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Mirrors > Home > MPE Home > Th. List > frlmlss | Structured version Visualization version GIF version |
Description: The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
frlmval.f | ⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
frlmpws.b | ⊢ 𝐵 = (Base‘𝐹) |
frlmlss.u | ⊢ 𝑈 = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) |
Ref | Expression |
---|---|
frlmlss | ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmpws.b | . . 3 ⊢ 𝐵 = (Base‘𝐹) | |
2 | frlmval.f | . . . . 5 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
3 | 2 | frlmval 20437 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
4 | 3 | fveq2d 6649 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘𝐹) = (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))) |
5 | 1, 4 | syl5eq 2845 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 = (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))) |
6 | simpr 488 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐼 ∈ 𝑊) | |
7 | simpl 486 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ Ring) | |
8 | rlmlmod 19970 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (ringLMod‘𝑅) ∈ LMod) |
10 | fconst6g 6542 | . . . . 5 ⊢ ((ringLMod‘𝑅) ∈ LMod → (𝐼 × {(ringLMod‘𝑅)}):𝐼⟶LMod) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × {(ringLMod‘𝑅)}):𝐼⟶LMod) |
12 | fvex 6658 | . . . . . . . 8 ⊢ (ringLMod‘𝑅) ∈ V | |
13 | 12 | fvconst2 6943 | . . . . . . 7 ⊢ (𝑖 ∈ 𝐼 → ((𝐼 × {(ringLMod‘𝑅)})‘𝑖) = (ringLMod‘𝑅)) |
14 | 13 | adantl 485 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → ((𝐼 × {(ringLMod‘𝑅)})‘𝑖) = (ringLMod‘𝑅)) |
15 | 14 | fveq2d 6649 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (Scalar‘((𝐼 × {(ringLMod‘𝑅)})‘𝑖)) = (Scalar‘(ringLMod‘𝑅))) |
16 | rlmsca 19965 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
17 | 16 | ad2antrr 725 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → 𝑅 = (Scalar‘(ringLMod‘𝑅))) |
18 | 15, 17 | eqtr4d 2836 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (Scalar‘((𝐼 × {(ringLMod‘𝑅)})‘𝑖)) = 𝑅) |
19 | eqid 2798 | . . . 4 ⊢ (𝑅Xs(𝐼 × {(ringLMod‘𝑅)})) = (𝑅Xs(𝐼 × {(ringLMod‘𝑅)})) | |
20 | eqid 2798 | . . . 4 ⊢ (LSubSp‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) = (LSubSp‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) | |
21 | eqid 2798 | . . . 4 ⊢ (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) = (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) | |
22 | 6, 7, 11, 18, 19, 20, 21 | dsmmlss 20433 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) ∈ (LSubSp‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)})))) |
23 | eqid 2798 | . . . . . . . . 9 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
24 | eqid 2798 | . . . . . . . . 9 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅)) | |
25 | 23, 24 | pwsval 16751 | . . . . . . . 8 ⊢ (((ringLMod‘𝑅) ∈ V ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)}))) |
26 | 12, 25 | mpan 689 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑊 → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)}))) |
27 | 26 | adantl 485 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)}))) |
28 | 16 | eqcomd 2804 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Scalar‘(ringLMod‘𝑅)) = 𝑅) |
29 | 28 | adantr 484 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Scalar‘(ringLMod‘𝑅)) = 𝑅) |
30 | 29 | oveq1d 7150 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})) = (𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) |
31 | 27, 30 | eqtr2d 2834 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝑅Xs(𝐼 × {(ringLMod‘𝑅)})) = ((ringLMod‘𝑅) ↑s 𝐼)) |
32 | 31 | fveq2d 6649 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (LSubSp‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) |
33 | frlmlss.u | . . . 4 ⊢ 𝑈 = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) | |
34 | 32, 33 | eqtr4di 2851 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (LSubSp‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) = 𝑈) |
35 | 22, 34 | eleqtrd 2892 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) ∈ 𝑈) |
36 | 5, 35 | eqeltrd 2890 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 × cxp 5517 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Scalarcsca 16560 Xscprds 16711 ↑s cpws 16712 Ringcrg 19290 LModclmod 19627 LSubSpclss 19696 ringLModcrglmod 19934 ⊕m cdsmm 20420 freeLMod cfrlm 20435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-mgp 19233 df-ur 19245 df-ring 19292 df-subrg 19526 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-dsmm 20421 df-frlm 20436 |
This theorem is referenced by: frlm0 20443 frlmsubgval 20454 frlmgsum 20461 frlmsplit2 20462 |
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