Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 20237 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})}) |
5 | 4 | eleq2d 2816 | . 2 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ 𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})) |
6 | eliun 4894 | . . 3 ⊢ (𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 ∈ {(𝐾‘{𝑔})}) | |
7 | fvex 6708 | . . . . 5 ⊢ (𝐾‘{𝑔}) ∈ V | |
8 | 7 | elsn2 4566 | . . . 4 ⊢ (𝐼 ∈ {(𝐾‘{𝑔})} ↔ 𝐼 = (𝐾‘{𝑔})) |
9 | 8 | rexbii 3160 | . . 3 ⊢ (∃𝑔 ∈ 𝐵 𝐼 ∈ {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔})) |
10 | 6, 9 | bitri 278 | . 2 ⊢ (𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔})) |
11 | 5, 10 | bitrdi 290 | 1 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 {csn 4527 ∪ ciun 4890 ‘cfv 6358 Basecbs 16666 Ringcrg 19516 RSpancrsp 20162 LPIdealclpidl 20233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fv 6366 df-lpidl 20235 |
This theorem is referenced by: lpi0 20239 lpi1 20240 lpiss 20242 lpigen 20248 ply1lpir 25030 lsmsnidl 31255 lpirlnr 40586 |
Copyright terms: Public domain | W3C validator |