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Mirrors > Home > MPE Home > Th. List > frlmpws | Structured version Visualization version GIF version |
Description: The free module as a restriction 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) |
Ref | Expression |
---|---|
frlmpws | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2799 | . . . 4 ⊢ (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) = (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) | |
2 | 1 | dsmmval2 20405 | . . 3 ⊢ (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) = ((𝑅Xs(𝐼 × {(ringLMod‘𝑅)})) ↾s (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))) |
3 | rlmsca 19523 | . . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
4 | 3 | adantr 473 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘(ringLMod‘𝑅))) |
5 | 4 | oveq1d 6893 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅Xs(𝐼 × {(ringLMod‘𝑅)})) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)}))) |
6 | frlmval.f | . . . . . . . 8 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
7 | 6 | frlmval 20417 | . . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
8 | 7 | eqcomd 2805 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) = 𝐹) |
9 | 8 | fveq2d 6415 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) = (Base‘𝐹)) |
10 | frlmpws.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐹) | |
11 | 9, 10 | syl6eqr 2851 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) = 𝐵) |
12 | 5, 11 | oveq12d 6896 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → ((𝑅Xs(𝐼 × {(ringLMod‘𝑅)})) ↾s (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))) = (((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})) ↾s 𝐵)) |
13 | 2, 12 | syl5eq 2845 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) = (((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})) ↾s 𝐵)) |
14 | fvex 6424 | . . . . 5 ⊢ (ringLMod‘𝑅) ∈ V | |
15 | eqid 2799 | . . . . . 6 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
16 | eqid 2799 | . . . . . 6 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅)) | |
17 | 15, 16 | pwsval 16461 | . . . . 5 ⊢ (((ringLMod‘𝑅) ∈ V ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)}))) |
18 | 14, 17 | mpan 682 | . . . 4 ⊢ (𝐼 ∈ 𝑊 → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)}))) |
19 | 18 | adantl 474 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)}))) |
20 | 19 | oveq1d 6893 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) = (((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})) ↾s 𝐵)) |
21 | 13, 7, 20 | 3eqtr4d 2843 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3385 {csn 4368 × cxp 5310 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 ↾s cress 16185 Scalarcsca 16270 Xscprds 16421 ↑s cpws 16422 ringLModcrglmod 19492 ⊕m cdsmm 20400 freeLMod cfrlm 20415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-sup 8590 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-hom 16291 df-cco 16292 df-prds 16423 df-pws 16425 df-sra 19495 df-rgmod 19496 df-dsmm 20401 df-frlm 20416 |
This theorem is referenced by: frlmsca 20422 frlm0 20423 frlmplusgval 20432 frlmsubgval 20433 frlmvscafval 20434 frlmgsum 20436 frlmsplit2 20437 frlmip 20442 rrxprds 23515 |
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