s 𝑠))
1712, 16syl6eq 2876 . . . . . 6 (𝜑 → (𝑠𝐵) = (Base‘(𝐿s 𝑠)))
18 elinel2 4176 . . . . . . . 8 (𝑥 ∈ (𝑠𝐵) → 𝑥𝐵)
19 elinel2 4176 . . . . . . . 8 (𝑦 ∈ (𝑠𝐵) → 𝑦𝐵)
2018, 19anim12i 612 . . . . . . 7 ((𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵)) → (𝑥𝐵𝑦𝐵))
21 eqid 2825 . . . . . . . . . . 11 (+g𝐾) = (+g𝐾)
227, 21ressplusg 16604 . . . . . . . . . 10 (𝑠 ∈ V → (+g𝐾) = (+g‘(𝐾s 𝑠)))
2322elv 3504 . . . . . . . . 9 (+g𝐾) = (+g‘(𝐾s 𝑠))
2423oveqi 7164 . . . . . . . 8 (𝑥(+g𝐾)𝑦) = (𝑥(+g‘(𝐾s 𝑠))𝑦)
25 eqid 2825 . . . . . . . . . . 11 (+g𝐿) = (+g𝐿)
2613, 25ressplusg 16604 . . . . . . . . . 10 (𝑠 ∈ V → (+g𝐿) = (+g‘(𝐿s 𝑠)))
2726elv 3504 . . . . . . . . 9 (+g𝐿) = (+g‘(𝐿s 𝑠))
2827oveqi 7164 . . . . . . . 8 (𝑥(+g𝐿)𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦)
293, 24, 283eqtr3g 2883 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g‘(𝐾s 𝑠))𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦))
3020, 29sylan2 592 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵))) → (𝑥(+g‘(𝐾s 𝑠))𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦))
31 eqid 2825 . . . . . . . . . . 11 (.r𝐾) = (.r𝐾)
327, 31ressmulr 16617 . . . . . . . . . 10 (𝑠 ∈ V → (.r𝐾) = (.r‘(𝐾s 𝑠)))
3332elv 3504 . . . . . . . . 9 (.r𝐾) = (.r‘(𝐾s 𝑠))
3433oveqi 7164 . . . . . . . 8 (𝑥(.r𝐾)𝑦) = (𝑥(.r‘(𝐾s 𝑠))𝑦)
35 eqid 2825 . . . . . . . . . . 11 (.r𝐿) = (.r𝐿)
3613, 35ressmulr 16617 . . . . . . . . . 10 (𝑠 ∈ V → (.r𝐿) = (.r‘(𝐿s 𝑠)))
3736elv 3504 . . . . . . . . 9 (.r𝐿) = (.r‘(𝐿s 𝑠))
3837oveqi 7164 . . . . . . . 8 (𝑥(.r𝐿)𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦)
394, 34, 383eqtr3g 2883 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r‘(𝐾s 𝑠))𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦))
4020, 39sylan2 592 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵))) → (𝑥(.r‘(𝐾s 𝑠))𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦))
4111, 17, 30, 40ringpropd 19254 . . . . 5 (𝜑 → ((𝐾s 𝑠) ∈ Ring ↔ (𝐿s 𝑠) ∈ Ring))
425, 41anbi12d 630 . . . 4 (𝜑 → ((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ↔ (𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring)))
431, 2eqtr3d 2862 . . . . . 6 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4443sseq2d 4002 . . . . 5 (𝜑 → (𝑠 ⊆ (Base‘𝐾) ↔ 𝑠 ⊆ (Base‘𝐿)))
451, 2, 4rngidpropd 19367 . . . . . 6 (𝜑 → (1r𝐾) = (1r𝐿))
4645eleq1d 2901 . . . . 5 (𝜑 → ((1r𝐾) ∈ 𝑠 ↔ (1r𝐿) ∈ 𝑠))
4744, 46anbi12d 630 . . . 4 (𝜑 → ((𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠)))
4842, 47anbi12d 630 . . 3 (𝜑 → (((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠)) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠))))
49 eqid 2825 . . . 4 (1r𝐾) = (1r𝐾)
508, 49issubrg 19457 . . 3 (𝑠 ∈ (SubRing‘𝐾) ↔ ((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠)))
51 eqid 2825 . . . 4 (1r𝐿) = (1r𝐿)
5214, 51issubrg 19457 . . 3 (𝑠 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠)))
5348, 50, 523bitr4g 315 . 2 (𝜑 → (𝑠 ∈ (SubRing‘𝐾) ↔ 𝑠 ∈ (SubRing‘𝐿)))
5453eqrdv 2823 1 (𝜑 → (SubRing‘𝐾) = (SubRing‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  Vcvv 3499  cin 3938  wss 3939  cfv 6351  (class class class)co 7151  Basecbs 16475  s cress 16476  +gcplusg 16557  .rcmulr 16558  1rcur 19173  Ringcrg 19219  SubRingcsubrg 19453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-0g 16707  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-grp 18038  df-mgp 19162  df-ur 19174  df-ring 19221  df-subrg 19455
This theorem is referenced by:  ply1subrg  20284  subrgply1  20320  srasubrg  30877
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