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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > igenss | Structured version Visualization version GIF version |
Description: A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
igenval.1 | ⊢ 𝐺 = (1st ‘𝑅) |
igenval.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
igenss | ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssintub 4527 | . 2 ⊢ 𝑆 ⊆ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} | |
2 | igenval.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
3 | igenval.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | 2, 3 | igenval 33990 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
5 | 1, 4 | syl5sseqr 3687 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 {crab 2945 ⊆ wss 3607 ∩ cint 4507 ran crn 5144 ‘cfv 5926 (class class class)co 6690 1st c1st 7208 RingOpscrngo 33823 Idlcidl 33936 IdlGen cigen 33988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fo 5932 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-grpo 27475 df-gid 27476 df-ablo 27527 df-rngo 33824 df-idl 33939 df-igen 33989 |
This theorem is referenced by: igenval2 33995 isfldidl 33997 ispridlc 33999 |
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