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Mirrors > Home > MPE Home > Th. List > Mathboxes > isrngoiso | Structured version Visualization version GIF version |
Description: The predicate "is a ring isomorphism between 𝑅 and 𝑆." (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 |
---|---|
isrngoiso | ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngisoval.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | rngisoval.2 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
3 | rngisoval.3 | . . . 4 ⊢ 𝐽 = (1st ‘𝑆) | |
4 | rngisoval.4 | . . . 4 ⊢ 𝑌 = ran 𝐽 | |
5 | 1, 2, 3, 4 | rngoisoval 32946 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌}) |
6 | 5 | eleq2d 2673 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})) |
7 | f1oeq1 6040 | . . 3 ⊢ (𝑓 = 𝐹 → (𝑓:𝑋–1-1-onto→𝑌 ↔ 𝐹:𝑋–1-1-onto→𝑌)) | |
8 | 7 | elrab 3331 | . 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌)) |
9 | 6, 8 | syl6bb 275 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (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: rngoiso1o 32948 rngoisohom 32949 rngoisocnv 32950 rngoisoco 32951 |
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