Theorem rngimcnv 20396

Description: The converse of an isomorphism of non-unital rings is an isomorphism of non-unital rings. (Contributed by AV, 27-Feb-2025.)

Assertion

Ref	Expression
rngimcnv	⊢ (𝐹 ∈ (𝑆 RngIso 𝑇) → ◡𝐹 ∈ (𝑇 RngIso 𝑆))

Proof of Theorem rngimcnv

Step	Hyp	Ref	Expression
1		rngimrcl 20386	. 2 ⊢ (𝐹 ∈ (𝑆 RngIso 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
2		isrngim 20385	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIso 𝑇) ↔ (𝐹 ∈ (𝑆 RngHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHom 𝑆))))
3		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
4		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
5	3, 4	rnghmf 20388	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6		frel 6668	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
7		dfrel2 6148	. . . . . . . 8 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
8	6, 7	sylib 218	. . . . . . 7 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → ◡◡𝐹 = 𝐹)
9	5, 8	syl 17	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → ◡◡𝐹 = 𝐹)
10		id 22	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹 ∈ (𝑆 RngHom 𝑇))
11	9, 10	eqeltrd 2837	. . . . 5 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → ◡◡𝐹 ∈ (𝑆 RngHom 𝑇))
12	11	anim1ci 617	. . . 4 ⊢ ((𝐹 ∈ (𝑆 RngHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHom 𝑆)) → (◡𝐹 ∈ (𝑇 RngHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHom 𝑇)))
13		isrngim 20385	. . . . 5 ⊢ ((𝑇 ∈ V ∧ 𝑆 ∈ V) → (◡𝐹 ∈ (𝑇 RngIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 RngHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHom 𝑇))))
14	13	ancoms 458	. . . 4 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (◡𝐹 ∈ (𝑇 RngIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 RngHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHom 𝑇))))
15	12, 14	imbitrrid 246	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → ((𝐹 ∈ (𝑆 RngHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHom 𝑆)) → ◡𝐹 ∈ (𝑇 RngIso 𝑆)))
16	2, 15	sylbid 240	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIso 𝑇) → ◡𝐹 ∈ (𝑇 RngIso 𝑆)))
17	1, 16	mpcom 38	1 ⊢ (𝐹 ∈ (𝑆 RngIso 𝑇) → ◡𝐹 ∈ (𝑇 RngIso 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3441  ^◡ccnv 5624  Rel wrel 5630  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  Basecbs 17140   RngHom crnghm 20374   RngIso crngim 20375

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8769  df-ghm 19146  df-abl 19716  df-rng 20092  df-rnghm 20376  df-rngim 20377

This theorem is referenced by:  rngisom1  20406  rngringbdlem2  21266  rngqiprngu  21277