Theorem isrngisom 44157

Description: An isomorphism of non-unital rings is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Feb-2020.)

Assertion

Ref	Expression
isrngisom	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHomo 𝑅))))

Proof of Theorem isrngisom

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

Step	Hyp	Ref	Expression
1		df-rngisom 44149	. . . . 5 ⊢ RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHomo 𝑟)})
2	1	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHomo 𝑟)}))
3		oveq12 7157	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
4	3	adantl 484	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
5		oveq12 7157	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
6	5	ancoms 461	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
7	6	adantl 484	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
8	7	eleq2d 2896	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (◡𝑓 ∈ (𝑠 RngHomo 𝑟) ↔ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)))
9	4, 8	rabeqbidv 3484	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHomo 𝑟)} = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)})
10		elex 3511	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
11	10	adantr 483	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 ∈ V)
12		elex 3511	. . . . 5 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
13	12	adantl 484	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ V)
14		ovex 7181	. . . . . 6 ⊢ (𝑅 RngHomo 𝑆) ∈ V
15	14	rabex 5226	. . . . 5 ⊢ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)} ∈ V
16	15	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)} ∈ V)
17	2, 9, 11, 13, 16	ovmpod 7294	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 RngIsom 𝑆) = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)})
18	17	eleq2d 2896	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)}))
19		cnveq 5737	. . . 4 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
20	19	eleq1d 2895	. . 3 ⊢ (𝑓 = 𝐹 → (◡𝑓 ∈ (𝑆 RngHomo 𝑅) ↔ ◡𝐹 ∈ (𝑆 RngHomo 𝑅)))
21	20	elrab 3678	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)} ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHomo 𝑅)))
22	18, 21	syl6bb 289	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHomo 𝑅))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108  {crab 3140  Vcvv 3493  ^◡ccnv 5547  (class class class)co 7148   ∈ cmpo 7150   RngHomo crngh 44146   RngIsom crngs 44147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-rngisom 44149

This theorem is referenced by:  isrngim  44165  rngcinv  44242  rngcinvALTV  44254