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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoisoval | Structured version Visualization version GIF version |
Description: The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
rngisoval.1 | ⊢ 𝐺 = (1st ‘𝑅) |
rngisoval.2 | ⊢ 𝑋 = ran 𝐺 |
rngisoval.3 | ⊢ 𝐽 = (1st ‘𝑆) |
rngisoval.4 | ⊢ 𝑌 = ran 𝐽 |
Ref | Expression |
---|---|
rngoisoval | ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7154 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆)) | |
2 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
3 | rngisoval.1 | . . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅) | |
4 | 2, 3 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
5 | 4 | rneqd 5801 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺) |
6 | rngisoval.2 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
7 | 5, 6 | syl6eqr 2871 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋) |
8 | 7 | f1oeq2d 6604 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→ran (1st ‘𝑠))) |
9 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1st ‘𝑠) = (1st ‘𝑆)) | |
10 | rngisoval.3 | . . . . . . . 8 ⊢ 𝐽 = (1st ‘𝑆) | |
11 | 9, 10 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (1st ‘𝑠) = 𝐽) |
12 | 11 | rneqd 5801 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ran (1st ‘𝑠) = ran 𝐽) |
13 | rngisoval.4 | . . . . . 6 ⊢ 𝑌 = ran 𝐽 | |
14 | 12, 13 | syl6eqr 2871 | . . . . 5 ⊢ (𝑠 = 𝑆 → ran (1st ‘𝑠) = 𝑌) |
15 | 14 | f1oeq3d 6605 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝑓:𝑋–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌)) |
16 | 8, 15 | sylan9bb 510 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌)) |
17 | 1, 16 | rabeqbidv 3483 | . 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠)} = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌}) |
18 | df-rngoiso 35135 | . 2 ⊢ RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠)}) | |
19 | ovex 7178 | . . 3 ⊢ (𝑅 RngHom 𝑆) ∈ V | |
20 | 19 | rabex 5226 | . 2 ⊢ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ∈ V |
21 | 17, 18, 20 | ovmpoa 7294 | 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 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 ran crn 5549 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 RingOpscrngo 35053 RngHom crnghom 35119 RngIso crngiso 35120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 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-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-rngoiso 35135 |
This theorem is referenced by: isrngoiso 35137 |
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