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

Theorem brric 18509
Description: The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)
Assertion
Ref Expression
brric (𝑅𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)

Proof of Theorem brric
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ric 18483 . 2 𝑟 = ( RingIso “ (V ∖ 1𝑜))
2 ovex 6551 . . . . 5 (𝑟 RingHom 𝑠) ∈ V
3 rabexg 4730 . . . . 5 ((𝑟 RingHom 𝑠) ∈ V → { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)} ∈ V)
42, 3mp1i 13 . . . 4 ((𝑟 ∈ V ∧ 𝑠 ∈ V) → { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)} ∈ V)
54rgen2a 2955 . . 3 𝑟 ∈ V ∀𝑠 ∈ V { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)} ∈ V
6 df-rngiso 18481 . . . 4 RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)})
76fnmpt2 7100 . . 3 (∀𝑟 ∈ V ∀𝑠 ∈ V { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)} ∈ V → RingIso Fn (V × V))
85, 7ax-mp 5 . 2 RingIso Fn (V × V)
91, 8brwitnlem 7447 1 (𝑅𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  wcel 1975  wne 2775  wral 2891  {crab 2895  Vcvv 3168  c0 3869   class class class wbr 4573   × cxp 5022  ccnv 5023   Fn wfn 5781  (class class class)co 6523   RingHom crh 18477   RingIso crs 18478  𝑟 cric 18479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-1o 7420  df-rngiso 18481  df-ric 18483
This theorem is referenced by:  brric2  18510  mat1ric  20050  scmatric  20100  matcpmric  20321  pmmpric  20385
  Copyright terms: Public domain W3C validator