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Mirrors > Home > MPE Home > Th. List > lidlval | Structured version Visualization version GIF version |
Description: Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
lidlval | ⊢ (LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lidl 19542 | . . 3 ⊢ LIdeal = (LSubSp ∘ ringLMod) | |
2 | 1 | fveq1i 6438 | . 2 ⊢ (LIdeal‘𝑊) = ((LSubSp ∘ ringLMod)‘𝑊) |
3 | 00lss 19305 | . . 3 ⊢ ∅ = (LSubSp‘∅) | |
4 | rlmfn 19558 | . . . 4 ⊢ ringLMod Fn V | |
5 | fnfun 6225 | . . . 4 ⊢ (ringLMod Fn V → Fun ringLMod) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ Fun ringLMod |
7 | 3, 6 | fvco4i 6527 | . 2 ⊢ ((LSubSp ∘ ringLMod)‘𝑊) = (LSubSp‘(ringLMod‘𝑊)) |
8 | 2, 7 | eqtri 2849 | 1 ⊢ (LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 Vcvv 3414 ∘ ccom 5350 Fun wfun 6121 Fn wfn 6122 ‘cfv 6127 LSubSpclss 19295 ringLModcrglmod 19537 LIdealclidl 19538 |
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 |
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-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-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-fv 6135 df-ov 6913 df-slot 16233 df-base 16235 df-lss 19296 df-rgmod 19541 df-lidl 19542 |
This theorem is referenced by: lidlss 19578 islidl 19579 lidl0cl 19580 lidlacl 19581 lidlnegcl 19582 lidlmcl 19585 lidl0 19587 lidl1 19588 lidlacs 19589 rspcl 19590 rspssp 19594 mrcrsp 19595 lidlrsppropd 19598 islnr2 38526 |
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