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Mirrors > Home > MPE Home > Th. List > pwsvscaval | Structured version Visualization version GIF version |
Description: Scalar multiplication of a single coordinate in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
Ref | Expression |
---|---|
pwsvscaval.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsvscaval.b | ⊢ 𝐵 = (Base‘𝑌) |
pwsvscaval.s | ⊢ · = ( ·𝑠 ‘𝑅) |
pwsvscaval.t | ⊢ ∙ = ( ·𝑠 ‘𝑌) |
pwsvscaval.f | ⊢ 𝐹 = (Scalar‘𝑅) |
pwsvscaval.k | ⊢ 𝐾 = (Base‘𝐹) |
pwsvscaval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
pwsvscaval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
pwsvscaval.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
pwsvscaval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
pwsvscaval.j | ⊢ (𝜑 → 𝐽 ∈ 𝐼) |
Ref | Expression |
---|---|
pwsvscaval | ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsvscaval.y | . . . 4 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
2 | pwsvscaval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
3 | pwsvscaval.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑅) | |
4 | pwsvscaval.t | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑌) | |
5 | pwsvscaval.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑅) | |
6 | pwsvscaval.k | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
7 | pwsvscaval.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
8 | pwsvscaval.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
9 | pwsvscaval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
10 | pwsvscaval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | pwsvscafval 17203 | . . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
12 | 11 | fveq1d 6773 | . 2 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (((𝐼 × {𝐴}) ∘f · 𝑋)‘𝐽)) |
13 | pwsvscaval.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼) | |
14 | eqid 2740 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
15 | 1, 14, 2, 7, 8, 10 | pwselbas 17198 | . . . . 5 ⊢ (𝜑 → 𝑋:𝐼⟶(Base‘𝑅)) |
16 | 15 | ffnd 6599 | . . . 4 ⊢ (𝜑 → 𝑋 Fn 𝐼) |
17 | eqidd 2741 | . . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐼) → (𝑋‘𝐽) = (𝑋‘𝐽)) | |
18 | 8, 9, 16, 17 | ofc1 7553 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐼) → (((𝐼 × {𝐴}) ∘f · 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) |
19 | 13, 18 | mpdan 684 | . 2 ⊢ (𝜑 → (((𝐼 × {𝐴}) ∘f · 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) |
20 | 12, 19 | eqtrd 2780 | 1 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 {csn 4567 × cxp 5588 ‘cfv 6432 (class class class)co 7271 ∘f cof 7525 Basecbs 16910 Scalarcsca 16963 ·𝑠 cvsca 16964 ↑s cpws 17155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-fz 13239 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-hom 16984 df-cco 16985 df-prds 17156 df-pws 17158 |
This theorem is referenced by: (None) |
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