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Mirrors > Home > ILE Home > Th. List > idomringd | GIF version |
Description: An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
Ref | Expression |
---|---|
idomringd.1 | ⊢ (𝜑 → 𝑅 ∈ IDomn) |
Ref | Expression |
---|---|
idomringd | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idomringd.1 | . . 3 ⊢ (𝜑 → 𝑅 ∈ IDomn) | |
2 | 1 | idomcringd 13758 | . 2 ⊢ (𝜑 → 𝑅 ∈ CRing) |
3 | 2 | crngringd 13489 | 1 ⊢ (𝜑 → 𝑅 ∈ Ring) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2164 Ringcrg 13476 IDomncidom 13737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-rex 2478 df-rab 2481 df-v 2762 df-un 3157 df-in 3159 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-iota 5207 df-fv 5254 df-cring 13479 df-idom 13740 |
This theorem is referenced by: lgseisenlem3 15136 |
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