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Mirrors > Home > MPE Home > Th. List > frlmvscafval | Structured version Visualization version GIF version |
Description: Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
Ref | Expression |
---|---|
frlmvscafval.y | ⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
frlmvscafval.b | ⊢ 𝐵 = (Base‘𝑌) |
frlmvscafval.k | ⊢ 𝐾 = (Base‘𝑅) |
frlmvscafval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
frlmvscafval.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
frlmvscafval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
frlmvscafval.v | ⊢ ∙ = ( ·𝑠 ‘𝑌) |
frlmvscafval.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
frlmvscafval | ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmvscafval.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | frlmvscafval.y | . . . . . . . 8 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
3 | frlmvscafval.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑌) | |
4 | 2, 3 | frlmrcl 20471 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑅 ∈ V) |
5 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V) |
6 | frlmvscafval.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
7 | 2, 3 | frlmpws 20464 | . . . . . 6 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑊) → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
8 | 5, 6, 7 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
9 | 8 | fveq2d 6441 | . . . 4 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
10 | frlmvscafval.v | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑌) | |
11 | 3 | fvexi 6451 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | eqid 2825 | . . . . . 6 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) | |
13 | eqid 2825 | . . . . . 6 ⊢ ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) | |
14 | 12, 13 | ressvsca 16398 | . . . . 5 ⊢ (𝐵 ∈ V → ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
15 | 11, 14 | ax-mp 5 | . . . 4 ⊢ ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
16 | 9, 10, 15 | 3eqtr4g 2886 | . . 3 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))) |
17 | 16 | oveqd 6927 | . 2 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))𝑋)) |
18 | eqid 2825 | . . 3 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
19 | eqid 2825 | . . 3 ⊢ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)) | |
20 | frlmvscafval.t | . . . 4 ⊢ · = (.r‘𝑅) | |
21 | rlmvsca 19570 | . . . 4 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
22 | 20, 21 | eqtri 2849 | . . 3 ⊢ · = ( ·𝑠 ‘(ringLMod‘𝑅)) |
23 | eqid 2825 | . . 3 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅)) | |
24 | eqid 2825 | . . 3 ⊢ (Base‘(Scalar‘(ringLMod‘𝑅))) = (Base‘(Scalar‘(ringLMod‘𝑅))) | |
25 | fvexd 6452 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ V) | |
26 | frlmvscafval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
27 | frlmvscafval.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
28 | rlmsca 19568 | . . . . . . 7 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
29 | 5, 28 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘(ringLMod‘𝑅))) |
30 | 29 | fveq2d 6441 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
31 | 27, 30 | syl5eq 2873 | . . . 4 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
32 | 26, 31 | eleqtrd 2908 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))) |
33 | 8 | fveq2d 6441 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
34 | 3, 33 | syl5eq 2873 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
35 | 12, 19 | ressbasss 16302 | . . . . 5 ⊢ (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) |
36 | 34, 35 | syl6eqss 3880 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
37 | 36, 1 | sseldd 3828 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
38 | 18, 19, 22, 13, 23, 24, 25, 6, 32, 37 | pwsvscafval 16514 | . 2 ⊢ (𝜑 → (𝐴( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
39 | 17, 38 | eqtrd 2861 | 1 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 Vcvv 3414 {csn 4399 × cxp 5344 ‘cfv 6127 (class class class)co 6910 ∘𝑓 cof 7160 Basecbs 16229 ↾s cress 16230 .rcmulr 16313 Scalarcsca 16315 ·𝑠 cvsca 16316 ↑s cpws 16467 ringLModcrglmod 19537 freeLMod cfrlm 20460 |
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-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 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-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 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-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-hom 16336 df-cco 16337 df-prds 16468 df-pws 16470 df-sra 19540 df-rgmod 19541 df-dsmm 20446 df-frlm 20461 |
This theorem is referenced by: frlmvscaval 20481 uvcresum 20506 matvsca2 20608 matunitlindflem1 33944 matunitlindflem2 33945 zlmodzxzscm 42996 aacllem 43453 |
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