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Mirrors > Home > MPE Home > Th. List > Mathboxes > igenidl2 | Structured version Visualization version GIF version |
Description: The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
igenidl2 | ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2738 | . . . 4 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
3 | 1, 2 | idlss 36160 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1st ‘𝑅)) |
4 | 1, 2 | igenval 36205 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1st ‘𝑅)) → (𝑅 IdlGen 𝐼) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗}) |
5 | 3, 4 | syldan 591 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗}) |
6 | intmin 4900 | . . 3 ⊢ (𝐼 ∈ (Idl‘𝑅) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗} = 𝐼) | |
7 | 6 | adantl 482 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗} = 𝐼) |
8 | 5, 7 | eqtrd 2778 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3887 ∩ cint 4880 ran crn 5586 ‘cfv 6427 (class class class)co 7268 1st c1st 7819 RingOpscrngo 36038 Idlcidl 36151 IdlGen cigen 36203 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-fo 6433 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-1st 7821 df-2nd 7822 df-grpo 28841 df-gid 28842 df-ablo 28893 df-rngo 36039 df-idl 36154 df-igen 36204 |
This theorem is referenced by: (None) |
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