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Mathbox for Jeff Madsen |
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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 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7440 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RingOpsHom 𝑠) = (𝑅 RingOpsHom 𝑆)) | |
2 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
3 | rngisoval.1 | . . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅) | |
4 | 2, 3 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
5 | 4 | rneqd 5952 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺) |
6 | rngisoval.2 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
7 | 5, 6 | eqtr4di 2793 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋) |
8 | 7 | f1oeq2d 6845 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→ran (1st ‘𝑠))) |
9 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1st ‘𝑠) = (1st ‘𝑆)) | |
10 | rngisoval.3 | . . . . . . . 8 ⊢ 𝐽 = (1st ‘𝑆) | |
11 | 9, 10 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (1st ‘𝑠) = 𝐽) |
12 | 11 | rneqd 5952 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ran (1st ‘𝑠) = ran 𝐽) |
13 | rngisoval.4 | . . . . . 6 ⊢ 𝑌 = ran 𝐽 | |
14 | 12, 13 | eqtr4di 2793 | . . . . 5 ⊢ (𝑠 = 𝑆 → ran (1st ‘𝑠) = 𝑌) |
15 | 14 | f1oeq3d 6846 | . . . 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 3452 | . 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 37963 | . 2 ⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠)}) | |
19 | ovex 7464 | . . 3 ⊢ (𝑅 RingOpsHom 𝑆) ∈ V | |
20 | 19 | rabex 5345 | . 2 ⊢ {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ∈ V |
21 | 17, 18, 20 | ovmpoa 7588 | 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 1537 ∈ wcel 2106 {crab 3433 ran crn 5690 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 RingOpscrngo 37881 RingOpsHom crngohom 37947 RingOpsIso crngoiso 37948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rngoiso 37963 |
This theorem is referenced by: isrngoiso 37965 |
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