rngoisoval - Mathbox for Jeff Madsen

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Theorem rngoisoval 34108

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 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 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 6802	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆))
2		fveq2 6332	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = (1^st ‘𝑅))
3		rngisoval.1	. . . . . . . 8 ⊢ 𝐺 = (1^st ‘𝑅)
4	2, 3	syl6eqr 2823	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = 𝐺)
5	4	rneqd 5491	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = ran 𝐺)
6		rngisoval.2	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	syl6eqr 2823	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = 𝑋)
8		f1oeq2 6269	. . . . 5 ⊢ (ran (1^st ‘𝑟) = 𝑋 → (𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→ran (1^st ‘𝑠)))
9	7, 8	syl 17	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→ran (1^st ‘𝑠)))
10		fveq2 6332	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1^st ‘𝑠) = (1^st ‘𝑆))
11		rngisoval.3	. . . . . . . 8 ⊢ 𝐽 = (1^st ‘𝑆)
12	10, 11	syl6eqr 2823	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (1^st ‘𝑠) = 𝐽)
13	12	rneqd 5491	. . . . . 6 ⊢ (𝑠 = 𝑆 → ran (1^st ‘𝑠) = ran 𝐽)
14		rngisoval.4	. . . . . 6 ⊢ 𝑌 = ran 𝐽
15	13, 14	syl6eqr 2823	. . . . 5 ⊢ (𝑠 = 𝑆 → ran (1^st ‘𝑠) = 𝑌)
16		f1oeq3 6270	. . . . 5 ⊢ (ran (1^st ‘𝑠) = 𝑌 → (𝑓:𝑋–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌))
17	15, 16	syl 17	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑓:𝑋–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌))
18	9, 17	sylan9bb 499	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌))
19	1, 18	rabeqbidv 3345	. 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)} = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})
20		df-rngoiso 34107	. 2 ⊢ RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)})
21		ovex 6823	. . 3 ⊢ (𝑅 RngHom 𝑆) ∈ V
22	21	rabex 4946	. 2 ⊢ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ∈ V
23	19, 20, 22	ovmpt2a 6938	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 {crab 3065 ran crn 5250 –1-1-onto→wf1o 6030 ‘cfv 6031 (class class class)co 6793 1^st c1st 7313 RingOpscrngo 34025 RngHom crnghom 34091 RngIso crngiso 34092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-rngoiso 34107

This theorem is referenced by: isrngoiso 34109

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