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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 7165 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆)) | |
2 | fveq2 6670 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
3 | rngisoval.1 | . . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅) | |
4 | 2, 3 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
5 | 4 | rneqd 5808 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺) |
6 | rngisoval.2 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
7 | 5, 6 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋) |
8 | 7 | f1oeq2d 6611 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→ran (1st ‘𝑠))) |
9 | fveq2 6670 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1st ‘𝑠) = (1st ‘𝑆)) | |
10 | rngisoval.3 | . . . . . . . 8 ⊢ 𝐽 = (1st ‘𝑆) | |
11 | 9, 10 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (1st ‘𝑠) = 𝐽) |
12 | 11 | rneqd 5808 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ran (1st ‘𝑠) = ran 𝐽) |
13 | rngisoval.4 | . . . . . 6 ⊢ 𝑌 = ran 𝐽 | |
14 | 12, 13 | syl6eqr 2874 | . . . . 5 ⊢ (𝑠 = 𝑆 → ran (1st ‘𝑠) = 𝑌) |
15 | 14 | f1oeq3d 6612 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝑓:𝑋–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌)) |
16 | 8, 15 | sylan9bb 512 | . . 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 3485 | . 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 35269 | . 2 ⊢ RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠)}) | |
19 | ovex 7189 | . . 3 ⊢ (𝑅 RngHom 𝑆) ∈ V | |
20 | 19 | rabex 5235 | . 2 ⊢ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ∈ V |
21 | 17, 18, 20 | ovmpoa 7305 | 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 398 = wceq 1537 ∈ wcel 2114 {crab 3142 ran crn 5556 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 RingOpscrngo 35187 RngHom crnghom 35253 RngIso crngiso 35254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-rngoiso 35269 |
This theorem is referenced by: isrngoiso 35271 |
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