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Theorem issubrg 19618
 Description: The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.)
Hypotheses
Ref Expression
issubrg.b 𝐵 = (Base‘𝑅)
issubrg.i 1 = (1r𝑅)
Assertion
Ref Expression
issubrg (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))

Proof of Theorem issubrg
Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subrg 19616 . . 3 SubRing = (𝑟 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠)})
21mptrcl 6774 . 2 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3 simpll 766 . 2 (((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)) → 𝑅 ∈ Ring)
4 fveq2 6664 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 issubrg.b . . . . . . . 8 𝐵 = (Base‘𝑅)
64, 5eqtr4di 2812 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
76pweqd 4517 . . . . . 6 (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 𝐵)
8 oveq1 7164 . . . . . . . 8 (𝑟 = 𝑅 → (𝑟s 𝑠) = (𝑅s 𝑠))
98eleq1d 2837 . . . . . . 7 (𝑟 = 𝑅 → ((𝑟s 𝑠) ∈ Ring ↔ (𝑅s 𝑠) ∈ Ring))
10 fveq2 6664 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
11 issubrg.i . . . . . . . . 9 1 = (1r𝑅)
1210, 11eqtr4di 2812 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = 1 )
1312eleq1d 2837 . . . . . . 7 (𝑟 = 𝑅 → ((1r𝑟) ∈ 𝑠1𝑠))
149, 13anbi12d 633 . . . . . 6 (𝑟 = 𝑅 → (((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠) ↔ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)))
157, 14rabeqbidv 3399 . . . . 5 (𝑟 = 𝑅 → {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠)} = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)})
165fvexi 6678 . . . . . . 7 𝐵 ∈ V
1716pwex 5254 . . . . . 6 𝒫 𝐵 ∈ V
1817rabex 5207 . . . . 5 {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ∈ V
1915, 1, 18fvmpt 6765 . . . 4 (𝑅 ∈ Ring → (SubRing‘𝑅) = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)})
2019eleq2d 2838 . . 3 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)}))
21 oveq2 7165 . . . . . . . 8 (𝑠 = 𝐴 → (𝑅s 𝑠) = (𝑅s 𝐴))
2221eleq1d 2837 . . . . . . 7 (𝑠 = 𝐴 → ((𝑅s 𝑠) ∈ Ring ↔ (𝑅s 𝐴) ∈ Ring))
23 eleq2 2841 . . . . . . 7 (𝑠 = 𝐴 → ( 1𝑠1𝐴))
2422, 23anbi12d 633 . . . . . 6 (𝑠 = 𝐴 → (((𝑅s 𝑠) ∈ Ring ∧ 1𝑠) ↔ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
2524elrab 3605 . . . . 5 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ (𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
2616elpw2 5220 . . . . . 6 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
2726anbi1i 626 . . . . 5 ((𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)) ↔ (𝐴𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
28 an12 644 . . . . 5 ((𝐴𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)) ↔ ((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)))
2925, 27, 283bitri 300 . . . 4 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ ((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)))
30 ibar 532 . . . . 5 (𝑅 ∈ Ring → ((𝑅s 𝐴) ∈ Ring ↔ (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring)))
3130anbi1d 632 . . . 4 (𝑅 ∈ Ring → (((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
3229, 31syl5bb 286 . . 3 (𝑅 ∈ Ring → (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
3320, 32bitrd 282 . 2 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
342, 3, 33pm5.21nii 383 1 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1539   ∈ wcel 2112  {crab 3075   ⊆ wss 3861  𝒫 cpw 4498  ‘cfv 6341  (class class class)co 7157  Basecbs 16556   ↾s cress 16557  1rcur 19334  Ringcrg 19380  SubRingcsubrg 19614 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 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  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-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fv 6349  df-ov 7160  df-subrg 19616 This theorem is referenced by:  subrgss  19619  subrgid  19620  subrgring  19621  subrgrcl  19623  subrg1cl  19626  issubrg2  19638  subsubrg  19644  subrgpropd  19653  issubassa  20646  subrgpsr  20762  cphsubrglem  23893
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