Theorem isrim0 14552

Description: A ring isomorphism is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 10-Jan-2025.)

Assertion

Ref	Expression
isrim0	⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))

Proof of Theorem isrim0

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

Step	Hyp	Ref	Expression
1		rimrcl 14551	. 2 ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
2		rhmrcl1 14546	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
3	2	elexd 2835	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ V)
4		rhmrcl2 14547	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
5	4	elexd 2835	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ V)
6	3, 5	jca 306	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
7	6	adantr 276	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
8		df-rim 14544	. . . . . 6 ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)})
9	8	a1i 9	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)}))
10		oveq12 6094	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RingHom 𝑠) = (𝑅 RingHom 𝑆))
11	10	adantl 277	. . . . . 6 ⊢ (((𝑅 ∈ V ∧ 𝑆 ∈ V) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑟 RingHom 𝑠) = (𝑅 RingHom 𝑆))
12		oveq12 6094	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
13	12	ancoms 268	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
14	13	adantl 277	. . . . . . 7 ⊢ (((𝑅 ∈ V ∧ 𝑆 ∈ V) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
15	14	eleq2d 2308	. . . . . 6 ⊢ (((𝑅 ∈ V ∧ 𝑆 ∈ V) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (◡𝑓 ∈ (𝑠 RingHom 𝑟) ↔ ◡𝑓 ∈ (𝑆 RingHom 𝑅)))
16	11, 15	rabeqbidv 2816	. . . . 5 ⊢ (((𝑅 ∈ V ∧ 𝑆 ∈ V) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)} = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RingHom 𝑅)})
17		simpl 109	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → 𝑅 ∈ V)
18		simpr 110	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
19		rhmex 14548	. . . . . . 7 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 RingHom 𝑆) ∈ V)
20	17, 18, 19	syl2anc 415	. . . . . 6 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 RingHom 𝑆) ∈ V)
21		rabexg 4279	. . . . . 6 ⊢ ((𝑅 RingHom 𝑆) ∈ V → {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RingHom 𝑅)} ∈ V)
22	20, 21	syl 14	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RingHom 𝑅)} ∈ V)
23	9, 16, 17, 18, 22	ovmpod 6216	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 RingIso 𝑆) = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RingHom 𝑅)})
24	23	eleq2d 2308	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RingHom 𝑅)}))
25		cnveq 4954	. . . . 5 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
26	25	eleq1d 2307	. . . 4 ⊢ (𝑓 = 𝐹 → (◡𝑓 ∈ (𝑆 RingHom 𝑅) ↔ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))
27	26	elrab 2982	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RingHom 𝑅)} ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))
28	24, 27	bitrdi 196	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅))))
29	1, 7, 28	pm5.21nii 716	1 ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {crab 2532  Vcvv 2821  ^◡ccnv 4773  (class class class)co 6085   ∈ cmpo 6087  Ringcrg 14384   RingHom crh 14541   RingIso crs 14542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-pre-ltirr 8292  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-pnf 8363  df-mnf 8364  df-ltxr 8366  df-inn 9308  df-2 9366  df-3 9367  df-ndx 13407  df-slot 13408  df-base 13410  df-sets 13411  df-plusg 13497  df-mulr 13498  df-mhm 13819  df-ghm 14097  df-mgp 14302  df-ur 14347  df-ring 14386  df-rhm 14543  df-rim 14544

This theorem is used by:  isrim  14560  rimrhm  14562