Theorem brric 19903

Description: The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)

Assertion

Ref	Expression
brric	⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)

Proof of Theorem brric

Dummy variables ℎ 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ric 19877	. 2 ⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1o))
2		ovex 7288	. . . . 5 ⊢ (𝑟 RingHom 𝑠) ∈ V
3		rabexg 5250	. . . . 5 ⊢ ((𝑟 RingHom 𝑠) ∈ V → {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V)
4	2, 3	mp1i 13	. . . 4 ⊢ ((𝑟 ∈ V ∧ 𝑠 ∈ V) → {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V)
5	4	rgen2 3126	. . 3 ⊢ ∀𝑟 ∈ V ∀𝑠 ∈ V {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V
6		df-rngiso 19875	. . . 4 ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)})
7	6	fnmpo 7882	. . 3 ⊢ (∀𝑟 ∈ V ∀𝑠 ∈ V {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V → RingIso Fn (V × V))
8	5, 7	ax-mp 5	. 2 ⊢ RingIso Fn (V × V)
9	1, 8	brwitnlem 8299	1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  {crab 3067  Vcvv 3422  ∅c0 4253   class class class wbr 5070   × cxp 5578  ^◡ccnv 5579   Fn wfn 6413  (class class class)co 7255   RingHom crh 19871   RingIso crs 19872   ≃_𝑟 cric 19873

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-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

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-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  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-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-1o 8267  df-rngiso 19875  df-ric 19877

This theorem is referenced by:  brric2  19904  mat1ric  21544  scmatric  21594  matcpmric  21816  pmmpric  21880