rngoisoval - Mathbox for Jeff Madsen

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Theorem rngoisoval 38222

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 7377	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RingOpsHom 𝑠) = (𝑅 RingOpsHom 𝑆))
2		fveq2 6842	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = (1^st ‘𝑅))
3		rngisoval.1	. . . . . . . 8 ⊢ 𝐺 = (1^st ‘𝑅)
4	2, 3	eqtr4di 2790	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = 𝐺)
5	4	rneqd 5895	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = ran 𝐺)
6		rngisoval.2	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	eqtr4di 2790	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = 𝑋)
8	7	f1oeq2d 6778	. . . 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 6842	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1^st ‘𝑠) = (1^st ‘𝑆))
10		rngisoval.3	. . . . . . . 8 ⊢ 𝐽 = (1^st ‘𝑆)
11	9, 10	eqtr4di 2790	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (1^st ‘𝑠) = 𝐽)
12	11	rneqd 5895	. . . . . 6 ⊢ (𝑠 = 𝑆 → ran (1^st ‘𝑠) = ran 𝐽)
13		rngisoval.4	. . . . . 6 ⊢ 𝑌 = ran 𝐽
14	12, 13	eqtr4di 2790	. . . . 5 ⊢ (𝑠 = 𝑆 → ran (1^st ‘𝑠) = 𝑌)
15	14	f1oeq3d 6779	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑓:𝑋–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌))
16	8, 15	sylan9bb 509	. . 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 3419	. 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 38221	. 2 ⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)})
19		ovex 7401	. . 3 ⊢ (𝑅 RingOpsHom 𝑆) ∈ V
20	19	rabex 5286	. 2 ⊢ {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ∈ V
21	17, 18, 20	ovmpoa 7523	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 395 = wceq 1542 ∈ wcel 2114 {crab 3401 ran crn 5633 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 RingOpscrngo 38139 RingOpsHom crngohom 38205 RingOpsIso crngoiso 38206

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 5243 ax-nul 5253 ax-pr 5379

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 3402 df-v 3444 df-sbc 3743 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-rngoiso 38221

This theorem is referenced by: isrngoiso 38223

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