|   | Mathbox for Jeff Madsen | < Previous  
      Next > Nearby theorems | |
| 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 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | oveq12 7441 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RingOpsHom 𝑠) = (𝑅 RingOpsHom 𝑆)) | |
| 2 | fveq2 6905 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
| 3 | rngisoval.1 | . . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅) | |
| 4 | 2, 3 | eqtr4di 2794 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) | 
| 5 | 4 | rneqd 5948 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺) | 
| 6 | rngisoval.2 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
| 7 | 5, 6 | eqtr4di 2794 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋) | 
| 8 | 7 | f1oeq2d 6843 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→ran (1st ‘𝑠))) | 
| 9 | fveq2 6905 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1st ‘𝑠) = (1st ‘𝑆)) | |
| 10 | rngisoval.3 | . . . . . . . 8 ⊢ 𝐽 = (1st ‘𝑆) | |
| 11 | 9, 10 | eqtr4di 2794 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (1st ‘𝑠) = 𝐽) | 
| 12 | 11 | rneqd 5948 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ran (1st ‘𝑠) = ran 𝐽) | 
| 13 | rngisoval.4 | . . . . . 6 ⊢ 𝑌 = ran 𝐽 | |
| 14 | 12, 13 | eqtr4di 2794 | . . . . 5 ⊢ (𝑠 = 𝑆 → ran (1st ‘𝑠) = 𝑌) | 
| 15 | 14 | f1oeq3d 6844 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝑓:𝑋–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌)) | 
| 16 | 8, 15 | sylan9bb 509 | . . 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 3454 | . 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠)} = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌}) | 
| 18 | df-rngoiso 37984 | . 2 ⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠)}) | |
| 19 | ovex 7465 | . . 3 ⊢ (𝑅 RingOpsHom 𝑆) ∈ V | |
| 20 | 19 | rabex 5338 | . 2 ⊢ {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ∈ V | 
| 21 | 17, 18, 20 | ovmpoa 7589 | 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 1539 ∈ wcel 2107 {crab 3435 ran crn 5685 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 1st c1st 8013 RingOpscrngo 37902 RingOpsHom crngohom 37968 RingOpsIso crngoiso 37969 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-rngoiso 37984 | 
| This theorem is referenced by: isrngoiso 37986 | 
| Copyright terms: Public domain | W3C validator |