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Mirrors > Home > MPE Home > Th. List > frlmvscaval | Structured version Visualization version GIF version |
Description: Coordinates of a scalar multiple with respect to a basis in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
Ref | Expression |
---|---|
frlmvscaval.y | ⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
frlmvscaval.b | ⊢ 𝐵 = (Base‘𝑌) |
frlmvscaval.k | ⊢ 𝐾 = (Base‘𝑅) |
frlmvscaval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
frlmvscaval.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
frlmvscaval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
frlmvscaval.j | ⊢ (𝜑 → 𝐽 ∈ 𝐼) |
frlmvscaval.v | ⊢ ∙ = ( ·𝑠 ‘𝑌) |
frlmvscaval.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
frlmvscaval | ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmvscaval.y | . . . 4 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
2 | frlmvscaval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
3 | frlmvscaval.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
4 | frlmvscaval.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | frlmvscaval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
6 | frlmvscaval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | frlmvscaval.v | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑌) | |
8 | frlmvscaval.t | . . . 4 ⊢ · = (.r‘𝑅) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | frlmvscafval 20509 | . . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
10 | 9 | fveq1d 6448 | . 2 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (((𝐼 × {𝐴}) ∘𝑓 · 𝑋)‘𝐽)) |
11 | fnconstg 6343 | . . . 4 ⊢ (𝐴 ∈ 𝐾 → (𝐼 × {𝐴}) Fn 𝐼) | |
12 | 5, 11 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼 × {𝐴}) Fn 𝐼) |
13 | 1, 3, 2 | frlmbasf 20503 | . . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋:𝐼⟶𝐾) |
14 | 4, 6, 13 | syl2anc 579 | . . . 4 ⊢ (𝜑 → 𝑋:𝐼⟶𝐾) |
15 | 14 | ffnd 6292 | . . 3 ⊢ (𝜑 → 𝑋 Fn 𝐼) |
16 | frlmvscaval.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼) | |
17 | fnfvof 7188 | . . 3 ⊢ ((((𝐼 × {𝐴}) Fn 𝐼 ∧ 𝑋 Fn 𝐼) ∧ (𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼)) → (((𝐼 × {𝐴}) ∘𝑓 · 𝑋)‘𝐽) = (((𝐼 × {𝐴})‘𝐽) · (𝑋‘𝐽))) | |
18 | 12, 15, 4, 16, 17 | syl22anc 829 | . 2 ⊢ (𝜑 → (((𝐼 × {𝐴}) ∘𝑓 · 𝑋)‘𝐽) = (((𝐼 × {𝐴})‘𝐽) · (𝑋‘𝐽))) |
19 | fvconst2g 6739 | . . . 4 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐽 ∈ 𝐼) → ((𝐼 × {𝐴})‘𝐽) = 𝐴) | |
20 | 5, 16, 19 | syl2anc 579 | . . 3 ⊢ (𝜑 → ((𝐼 × {𝐴})‘𝐽) = 𝐴) |
21 | 20 | oveq1d 6937 | . 2 ⊢ (𝜑 → (((𝐼 × {𝐴})‘𝐽) · (𝑋‘𝐽)) = (𝐴 · (𝑋‘𝐽))) |
22 | 10, 18, 21 | 3eqtrd 2817 | 1 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 {csn 4397 × cxp 5353 Fn wfn 6130 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ∘𝑓 cof 7172 Basecbs 16255 .rcmulr 16339 ·𝑠 cvsca 16342 freeLMod cfrlm 20489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-hom 16362 df-cco 16363 df-0g 16488 df-prds 16494 df-pws 16496 df-sra 19569 df-rgmod 19570 df-dsmm 20475 df-frlm 20490 |
This theorem is referenced by: frlmvscavalb 20513 frlmvplusgscavalb 20514 frlmphl 20524 frlmssuvc2 20538 frlmup1 20541 rrxvsca 23600 |
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