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

Theorem isrngim 20462
Description: An isomorphism of non-unital rings is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Feb-2020.)
Assertion
Ref Expression
isrngim ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹 ∈ (𝑆 RngHom 𝑅))))

Proof of Theorem isrngim
Dummy variables 𝑓 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rngim 20454 . . . . 5 RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓 ∈ (𝑠 RngHom 𝑟)})
21a1i 11 . . . 4 ((𝑅𝑉𝑆𝑊) → RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓 ∈ (𝑠 RngHom 𝑟)}))
3 oveq12 7440 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆))
43adantl 481 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆))
5 oveq12 7440 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑠 RngHom 𝑟) = (𝑆 RngHom 𝑅))
65ancoms 458 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠 RngHom 𝑟) = (𝑆 RngHom 𝑅))
76adantl 481 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑠 RngHom 𝑟) = (𝑆 RngHom 𝑅))
87eleq2d 2825 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑓 ∈ (𝑠 RngHom 𝑟) ↔ 𝑓 ∈ (𝑆 RngHom 𝑅)))
94, 8rabeqbidv 3452 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓 ∈ (𝑠 RngHom 𝑟)} = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓 ∈ (𝑆 RngHom 𝑅)})
10 elex 3499 . . . . 5 (𝑅𝑉𝑅 ∈ V)
1110adantr 480 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑅 ∈ V)
12 elex 3499 . . . . 5 (𝑆𝑊𝑆 ∈ V)
1312adantl 481 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑆 ∈ V)
14 ovex 7464 . . . . . 6 (𝑅 RngHom 𝑆) ∈ V
1514rabex 5345 . . . . 5 {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓 ∈ (𝑆 RngHom 𝑅)} ∈ V
1615a1i 11 . . . 4 ((𝑅𝑉𝑆𝑊) → {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓 ∈ (𝑆 RngHom 𝑅)} ∈ V)
172, 9, 11, 13, 16ovmpod 7585 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓 ∈ (𝑆 RngHom 𝑅)})
1817eleq2d 2825 . 2 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓 ∈ (𝑆 RngHom 𝑅)}))
19 cnveq 5887 . . . 4 (𝑓 = 𝐹𝑓 = 𝐹)
2019eleq1d 2824 . . 3 (𝑓 = 𝐹 → (𝑓 ∈ (𝑆 RngHom 𝑅) ↔ 𝐹 ∈ (𝑆 RngHom 𝑅)))
2120elrab 3695 . 2 (𝐹 ∈ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓 ∈ (𝑆 RngHom 𝑅)} ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹 ∈ (𝑆 RngHom 𝑅)))
2218, 21bitrdi 287 1 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹 ∈ (𝑆 RngHom 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  ccnv 5688  (class class class)co 7431  cmpo 7433   RngHom crnghm 20451   RngIso crngim 20452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rngim 20454
This theorem is referenced by:  isrngim2  20470  rngimcnv  20473  rngcinv  20654  rngcinvALTV  48120
  Copyright terms: Public domain W3C validator