rngoisoval - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoisoval		Structured version Visualization version GIF version

Theorem rngoisoval 38357

Description: The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)

**Hypotheses**
Ref	Expression
rngisoval.1	⊢ 𝐺 = (1^st ‘𝑅)
rngisoval.2	⊢ 𝑋 = ran 𝐺
rngisoval.3	⊢ 𝐽 = (1^st ‘𝑆)
rngisoval.4	⊢ 𝑌 = ran 𝐽

**Assertion**
Ref	Expression
rngoisoval	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})

Distinct variable groups: 𝑅,𝑓 𝑆,𝑓 𝑓,𝑋 𝑓,𝑌 Allowed substitution hints: 𝐺(𝑓) 𝐽(𝑓)

Proof of Theorem rngoisoval Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq12 7368	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RingOpsHom 𝑠) = (𝑅 RingOpsHom 𝑆))
2		fveq2 6830	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = (1^st ‘𝑅))
3		rngisoval.1	. . . . . . . 8 ⊢ 𝐺 = (1^st ‘𝑅)
4	2, 3	eqtr4di 2794	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = 𝐺)
5	4	rneqd 5886	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = ran 𝐺)
6		rngisoval.2	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	eqtr4di 2794	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = 𝑋)
8	7	f1oeq2d 6766	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→ran (1^st ‘𝑠)))
9		fveq2 6830	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1^st ‘𝑠) = (1^st ‘𝑆))
10		rngisoval.3	. . . . . . . 8 ⊢ 𝐽 = (1^st ‘𝑆)
11	9, 10	eqtr4di 2794	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (1^st ‘𝑠) = 𝐽)
12	11	rneqd 5886	. . . . . 6 ⊢ (𝑠 = 𝑆 → ran (1^st ‘𝑠) = ran 𝐽)
13		rngisoval.4	. . . . . 6 ⊢ 𝑌 = ran 𝐽
14	12, 13	eqtr4di 2794	. . . . 5 ⊢ (𝑠 = 𝑆 → ran (1^st ‘𝑠) = 𝑌)
15	14	f1oeq3d 6767	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑓:𝑋–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌))
16	8, 15	sylan9bb 515	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌))
17	1, 16	rabeqbidv 3411	. 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)} = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})
18		df-rngoiso 38356	. 2 ⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)})
19		ovex 7392	. . 3 ⊢ (𝑅 RingOpsHom 𝑆) ∈ V
20	19	rabex 5269	. 2 ⊢ {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ∈ V
21	17, 18, 20	ovmpoa 7514	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 {crab 3393 ran crn 5621 –1-1-onto→wf1o 6487 ‘cfv 6488 (class class class)co 7359 1^st c1st 7931 RingOpscrngo 38274 RingOpsHom crngohom 38340 RingOpsIso crngoiso 38341

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5220 ax-nul 5230 ax-pr 5364

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-sbc 3725 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-br 5075 df-opab 5137 df-id 5515 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-ov 7362 df-oprab 7363 df-mpo 7364 df-rngoiso 38356

This theorem is referenced by: isrngoiso 38358

W3C validator