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Theorem subrgpropd 18735
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 18503 . . . . 5 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
61ineq2d 3792 . . . . . . 7 (𝜑 → (𝑠𝐵) = (𝑠 ∩ (Base‘𝐾)))
7 vex 3189 . . . . . . . 8 𝑠 ∈ V
8 eqid 2621 . . . . . . . . 9 (𝐾s 𝑠) = (𝐾s 𝑠)
9 eqid 2621 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
108, 9ressbas 15851 . . . . . . . 8 (𝑠 ∈ V → (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝑠)))
117, 10ax-mp 5 . . . . . . 7 (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝑠))
126, 11syl6eq 2671 . . . . . 6 (𝜑 → (𝑠𝐵) = (Base‘(𝐾s 𝑠)))
132ineq2d 3792 . . . . . . 7 (𝜑 → (𝑠𝐵) = (𝑠 ∩ (Base‘𝐿)))
14 eqid 2621 . . . . . . . . 9 (𝐿s 𝑠) = (𝐿s 𝑠)
15 eqid 2621 . . . . . . . . 9 (Base‘𝐿) = (Base‘𝐿)
1614, 15ressbas 15851 . . . . . . . 8 (𝑠 ∈ V → (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿s 𝑠)))
177, 16ax-mp 5 . . . . . . 7 (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿s 𝑠))
1813, 17syl6eq 2671 . . . . . 6 (𝜑 → (𝑠𝐵) = (Base‘(𝐿s 𝑠)))
19 inss2 3812 . . . . . . . . 9 (𝑠𝐵) ⊆ 𝐵
2019sseli 3579 . . . . . . . 8 (𝑥 ∈ (𝑠𝐵) → 𝑥𝐵)
2119sseli 3579 . . . . . . . 8 (𝑦 ∈ (𝑠𝐵) → 𝑦𝐵)
2220, 21anim12i 589 . . . . . . 7 ((𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵)) → (𝑥𝐵𝑦𝐵))
23 eqid 2621 . . . . . . . . . . 11 (+g𝐾) = (+g𝐾)
248, 23ressplusg 15914 . . . . . . . . . 10 (𝑠 ∈ V → (+g𝐾) = (+g‘(𝐾s 𝑠)))
257, 24ax-mp 5 . . . . . . . . 9 (+g𝐾) = (+g‘(𝐾s 𝑠))
2625oveqi 6617 . . . . . . . 8 (𝑥(+g𝐾)𝑦) = (𝑥(+g‘(𝐾s 𝑠))𝑦)
27 eqid 2621 . . . . . . . . . . 11 (+g𝐿) = (+g𝐿)
2814, 27ressplusg 15914 . . . . . . . . . 10 (𝑠 ∈ V → (+g𝐿) = (+g‘(𝐿s 𝑠)))
297, 28ax-mp 5 . . . . . . . . 9 (+g𝐿) = (+g‘(𝐿s 𝑠))
3029oveqi 6617 . . . . . . . 8 (𝑥(+g𝐿)𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦)
313, 26, 303eqtr3g 2678 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g‘(𝐾s 𝑠))𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦))
3222, 31sylan2 491 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵))) → (𝑥(+g‘(𝐾s 𝑠))𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦))
33 eqid 2621 . . . . . . . . . . 11 (.r𝐾) = (.r𝐾)
348, 33ressmulr 15927 . . . . . . . . . 10 (𝑠 ∈ V → (.r𝐾) = (.r‘(𝐾s 𝑠)))
357, 34ax-mp 5 . . . . . . . . 9 (.r𝐾) = (.r‘(𝐾s 𝑠))
3635oveqi 6617 . . . . . . . 8 (𝑥(.r𝐾)𝑦) = (𝑥(.r‘(𝐾s 𝑠))𝑦)
37 eqid 2621 . . . . . . . . . . 11 (.r𝐿) = (.r𝐿)
3814, 37ressmulr 15927 . . . . . . . . . 10 (𝑠 ∈ V → (.r𝐿) = (.r‘(𝐿s 𝑠)))
397, 38ax-mp 5 . . . . . . . . 9 (.r𝐿) = (.r‘(𝐿s 𝑠))
4039oveqi 6617 . . . . . . . 8 (𝑥(.r𝐿)𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦)
414, 36, 403eqtr3g 2678 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r‘(𝐾s 𝑠))𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦))
4222, 41sylan2 491 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵))) → (𝑥(.r‘(𝐾s 𝑠))𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦))
4312, 18, 32, 42ringpropd 18503 . . . . 5 (𝜑 → ((𝐾s 𝑠) ∈ Ring ↔ (𝐿s 𝑠) ∈ Ring))
445, 43anbi12d 746 . . . 4 (𝜑 → ((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ↔ (𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring)))
451, 2eqtr3d 2657 . . . . . 6 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4645sseq2d 3612 . . . . 5 (𝜑 → (𝑠 ⊆ (Base‘𝐾) ↔ 𝑠 ⊆ (Base‘𝐿)))
471, 2, 4rngidpropd 18616 . . . . . 6 (𝜑 → (1r𝐾) = (1r𝐿))
4847eleq1d 2683 . . . . 5 (𝜑 → ((1r𝐾) ∈ 𝑠 ↔ (1r𝐿) ∈ 𝑠))
4946, 48anbi12d 746 . . . 4 (𝜑 → ((𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠)))
5044, 49anbi12d 746 . . 3 (𝜑 → (((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠)) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠))))
51 eqid 2621 . . . 4 (1r𝐾) = (1r𝐾)
529, 51issubrg 18701 . . 3 (𝑠 ∈ (SubRing‘𝐾) ↔ ((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠)))
53 eqid 2621 . . . 4 (1r𝐿) = (1r𝐿)
5415, 53issubrg 18701 . . 3 (𝑠 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠)))
5550, 52, 543bitr4g 303 . 2 (𝜑 → (𝑠 ∈ (SubRing‘𝐾) ↔ 𝑠 ∈ (SubRing‘𝐿)))
5655eqrdv 2619 1 (𝜑 → (SubRing‘𝐾) = (SubRing‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cin 3554  wss 3555  cfv 5847  (class class class)co 6604  Basecbs 15781  s cress 15782  +gcplusg 15862  .rcmulr 15863  1rcur 18422  Ringcrg 18468  SubRingcsubrg 18697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-mgp 18411  df-ur 18423  df-ring 18470  df-subrg 18699
This theorem is referenced by:  ply1subrg  19486  subrgply1  19522
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