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Theorem issubrg 20461
Description: The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.)
Hypotheses
Ref Expression
issubrg.b 𝐡 = (Baseβ€˜π‘…)
issubrg.i 1 = (1rβ€˜π‘…)
Assertion
Ref Expression
issubrg (𝐴 ∈ (SubRingβ€˜π‘…) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴)))
Proof of Theorem issubrg
Dummy variables 𝑠 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subrg 20459 . . 3 SubRing = (π‘Ÿ ∈ Ring ↦ {𝑠 ∈ 𝒫 (Baseβ€˜π‘Ÿ) ∣ ((π‘Ÿ β†Ύs 𝑠) ∈ Ring ∧ (1rβ€˜π‘Ÿ) ∈ 𝑠)})
21mptrcl 7007 . 2 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ Ring)
3 simpll 765 . 2 (((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴)) β†’ 𝑅 ∈ Ring)
4 fveq2 6891 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
5 issubrg.b . . . . . . . 8 𝐡 = (Baseβ€˜π‘…)
64, 5eqtr4di 2790 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = 𝐡)
76pweqd 4619 . . . . . 6 (π‘Ÿ = 𝑅 β†’ 𝒫 (Baseβ€˜π‘Ÿ) = 𝒫 𝐡)
8 oveq1 7418 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ β†Ύs 𝑠) = (𝑅 β†Ύs 𝑠))
98eleq1d 2818 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((π‘Ÿ β†Ύs 𝑠) ∈ Ring ↔ (𝑅 β†Ύs 𝑠) ∈ Ring))
10 fveq2 6891 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = (1rβ€˜π‘…))
11 issubrg.i . . . . . . . . 9 1 = (1rβ€˜π‘…)
1210, 11eqtr4di 2790 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = 1 )
1312eleq1d 2818 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((1rβ€˜π‘Ÿ) ∈ 𝑠 ↔ 1 ∈ 𝑠))
149, 13anbi12d 631 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (((π‘Ÿ β†Ύs 𝑠) ∈ Ring ∧ (1rβ€˜π‘Ÿ) ∈ 𝑠) ↔ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)))
157, 14rabeqbidv 3449 . . . . 5 (π‘Ÿ = 𝑅 β†’ {𝑠 ∈ 𝒫 (Baseβ€˜π‘Ÿ) ∣ ((π‘Ÿ β†Ύs 𝑠) ∈ Ring ∧ (1rβ€˜π‘Ÿ) ∈ 𝑠)} = {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)})
165fvexi 6905 . . . . . . 7 𝐡 ∈ V
1716pwex 5378 . . . . . 6 𝒫 𝐡 ∈ V
1817rabex 5332 . . . . 5 {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ∈ V
1915, 1, 18fvmpt 6998 . . . 4 (𝑅 ∈ Ring β†’ (SubRingβ€˜π‘…) = {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)})
2019eleq2d 2819 . . 3 (𝑅 ∈ Ring β†’ (𝐴 ∈ (SubRingβ€˜π‘…) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)}))
21 oveq2 7419 . . . . . . . 8 (𝑠 = 𝐴 β†’ (𝑅 β†Ύs 𝑠) = (𝑅 β†Ύs 𝐴))
2221eleq1d 2818 . . . . . . 7 (𝑠 = 𝐴 β†’ ((𝑅 β†Ύs 𝑠) ∈ Ring ↔ (𝑅 β†Ύs 𝐴) ∈ Ring))
23 eleq2 2822 . . . . . . 7 (𝑠 = 𝐴 β†’ ( 1 ∈ 𝑠 ↔ 1 ∈ 𝐴))
2422, 23anbi12d 631 . . . . . 6 (𝑠 = 𝐴 β†’ (((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠) ↔ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
2524elrab 3683 . . . . 5 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ (𝐴 ∈ 𝒫 𝐡 ∧ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
2616elpw2 5345 . . . . . 6 (𝐴 ∈ 𝒫 𝐡 ↔ 𝐴 βŠ† 𝐡)
2726anbi1i 624 . . . . 5 ((𝐴 ∈ 𝒫 𝐡 ∧ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)) ↔ (𝐴 βŠ† 𝐡 ∧ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
28 an12 643 . . . . 5 ((𝐴 βŠ† 𝐡 ∧ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)) ↔ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴)))
2925, 27, 283bitri 296 . . . 4 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴)))
30 ibar 529 . . . . 5 (𝑅 ∈ Ring β†’ ((𝑅 β†Ύs 𝐴) ∈ Ring ↔ (𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring)))
3130anbi1d 630 . . . 4 (𝑅 ∈ Ring β†’ (((𝑅 β†Ύs 𝐴) ∈ Ring ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴)) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴))))
3229, 31bitrid 282 . . 3 (𝑅 ∈ Ring β†’ (𝐴 ∈ {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴))))
3320, 32bitrd 278 . 2 (𝑅 ∈ Ring β†’ (𝐴 ∈ (SubRingβ€˜π‘…) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴))))
342, 3, 33pm5.21nii 379 1 (𝐴 ∈ (SubRingβ€˜π‘…) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432   βŠ† wss 3948  π’« cpw 4602  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148   β†Ύs cress 17177  1rcur 20075  Ringcrg 20127  SubRingcsubrg 20457
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-subrg 20459
This theorem is referenced by:  subrgss  20462  subrgid  20463  subrgring  20464  subrgrcl  20466  subrgsubrng  20468  subrg1cl  20470  issubrg2  20482  subsubrg  20488  subrgpropd  20498  issubassa  21640  subrgpsr  21758  cphsubrglem  24918  fldgensdrg  32662
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