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Theorem issubrg 20319
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 20317 . . 3 SubRing = (π‘Ÿ ∈ Ring ↦ {𝑠 ∈ 𝒫 (Baseβ€˜π‘Ÿ) ∣ ((π‘Ÿ β†Ύs 𝑠) ∈ Ring ∧ (1rβ€˜π‘Ÿ) ∈ 𝑠)})
21mptrcl 7008 . 2 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ Ring)
3 simpll 766 . 2 (((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴)) β†’ 𝑅 ∈ Ring)
4 fveq2 6892 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
5 issubrg.b . . . . . . . 8 𝐡 = (Baseβ€˜π‘…)
64, 5eqtr4di 2791 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = 𝐡)
76pweqd 4620 . . . . . 6 (π‘Ÿ = 𝑅 β†’ 𝒫 (Baseβ€˜π‘Ÿ) = 𝒫 𝐡)
8 oveq1 7416 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ β†Ύs 𝑠) = (𝑅 β†Ύs 𝑠))
98eleq1d 2819 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((π‘Ÿ β†Ύs 𝑠) ∈ Ring ↔ (𝑅 β†Ύs 𝑠) ∈ Ring))
10 fveq2 6892 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = (1rβ€˜π‘…))
11 issubrg.i . . . . . . . . 9 1 = (1rβ€˜π‘…)
1210, 11eqtr4di 2791 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = 1 )
1312eleq1d 2819 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((1rβ€˜π‘Ÿ) ∈ 𝑠 ↔ 1 ∈ 𝑠))
149, 13anbi12d 632 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (((π‘Ÿ β†Ύs 𝑠) ∈ Ring ∧ (1rβ€˜π‘Ÿ) ∈ 𝑠) ↔ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)))
157, 14rabeqbidv 3450 . . . . 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 6906 . . . . . . 7 𝐡 ∈ V
1716pwex 5379 . . . . . 6 𝒫 𝐡 ∈ V
1817rabex 5333 . . . . 5 {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ∈ V
1915, 1, 18fvmpt 6999 . . . 4 (𝑅 ∈ Ring β†’ (SubRingβ€˜π‘…) = {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)})
2019eleq2d 2820 . . 3 (𝑅 ∈ Ring β†’ (𝐴 ∈ (SubRingβ€˜π‘…) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)}))
21 oveq2 7417 . . . . . . . 8 (𝑠 = 𝐴 β†’ (𝑅 β†Ύs 𝑠) = (𝑅 β†Ύs 𝐴))
2221eleq1d 2819 . . . . . . 7 (𝑠 = 𝐴 β†’ ((𝑅 β†Ύs 𝑠) ∈ Ring ↔ (𝑅 β†Ύs 𝐴) ∈ Ring))
23 eleq2 2823 . . . . . . 7 (𝑠 = 𝐴 β†’ ( 1 ∈ 𝑠 ↔ 1 ∈ 𝐴))
2422, 23anbi12d 632 . . . . . 6 (𝑠 = 𝐴 β†’ (((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠) ↔ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
2524elrab 3684 . . . . 5 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ (𝐴 ∈ 𝒫 𝐡 ∧ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
2616elpw2 5346 . . . . . 6 (𝐴 ∈ 𝒫 𝐡 ↔ 𝐴 βŠ† 𝐡)
2726anbi1i 625 . . . . 5 ((𝐴 ∈ 𝒫 𝐡 ∧ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)) ↔ (𝐴 βŠ† 𝐡 ∧ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
28 an12 644 . . . . 5 ((𝐴 βŠ† 𝐡 ∧ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)) ↔ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴)))
2925, 27, 283bitri 297 . . . 4 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐡 ∣ ((𝑅 β†Ύs 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴)))
30 ibar 530 . . . . 5 (𝑅 ∈ Ring β†’ ((𝑅 β†Ύs 𝐴) ∈ Ring ↔ (𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring)))
3130anbi1d 631 . . . 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 283 . . 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 279 . 2 (𝑅 ∈ Ring β†’ (𝐴 ∈ (SubRingβ€˜π‘…) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴))))
342, 3, 33pm5.21nii 380 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 397   = wceq 1542   ∈ wcel 2107  {crab 3433   βŠ† wss 3949  π’« cpw 4603  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144   β†Ύs cress 17173  1rcur 20004  Ringcrg 20056  SubRingcsubrg 20315
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-subrg 20317
This theorem is referenced by:  subrgss  20320  subrgid  20321  subrgring  20322  subrgrcl  20324  subrg1cl  20327  issubrg2  20339  subsubrg  20345  subrgpropd  20355  issubassa  21421  subrgpsr  21539  cphsubrglem  24694  fldgensdrg  32404
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