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Theorem subrgpropd 19631
 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 19396 . . . . 5 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
61ineq2d 4118 . . . . . . 7 (𝜑 → (𝑠𝐵) = (𝑠 ∩ (Base‘𝐾)))
7 eqid 2759 . . . . . . . . 9 (𝐾s 𝑠) = (𝐾s 𝑠)
8 eqid 2759 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
97, 8ressbas 16605 . . . . . . . 8 (𝑠 ∈ V → (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝑠)))
109elv 3416 . . . . . . 7 (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝑠))
116, 10eqtrdi 2810 . . . . . 6 (𝜑 → (𝑠𝐵) = (Base‘(𝐾s 𝑠)))
122ineq2d 4118 . . . . . . 7 (𝜑 → (𝑠𝐵) = (𝑠 ∩ (Base‘𝐿)))
13 eqid 2759 . . . . . . . . 9 (𝐿s 𝑠) = (𝐿s 𝑠)
14 eqid 2759 . . . . . . . . 9 (Base‘𝐿) = (Base‘𝐿)
1513, 14ressbas 16605 . . . . . . . 8 (𝑠 ∈ V → (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿s 𝑠)))
1615elv 3416 . . . . . . 7 (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿s 𝑠))
1712, 16eqtrdi 2810 . . . . . 6 (𝜑 → (𝑠𝐵) = (Base‘(𝐿s 𝑠)))
18 elinel2 4102 . . . . . . . 8 (𝑥 ∈ (𝑠𝐵) → 𝑥𝐵)
19 elinel2 4102 . . . . . . . 8 (𝑦 ∈ (𝑠𝐵) → 𝑦𝐵)
2018, 19anim12i 616 . . . . . . 7 ((𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵)) → (𝑥𝐵𝑦𝐵))
21 eqid 2759 . . . . . . . . . . 11 (+g𝐾) = (+g𝐾)
227, 21ressplusg 16663 . . . . . . . . . 10 (𝑠 ∈ V → (+g𝐾) = (+g‘(𝐾s 𝑠)))
2322elv 3416 . . . . . . . . 9 (+g𝐾) = (+g‘(𝐾s 𝑠))
2423oveqi 7164 . . . . . . . 8 (𝑥(+g𝐾)𝑦) = (𝑥(+g‘(𝐾s 𝑠))𝑦)
25 eqid 2759 . . . . . . . . . . 11 (+g𝐿) = (+g𝐿)
2613, 25ressplusg 16663 . . . . . . . . . 10 (𝑠 ∈ V → (+g𝐿) = (+g‘(𝐿s 𝑠)))
2726elv 3416 . . . . . . . . 9 (+g𝐿) = (+g‘(𝐿s 𝑠))
2827oveqi 7164 . . . . . . . 8 (𝑥(+g𝐿)𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦)
293, 24, 283eqtr3g 2817 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g‘(𝐾s 𝑠))𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦))
3020, 29sylan2 596 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵))) → (𝑥(+g‘(𝐾s 𝑠))𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦))
31 eqid 2759 . . . . . . . . . . 11 (.r𝐾) = (.r𝐾)
327, 31ressmulr 16676 . . . . . . . . . 10 (𝑠 ∈ V → (.r𝐾) = (.r‘(𝐾s 𝑠)))
3332elv 3416 . . . . . . . . 9 (.r𝐾) = (.r‘(𝐾s 𝑠))
3433oveqi 7164 . . . . . . . 8 (𝑥(.r𝐾)𝑦) = (𝑥(.r‘(𝐾s 𝑠))𝑦)
35 eqid 2759 . . . . . . . . . . 11 (.r𝐿) = (.r𝐿)
3613, 35ressmulr 16676 . . . . . . . . . 10 (𝑠 ∈ V → (.r𝐿) = (.r‘(𝐿s 𝑠)))
3736elv 3416 . . . . . . . . 9 (.r𝐿) = (.r‘(𝐿s 𝑠))
3837oveqi 7164 . . . . . . . 8 (𝑥(.r𝐿)𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦)
394, 34, 383eqtr3g 2817 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r‘(𝐾s 𝑠))𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦))
4020, 39sylan2 596 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵))) → (𝑥(.r‘(𝐾s 𝑠))𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦))
4111, 17, 30, 40ringpropd 19396 . . . . 5 (𝜑 → ((𝐾s 𝑠) ∈ Ring ↔ (𝐿s 𝑠) ∈ Ring))
425, 41anbi12d 634 . . . 4 (𝜑 → ((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ↔ (𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring)))
431, 2eqtr3d 2796 . . . . . 6 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4443sseq2d 3925 . . . . 5 (𝜑 → (𝑠 ⊆ (Base‘𝐾) ↔ 𝑠 ⊆ (Base‘𝐿)))
451, 2, 4rngidpropd 19509 . . . . . 6 (𝜑 → (1r𝐾) = (1r𝐿))
4645eleq1d 2837 . . . . 5 (𝜑 → ((1r𝐾) ∈ 𝑠 ↔ (1r𝐿) ∈ 𝑠))
4744, 46anbi12d 634 . . . 4 (𝜑 → ((𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠)))
4842, 47anbi12d 634 . . 3 (𝜑 → (((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠)) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠))))
49 eqid 2759 . . . 4 (1r𝐾) = (1r𝐾)
508, 49issubrg 19596 . . 3 (𝑠 ∈ (SubRing‘𝐾) ↔ ((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠)))
51 eqid 2759 . . . 4 (1r𝐿) = (1r𝐿)
5214, 51issubrg 19596 . . 3 (𝑠 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠)))
5348, 50, 523bitr4g 318 . 2 (𝜑 → (𝑠 ∈ (SubRing‘𝐾) ↔ 𝑠 ∈ (SubRing‘𝐿)))
5453eqrdv 2757 1 (𝜑 → (SubRing‘𝐾) = (SubRing‘𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  Vcvv 3410   ∩ cin 3858   ⊆ wss 3859  ‘cfv 6336  (class class class)co 7151  Basecbs 16534   ↾s cress 16535  +gcplusg 16616  .rcmulr 16617  1rcur 19312  Ringcrg 19358  SubRingcsubrg 19592 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10624  ax-resscn 10625  ax-1cn 10626  ax-icn 10627  ax-addcl 10628  ax-addrcl 10629  ax-mulcl 10630  ax-mulrcl 10631  ax-mulcom 10632  ax-addass 10633  ax-mulass 10634  ax-distr 10635  ax-i2m1 10636  ax-1ne0 10637  ax-1rid 10638  ax-rnegex 10639  ax-rrecex 10640  ax-cnre 10641  ax-pre-lttri 10642  ax-pre-lttrn 10643  ax-pre-ltadd 10644  ax-pre-mulgt0 10645 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-pnf 10708  df-mnf 10709  df-xr 10710  df-ltxr 10711  df-le 10712  df-sub 10903  df-neg 10904  df-nn 11668  df-2 11730  df-3 11731  df-ndx 16537  df-slot 16538  df-base 16540  df-sets 16541  df-ress 16542  df-plusg 16629  df-mulr 16630  df-0g 16766  df-mgm 17911  df-sgrp 17960  df-mnd 17971  df-grp 18165  df-mgp 19301  df-ur 19313  df-ring 19360  df-subrg 19594 This theorem is referenced by:  ply1subrg  20914  subrgply1  20950  srasubrg  31188
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