MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subrngid Structured version   Visualization version   GIF version

Theorem subrngid 20449
Description: Every non-unital ring is a subring of itself. (Contributed by AV, 14-Feb-2025.)
Hypothesis
Ref Expression
subrngss.1 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
subrngid (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅))

Proof of Theorem subrngid
StepHypRef Expression
1 id 22 . 2 (𝑅 ∈ Rng → 𝑅 ∈ Rng)
2 subrngss.1 . . . 4 𝐵 = (Base‘𝑅)
32ressid 17198 . . 3 (𝑅 ∈ Rng → (𝑅s 𝐵) = 𝑅)
43, 1eqeltrd 2827 . 2 (𝑅 ∈ Rng → (𝑅s 𝐵) ∈ Rng)
5 ssidd 4000 . 2 (𝑅 ∈ Rng → 𝐵𝐵)
62issubrng 20447 . 2 (𝐵 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐵) ∈ Rng ∧ 𝐵𝐵))
71, 4, 5, 6syl3anbrc 1340 1 (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wss 3943  cfv 6537  (class class class)co 7405  Basecbs 17153  s cress 17182  Rngcrng 20057  SubRngcsubrng 20445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-ress 17183  df-subrng 20446
This theorem is referenced by:  subrngmre  20462
  Copyright terms: Public domain W3C validator