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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 6558 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆)) | |
2 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
3 | rngisoval.1 | . . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅) | |
4 | 2, 3 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
5 | 4 | rneqd 5274 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺) |
6 | rngisoval.2 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
7 | 5, 6 | syl6eqr 2662 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋) |
8 | f1oeq2 6041 | . . . . 5 ⊢ (ran (1st ‘𝑟) = 𝑋 → (𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→ran (1st ‘𝑠))) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→ran (1st ‘𝑠))) |
10 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1st ‘𝑠) = (1st ‘𝑆)) | |
11 | rngisoval.3 | . . . . . . . 8 ⊢ 𝐽 = (1st ‘𝑆) | |
12 | 10, 11 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (1st ‘𝑠) = 𝐽) |
13 | 12 | rneqd 5274 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ran (1st ‘𝑠) = ran 𝐽) |
14 | rngisoval.4 | . . . . . 6 ⊢ 𝑌 = ran 𝐽 | |
15 | 13, 14 | syl6eqr 2662 | . . . . 5 ⊢ (𝑠 = 𝑆 → ran (1st ‘𝑠) = 𝑌) |
16 | f1oeq3 6042 | . . . . 5 ⊢ (ran (1st ‘𝑠) = 𝑌 → (𝑓:𝑋–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝑓:𝑋–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌)) |
18 | 9, 17 | sylan9bb 732 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌)) |
19 | 1, 18 | rabeqbidv 3168 | . 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}) |
20 | df-rngoiso 32945 | . 2 ⊢ RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠)}) | |
21 | ovex 6577 | . . 3 ⊢ (𝑅 RngHom 𝑆) ∈ V | |
22 | 21 | rabex 4740 | . 2 ⊢ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ∈ V |
23 | 19, 20, 22 | ovmpt2a 6689 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ran crn 5039 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 RingOpscrngo 32863 RngHom crnghom 32929 RngIso crngiso 32930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-rngoiso 32945 |
This theorem is referenced by: isrngoiso 32947 |
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