![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pwsvscafval | Structured version Visualization version GIF version |
Description: Scalar multiplication in a structure power is pointwise. (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) |
Ref | Expression |
---|---|
pwsvscafval | ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsvscaval.t | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑌) | |
2 | pwsvscaval.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
3 | pwsvscaval.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
4 | pwsvscaval.y | . . . . . . 7 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
5 | pwsvscaval.f | . . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑅) | |
6 | 4, 5 | pwsval 16499 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
7 | 2, 3, 6 | syl2anc 581 | . . . . 5 ⊢ (𝜑 → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
8 | 7 | fveq2d 6437 | . . . 4 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))) |
9 | 1, 8 | syl5eq 2873 | . . 3 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))) |
10 | 9 | oveqd 6922 | . 2 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))𝑋)) |
11 | eqid 2825 | . . 3 ⊢ (𝐹Xs(𝐼 × {𝑅})) = (𝐹Xs(𝐼 × {𝑅})) | |
12 | eqid 2825 | . . 3 ⊢ (Base‘(𝐹Xs(𝐼 × {𝑅}))) = (Base‘(𝐹Xs(𝐼 × {𝑅}))) | |
13 | eqid 2825 | . . 3 ⊢ ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅}))) = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅}))) | |
14 | pwsvscaval.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
15 | 5 | fvexi 6447 | . . . 4 ⊢ 𝐹 ∈ V |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
17 | fnconstg 6330 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝐼 × {𝑅}) Fn 𝐼) | |
18 | 2, 17 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼 × {𝑅}) Fn 𝐼) |
19 | pwsvscaval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
20 | pwsvscaval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
21 | pwsvscaval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
22 | 7 | fveq2d 6437 | . . . . 5 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
23 | 21, 22 | syl5eq 2873 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
24 | 20, 23 | eleqtrd 2908 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
25 | 11, 12, 13, 14, 16, 3, 18, 19, 24 | prdsvscaval 16492 | . 2 ⊢ (𝜑 → (𝐴( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))𝑋) = (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥)))) |
26 | fvconst2g 6723 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) | |
27 | 2, 26 | sylan 577 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) |
28 | 27 | fveq2d 6437 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥)) = ( ·𝑠 ‘𝑅)) |
29 | pwsvscaval.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑅) | |
30 | 28, 29 | syl6eqr 2879 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥)) = · ) |
31 | 30 | oveqd 6922 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥)) = (𝐴 · (𝑋‘𝑥))) |
32 | 31 | mpteq2dva 4967 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝐴 · (𝑋‘𝑥)))) |
33 | 19 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ 𝐾) |
34 | fvexd 6448 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑋‘𝑥) ∈ V) | |
35 | fconstmpt 5398 | . . . . 5 ⊢ (𝐼 × {𝐴}) = (𝑥 ∈ 𝐼 ↦ 𝐴) | |
36 | 35 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐼 × {𝐴}) = (𝑥 ∈ 𝐼 ↦ 𝐴)) |
37 | eqid 2825 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
38 | 4, 37, 21, 2, 3, 20 | pwselbas 16502 | . . . . 5 ⊢ (𝜑 → 𝑋:𝐼⟶(Base‘𝑅)) |
39 | 38 | feqmptd 6496 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐼 ↦ (𝑋‘𝑥))) |
40 | 3, 33, 34, 36, 39 | offval2 7174 | . . 3 ⊢ (𝜑 → ((𝐼 × {𝐴}) ∘𝑓 · 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝐴 · (𝑋‘𝑥)))) |
41 | 32, 40 | eqtr4d 2864 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥))) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
42 | 10, 25, 41 | 3eqtrd 2865 | 1 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 Vcvv 3414 {csn 4397 ↦ cmpt 4952 × cxp 5340 Fn wfn 6118 ‘cfv 6123 (class class class)co 6905 ∘𝑓 cof 7155 Basecbs 16222 Scalarcsca 16308 ·𝑠 cvsca 16309 Xscprds 16459 ↑s cpws 16460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 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 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-hom 16329 df-cco 16330 df-prds 16461 df-pws 16463 |
This theorem is referenced by: pwsvscaval 16508 pwsdiaglmhm 19416 pwssplit3 19420 frlmvscafval 20472 |
Copyright terms: Public domain | W3C validator |