rngoisoval - Mathbox for Jeff Madsen

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Theorem rngoisoval 38440

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 7401	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RingOpsHom 𝑠) = (𝑅 RingOpsHom 𝑆))
2		fveq2 6863	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = (1^st ‘𝑅))
3		rngisoval.1	. . . . . . . 8 ⊢ 𝐺 = (1^st ‘𝑅)
4	2, 3	eqtr4di 2814	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = 𝐺)
5	4	rneqd 5912	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = ran 𝐺)
6		rngisoval.2	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	eqtr4di 2814	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = 𝑋)
8	7	f1oeq2d 6798	. . . 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 6863	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1^st ‘𝑠) = (1^st ‘𝑆))
10		rngisoval.3	. . . . . . . 8 ⊢ 𝐽 = (1^st ‘𝑆)
11	9, 10	eqtr4di 2814	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (1^st ‘𝑠) = 𝐽)
12	11	rneqd 5912	. . . . . 6 ⊢ (𝑠 = 𝑆 → ran (1^st ‘𝑠) = ran 𝐽)
13		rngisoval.4	. . . . . 6 ⊢ 𝑌 = ran 𝐽
14	12, 13	eqtr4di 2814	. . . . 5 ⊢ (𝑠 = 𝑆 → ran (1^st ‘𝑠) = 𝑌)
15	14	f1oeq3d 6799	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑓:𝑋–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌))
16	8, 15	sylan9bb 517	. . 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 3431	. 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 38439	. 2 ⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)})
19		ovex 7425	. . 3 ⊢ (𝑅 RingOpsHom 𝑆) ∈ V
20	19	rabex 5294	. 2 ⊢ {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ∈ V
21	17, 18, 20	ovmpoa 7547	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 399 = wceq 1559 ∈ wcel 2141 {crab 3413 ran crn 5646 –1-1-onto→wf1o 6516 ‘cfv 6517 (class class class)co 7392 1^st c1st 7964 RingOpscrngo 38357 RingOpsHom crngohom 38423 RingOpsIso crngoiso 38424

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pr 5389

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3745 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-ov 7395 df-oprab 7396 df-mpo 7397 df-rngoiso 38439

This theorem is referenced by: isrngoiso 38441

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