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| Mirrors > Home > MPE Home > Th. List > islpidl | Structured version Visualization version GIF version | ||
| Description: Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
| Ref | Expression |
|---|---|
| lpival.p | ⊢ 𝑃 = (LPIdeal‘𝑅) |
| lpival.k | ⊢ 𝐾 = (RSpan‘𝑅) |
| lpival.b | ⊢ 𝐵 = (Base‘𝑅) |
| Ref | Expression |
|---|---|
| islpidl | ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lpival.p | . . . 4 ⊢ 𝑃 = (LPIdeal‘𝑅) | |
| 2 | lpival.k | . . . 4 ⊢ 𝐾 = (RSpan‘𝑅) | |
| 3 | lpival.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
| 4 | 1, 2, 3 | lpival 21334 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})}) |
| 5 | 4 | eleq2d 2827 | . 2 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ 𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})) |
| 6 | eliun 4995 | . . 3 ⊢ (𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 ∈ {(𝐾‘{𝑔})}) | |
| 7 | fvex 6919 | . . . . 5 ⊢ (𝐾‘{𝑔}) ∈ V | |
| 8 | 7 | elsn2 4665 | . . . 4 ⊢ (𝐼 ∈ {(𝐾‘{𝑔})} ↔ 𝐼 = (𝐾‘{𝑔})) |
| 9 | 8 | rexbii 3094 | . . 3 ⊢ (∃𝑔 ∈ 𝐵 𝐼 ∈ {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔})) |
| 10 | 6, 9 | bitri 275 | . 2 ⊢ (𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔})) |
| 11 | 5, 10 | bitrdi 287 | 1 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔}))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {csn 4626 ∪ ciun 4991 ‘cfv 6561 Basecbs 17247 Ringcrg 20230 RSpancrsp 21217 LPIdealclpidl 21330 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-lpidl 21332 |
| This theorem is referenced by: lpi0 21336 lpi1 21337 lpiss 21339 lpigen 21345 ply1lpir 26221 lpirlidllpi 33402 lsmsnidl 33427 mxidlirred 33500 lpirlnr 43129 |
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