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Mirrors > Home > MPE Home > Th. List > frlmlmod | Structured version Visualization version GIF version |
Description: The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
frlmval.f | ⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
Ref | Expression |
---|---|
frlmlmod | ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmval.f | . . 3 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
2 | 1 | frlmval 20567 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
3 | simpr 488 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐼 ∈ 𝑊) | |
4 | simpl 486 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ Ring) | |
5 | rlmlmod 20099 | . . . . 5 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
6 | 5 | adantr 484 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (ringLMod‘𝑅) ∈ LMod) |
7 | fconst6g 6568 | . . . 4 ⊢ ((ringLMod‘𝑅) ∈ LMod → (𝐼 × {(ringLMod‘𝑅)}):𝐼⟶LMod) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × {(ringLMod‘𝑅)}):𝐼⟶LMod) |
9 | fvex 6690 | . . . . . . 7 ⊢ (ringLMod‘𝑅) ∈ V | |
10 | 9 | fvconst2 6979 | . . . . . 6 ⊢ (𝑖 ∈ 𝐼 → ((𝐼 × {(ringLMod‘𝑅)})‘𝑖) = (ringLMod‘𝑅)) |
11 | 10 | adantl 485 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → ((𝐼 × {(ringLMod‘𝑅)})‘𝑖) = (ringLMod‘𝑅)) |
12 | 11 | fveq2d 6681 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (Scalar‘((𝐼 × {(ringLMod‘𝑅)})‘𝑖)) = (Scalar‘(ringLMod‘𝑅))) |
13 | rlmsca 20094 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
14 | 13 | ad2antrr 726 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → 𝑅 = (Scalar‘(ringLMod‘𝑅))) |
15 | 12, 14 | eqtr4d 2777 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (Scalar‘((𝐼 × {(ringLMod‘𝑅)})‘𝑖)) = 𝑅) |
16 | eqid 2739 | . . 3 ⊢ (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) | |
17 | 3, 4, 8, 15, 16 | dsmmlmod 20564 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) ∈ LMod) |
18 | 2, 17 | eqeltrd 2834 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {csn 4517 × cxp 5524 ⟶wf 6336 ‘cfv 6340 (class class class)co 7173 Scalarcsca 16674 Ringcrg 19419 LModclmod 19756 ringLModcrglmod 20063 ⊕m cdsmm 20550 freeLMod cfrlm 20565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-ixp 8511 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-sup 8982 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-dec 12183 df-uz 12328 df-fz 12985 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-sca 16687 df-vsca 16688 df-ip 16689 df-tset 16690 df-ple 16691 df-ds 16693 df-hom 16695 df-cco 16696 df-0g 16821 df-prds 16827 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-grp 18225 df-minusg 18226 df-sbg 18227 df-subg 18397 df-mgp 19362 df-ur 19374 df-ring 19421 df-subrg 19655 df-lmod 19758 df-lss 19826 df-sra 20066 df-rgmod 20067 df-dsmm 20551 df-frlm 20566 |
This theorem is referenced by: frlmlvec 20580 frlmplusgvalb 20588 frlmvscavalb 20589 frlmvplusgscavalb 20590 frlmphl 20600 uvcresum 20612 frlmssuvc1 20613 frlmssuvc2 20614 frlmsslsp 20615 frlmup1 20617 frlmisfrlm 20667 matlmod 21183 rrxnm 24146 rrxds 24148 lindsdom 35417 lindsenlbs 35418 matunitlindflem1 35419 matunitlindflem2 35420 frlmsnic 39867 isnumbasgrplem3 40525 mnringlmodd 41409 zlmodzxzlmod 45254 aacllem 45988 |
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