MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isrim0 Structured version   Visualization version   GIF version

Theorem isrim0 19965
Description: An isomorphism of rings is a homomorphism whose converse is also a homomorphism . (Contributed by AV, 22-Oct-2019.)
Assertion
Ref Expression
isrim0 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹 ∈ (𝑆 RingHom 𝑅))))

Proof of Theorem isrim0
Dummy variables 𝑓 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rngiso 19958 . . . . 5 RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)})
21a1i 11 . . . 4 ((𝑅𝑉𝑆𝑊) → RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)}))
3 oveq12 7286 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RingHom 𝑠) = (𝑅 RingHom 𝑆))
43adantl 482 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑟 RingHom 𝑠) = (𝑅 RingHom 𝑆))
5 oveq12 7286 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
65ancoms 459 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
76adantl 482 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
87eleq2d 2824 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑓 ∈ (𝑠 RingHom 𝑟) ↔ 𝑓 ∈ (𝑆 RingHom 𝑅)))
94, 8rabeqbidv 3419 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)} = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)})
10 elex 3449 . . . . 5 (𝑅𝑉𝑅 ∈ V)
1110adantr 481 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑅 ∈ V)
12 elex 3449 . . . . 5 (𝑆𝑊𝑆 ∈ V)
1312adantl 482 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑆 ∈ V)
14 ovex 7310 . . . . . 6 (𝑅 RingHom 𝑆) ∈ V
1514rabex 5258 . . . . 5 {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)} ∈ V
1615a1i 11 . . . 4 ((𝑅𝑉𝑆𝑊) → {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)} ∈ V)
172, 9, 11, 13, 16ovmpod 7425 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 RingIso 𝑆) = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)})
1817eleq2d 2824 . 2 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)}))
19 cnveq 5784 . . . 4 (𝑓 = 𝐹𝑓 = 𝐹)
2019eleq1d 2823 . . 3 (𝑓 = 𝐹 → (𝑓 ∈ (𝑆 RingHom 𝑅) ↔ 𝐹 ∈ (𝑆 RingHom 𝑅)))
2120elrab 3625 . 2 (𝐹 ∈ {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)} ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹 ∈ (𝑆 RingHom 𝑅)))
2218, 21bitrdi 287 1 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹 ∈ (𝑆 RingHom 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  {crab 3068  Vcvv 3431  ccnv 5590  (class class class)co 7277  cmpo 7279   RingHom crh 19954   RingIso crs 19955
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pr 5354
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-sbc 3718  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5077  df-opab 5139  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-iota 6393  df-fun 6437  df-fv 6443  df-ov 7280  df-oprab 7281  df-mpo 7282  df-rngiso 19958
This theorem is referenced by:  isrim  19975  brric2  19987  ringcinv  45557  ringcinvALTV  45581
  Copyright terms: Public domain W3C validator