Theorem rngimcnv 20425

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 20415	. 2 ⊢ (𝐹 ∈ (𝑆 RngIso 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
2		isrngim 20414	. . 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 20417	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6		frel 6665	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
7		dfrel2 6145	. . . . . . . 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 20414	. . . . 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 3430  ^◡ccnv 5621  Rel wrel 5627  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358  Basecbs 17168   RngHom crnghm 20403   RngIso crngim 20404

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 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

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 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8766  df-ghm 19177  df-abl 19747  df-rng 20123  df-rnghm 20405  df-rngim 20406

This theorem is referenced by:  rngisom1  20435  rngringbdlem2  21295  rngqiprngu  21306