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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 19613 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})}) |
5 | 4 | eleq2d 2892 | . 2 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ 𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})) |
6 | eliun 4746 | . . 3 ⊢ (𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 ∈ {(𝐾‘{𝑔})}) | |
7 | fvex 6450 | . . . . 5 ⊢ (𝐾‘{𝑔}) ∈ V | |
8 | 7 | elsn2 4434 | . . . 4 ⊢ (𝐼 ∈ {(𝐾‘{𝑔})} ↔ 𝐼 = (𝐾‘{𝑔})) |
9 | 8 | rexbii 3251 | . . 3 ⊢ (∃𝑔 ∈ 𝐵 𝐼 ∈ {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔})) |
10 | 6, 9 | bitri 267 | . 2 ⊢ (𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔})) |
11 | 5, 10 | syl6bb 279 | 1 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 {csn 4399 ∪ ciun 4742 ‘cfv 6127 Basecbs 16229 Ringcrg 18908 RSpancrsp 19539 LPIdealclpidl 19609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-iota 6090 df-fun 6129 df-fv 6135 df-lpidl 19611 |
This theorem is referenced by: lpi0 19615 lpi1 19616 lpiss 19618 lpigen 19624 ply1lpir 24344 lpirlnr 38529 |
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