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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > inidl | Structured version Visualization version GIF version |
Description: The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
inidl | ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (Idl‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intprg 4943 | . . 3 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} = (𝐼 ∩ 𝐽)) | |
2 | 1 | 3adant1 1131 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} = (𝐼 ∩ 𝐽)) |
3 | prnzg 4740 | . . . . . 6 ⊢ (𝐼 ∈ (Idl‘𝑅) → {𝐼, 𝐽} ≠ ∅) | |
4 | 3 | adantr 482 | . . . . 5 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ≠ ∅) |
5 | prssi 4782 | . . . . 5 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ⊆ (Idl‘𝑅)) | |
6 | 4, 5 | jca 513 | . . . 4 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ({𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅))) |
7 | intidl 36491 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅)) | |
8 | 7 | 3expb 1121 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ({𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅))) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅)) |
9 | 6, 8 | sylan2 594 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅))) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅)) |
10 | 9 | 3impb 1116 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅)) |
11 | 2, 10 | eqeltrrd 2839 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (Idl‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∩ cin 3910 ⊆ wss 3911 ∅c0 4283 {cpr 4589 ∩ cint 4908 ‘cfv 6497 RingOpscrngo 36356 Idlcidl 36469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-idl 36472 |
This theorem is referenced by: (None) |
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