Theorem risci 36072

Description: Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)

Assertion

Ref	Expression
risci	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑅 ≃𝑟 𝑆)

Proof of Theorem risci

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex2 3443	. . 3 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))
2		risc 36071	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 ≃𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))
3	1, 2	syl5ibr 245	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝑅 ≃𝑟 𝑆))
4	3	3impia 1115	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑅 ≃𝑟 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085  ∃wex 1783   ∈ wcel 2108   class class class wbr 5070  (class class class)co 7255  RingOpscrngo 35979   RngIso crngiso 36046   ≃_𝑟 crisc 36047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-iota 6376  df-fv 6426  df-ov 7258  df-risc 36068

This theorem is referenced by:  riscer  36073