Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isrngisom Structured version   Visualization version   GIF version

Theorem isrngisom 44688
 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
isrngisom ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐹 ∈ (𝑆 RngHomo 𝑅))))

Proof of Theorem isrngisom
Dummy variables 𝑓 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rngisom 44680 . . . . 5 RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ 𝑓 ∈ (𝑠 RngHomo 𝑟)})
21a1i 11 . . . 4 ((𝑅𝑉𝑆𝑊) → RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ 𝑓 ∈ (𝑠 RngHomo 𝑟)}))
3 oveq12 7154 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
43adantl 485 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
5 oveq12 7154 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
65ancoms 462 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
76adantl 485 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
87eleq2d 2875 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑓 ∈ (𝑠 RngHomo 𝑟) ↔ 𝑓 ∈ (𝑆 RngHomo 𝑅)))
94, 8rabeqbidv 3434 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ 𝑓 ∈ (𝑠 RngHomo 𝑟)} = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)})
10 elex 3460 . . . . 5 (𝑅𝑉𝑅 ∈ V)
1110adantr 484 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑅 ∈ V)
12 elex 3460 . . . . 5 (𝑆𝑊𝑆 ∈ V)
1312adantl 485 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑆 ∈ V)
14 ovex 7178 . . . . . 6 (𝑅 RngHomo 𝑆) ∈ V
1514rabex 5203 . . . . 5 {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)} ∈ V
1615a1i 11 . . . 4 ((𝑅𝑉𝑆𝑊) → {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)} ∈ V)
172, 9, 11, 13, 16ovmpod 7292 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 RngIsom 𝑆) = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)})
1817eleq2d 2875 . 2 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)}))
19 cnveq 5712 . . . 4 (𝑓 = 𝐹𝑓 = 𝐹)
2019eleq1d 2874 . . 3 (𝑓 = 𝐹 → (𝑓 ∈ (𝑆 RngHomo 𝑅) ↔ 𝐹 ∈ (𝑆 RngHomo 𝑅)))
2120elrab 3630 . 2 (𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)} ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐹 ∈ (𝑆 RngHomo 𝑅)))
2218, 21syl6bb 290 1 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐹 ∈ (𝑆 RngHomo 𝑅))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3442  ◡ccnv 5522  (class class class)co 7145   ∈ cmpo 7147   RngHomo crngh 44677   RngIsom crngs 44678 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6291  df-fun 6334  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rngisom 44680 This theorem is referenced by:  isrngim  44696  rngcinv  44773  rngcinvALTV  44785
 Copyright terms: Public domain W3C validator