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Theorem subrgpropd 18637
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 18405 . . . . 5 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
61ineq2d 3776 . . . . . . 7 (𝜑 → (𝑠𝐵) = (𝑠 ∩ (Base‘𝐾)))
7 vex 3176 . . . . . . . 8 𝑠 ∈ V
8 eqid 2610 . . . . . . . . 9 (𝐾s 𝑠) = (𝐾s 𝑠)
9 eqid 2610 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
108, 9ressbas 15757 . . . . . . . 8 (𝑠 ∈ V → (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝑠)))
117, 10ax-mp 5 . . . . . . 7 (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝑠))
126, 11syl6eq 2660 . . . . . 6 (𝜑 → (𝑠𝐵) = (Base‘(𝐾s 𝑠)))
132ineq2d 3776 . . . . . . 7 (𝜑 → (𝑠𝐵) = (𝑠 ∩ (Base‘𝐿)))
14 eqid 2610 . . . . . . . . 9 (𝐿s 𝑠) = (𝐿s 𝑠)
15 eqid 2610 . . . . . . . . 9 (Base‘𝐿) = (Base‘𝐿)
1614, 15ressbas 15757 . . . . . . . 8 (𝑠 ∈ V → (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿s 𝑠)))
177, 16ax-mp 5 . . . . . . 7 (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿s 𝑠))
1813, 17syl6eq 2660 . . . . . 6 (𝜑 → (𝑠𝐵) = (Base‘(𝐿s 𝑠)))
19 inss2 3796 . . . . . . . . 9 (𝑠𝐵) ⊆ 𝐵
2019sseli 3564 . . . . . . . 8 (𝑥 ∈ (𝑠𝐵) → 𝑥𝐵)
2119sseli 3564 . . . . . . . 8 (𝑦 ∈ (𝑠𝐵) → 𝑦𝐵)
2220, 21anim12i 588 . . . . . . 7 ((𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵)) → (𝑥𝐵𝑦𝐵))
23 eqid 2610 . . . . . . . . . . 11 (+g𝐾) = (+g𝐾)
248, 23ressplusg 15818 . . . . . . . . . 10 (𝑠 ∈ V → (+g𝐾) = (+g‘(𝐾s 𝑠)))
257, 24ax-mp 5 . . . . . . . . 9 (+g𝐾) = (+g‘(𝐾s 𝑠))
2625oveqi 6562 . . . . . . . 8 (𝑥(+g𝐾)𝑦) = (𝑥(+g‘(𝐾s 𝑠))𝑦)
27 eqid 2610 . . . . . . . . . . 11 (+g𝐿) = (+g𝐿)
2814, 27ressplusg 15818 . . . . . . . . . 10 (𝑠 ∈ V → (+g𝐿) = (+g‘(𝐿s 𝑠)))
297, 28ax-mp 5 . . . . . . . . 9 (+g𝐿) = (+g‘(𝐿s 𝑠))
3029oveqi 6562 . . . . . . . 8 (𝑥(+g𝐿)𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦)
313, 26, 303eqtr3g 2667 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g‘(𝐾s 𝑠))𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦))
3222, 31sylan2 490 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵))) → (𝑥(+g‘(𝐾s 𝑠))𝑦) = (𝑥(+g‘(𝐿s 𝑠))𝑦))
33 eqid 2610 . . . . . . . . . . 11 (.r𝐾) = (.r𝐾)
348, 33ressmulr 15829 . . . . . . . . . 10 (𝑠 ∈ V → (.r𝐾) = (.r‘(𝐾s 𝑠)))
357, 34ax-mp 5 . . . . . . . . 9 (.r𝐾) = (.r‘(𝐾s 𝑠))
3635oveqi 6562 . . . . . . . 8 (𝑥(.r𝐾)𝑦) = (𝑥(.r‘(𝐾s 𝑠))𝑦)
37 eqid 2610 . . . . . . . . . . 11 (.r𝐿) = (.r𝐿)
3814, 37ressmulr 15829 . . . . . . . . . 10 (𝑠 ∈ V → (.r𝐿) = (.r‘(𝐿s 𝑠)))
397, 38ax-mp 5 . . . . . . . . 9 (.r𝐿) = (.r‘(𝐿s 𝑠))
4039oveqi 6562 . . . . . . . 8 (𝑥(.r𝐿)𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦)
414, 36, 403eqtr3g 2667 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r‘(𝐾s 𝑠))𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦))
4222, 41sylan2 490 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑠𝐵) ∧ 𝑦 ∈ (𝑠𝐵))) → (𝑥(.r‘(𝐾s 𝑠))𝑦) = (𝑥(.r‘(𝐿s 𝑠))𝑦))
4312, 18, 32, 42ringpropd 18405 . . . . 5 (𝜑 → ((𝐾s 𝑠) ∈ Ring ↔ (𝐿s 𝑠) ∈ Ring))
445, 43anbi12d 743 . . . 4 (𝜑 → ((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ↔ (𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring)))
451, 2eqtr3d 2646 . . . . . 6 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4645sseq2d 3596 . . . . 5 (𝜑 → (𝑠 ⊆ (Base‘𝐾) ↔ 𝑠 ⊆ (Base‘𝐿)))
471, 2, 4rngidpropd 18518 . . . . . 6 (𝜑 → (1r𝐾) = (1r𝐿))
4847eleq1d 2672 . . . . 5 (𝜑 → ((1r𝐾) ∈ 𝑠 ↔ (1r𝐿) ∈ 𝑠))
4946, 48anbi12d 743 . . . 4 (𝜑 → ((𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠)))
5044, 49anbi12d 743 . . 3 (𝜑 → (((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠)) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠))))
51 eqid 2610 . . . 4 (1r𝐾) = (1r𝐾)
529, 51issubrg 18603 . . 3 (𝑠 ∈ (SubRing‘𝐾) ↔ ((𝐾 ∈ Ring ∧ (𝐾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r𝐾) ∈ 𝑠)))
53 eqid 2610 . . . 4 (1r𝐿) = (1r𝐿)
5415, 53issubrg 18603 . . 3 (𝑠 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝑠)))
5550, 52, 543bitr4g 302 . 2 (𝜑 → (𝑠 ∈ (SubRing‘𝐾) ↔ 𝑠 ∈ (SubRing‘𝐿)))
5655eqrdv 2608 1 (𝜑 → (SubRing‘𝐾) = (SubRing‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cin 3539  wss 3540  cfv 5804  (class class class)co 6549  Basecbs 15695  s cress 15696  +gcplusg 15768  .rcmulr 15769  1rcur 18324  Ringcrg 18370  SubRingcsubrg 18599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601
This theorem is referenced by:  ply1subrg  19388  subrgply1  19424
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