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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 20954 | . . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
10 | 9 | fveq1d 6770 | . 2 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (((𝐼 × {𝐴}) ∘f · 𝑋)‘𝐽)) |
11 | fnconstg 6658 | . . . 4 ⊢ (𝐴 ∈ 𝐾 → (𝐼 × {𝐴}) Fn 𝐼) | |
12 | 5, 11 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼 × {𝐴}) Fn 𝐼) |
13 | 1, 3, 2 | frlmbasf 20948 | . . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋:𝐼⟶𝐾) |
14 | 4, 6, 13 | syl2anc 583 | . . . 4 ⊢ (𝜑 → 𝑋:𝐼⟶𝐾) |
15 | 14 | ffnd 6597 | . . 3 ⊢ (𝜑 → 𝑋 Fn 𝐼) |
16 | frlmvscaval.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼) | |
17 | fnfvof 7541 | . . 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 835 | . 2 ⊢ (𝜑 → (((𝐼 × {𝐴}) ∘f · 𝑋)‘𝐽) = (((𝐼 × {𝐴})‘𝐽) · (𝑋‘𝐽))) |
19 | fvconst2g 7071 | . . . 4 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐽 ∈ 𝐼) → ((𝐼 × {𝐴})‘𝐽) = 𝐴) | |
20 | 5, 16, 19 | syl2anc 583 | . . 3 ⊢ (𝜑 → ((𝐼 × {𝐴})‘𝐽) = 𝐴) |
21 | 20 | oveq1d 7283 | . 2 ⊢ (𝜑 → (((𝐼 × {𝐴})‘𝐽) · (𝑋‘𝐽)) = (𝐴 · (𝑋‘𝐽))) |
22 | 10, 18, 21 | 3eqtrd 2783 | 1 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2109 {csn 4566 × cxp 5586 Fn wfn 6425 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 ∘f cof 7522 Basecbs 16893 .rcmulr 16944 ·𝑠 cvsca 16947 freeLMod cfrlm 20934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-sup 9162 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-fz 13222 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-hom 16967 df-cco 16968 df-0g 17133 df-prds 17139 df-pws 17141 df-sra 20415 df-rgmod 20416 df-dsmm 20920 df-frlm 20935 |
This theorem is referenced by: frlmvscavalb 20958 frlmvplusgscavalb 20959 frlmphl 20969 frlmssuvc2 20983 frlmup1 20986 rrxvsca 24539 frlmsnic 40243 prjspnfv01 40441 prjspner1 40443 |
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