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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 17533 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
7 | 2, 3, 6 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
8 | 7 | fveq2d 6911 | . . . 4 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))) |
9 | 1, 8 | eqtrid 2787 | . . 3 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))) |
10 | 9 | oveqd 7448 | . 2 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))𝑋)) |
11 | eqid 2735 | . . 3 ⊢ (𝐹Xs(𝐼 × {𝑅})) = (𝐹Xs(𝐼 × {𝑅})) | |
12 | eqid 2735 | . . 3 ⊢ (Base‘(𝐹Xs(𝐼 × {𝑅}))) = (Base‘(𝐹Xs(𝐼 × {𝑅}))) | |
13 | eqid 2735 | . . 3 ⊢ ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅}))) = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅}))) | |
14 | pwsvscaval.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
15 | 5 | fvexi 6921 | . . . 4 ⊢ 𝐹 ∈ V |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
17 | fnconstg 6797 | . . . 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 6911 | . . . . 5 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
23 | 21, 22 | eqtrid 2787 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
24 | 20, 23 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
25 | 11, 12, 13, 14, 16, 3, 18, 19, 24 | prdsvscaval 17526 | . 2 ⊢ (𝜑 → (𝐴( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))𝑋) = (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥)))) |
26 | fvconst2g 7222 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) | |
27 | 2, 26 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) |
28 | 27 | fveq2d 6911 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥)) = ( ·𝑠 ‘𝑅)) |
29 | pwsvscaval.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑅) | |
30 | 28, 29 | eqtr4di 2793 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥)) = · ) |
31 | 30 | oveqd 7448 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥)) = (𝐴 · (𝑋‘𝑥))) |
32 | 31 | mpteq2dva 5248 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝐴 · (𝑋‘𝑥)))) |
33 | 19 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ 𝐾) |
34 | fvexd 6922 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑋‘𝑥) ∈ V) | |
35 | fconstmpt 5751 | . . . . 5 ⊢ (𝐼 × {𝐴}) = (𝑥 ∈ 𝐼 ↦ 𝐴) | |
36 | 35 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐼 × {𝐴}) = (𝑥 ∈ 𝐼 ↦ 𝐴)) |
37 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
38 | 4, 37, 21, 2, 3, 20 | pwselbas 17536 | . . . . 5 ⊢ (𝜑 → 𝑋:𝐼⟶(Base‘𝑅)) |
39 | 38 | feqmptd 6977 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐼 ↦ (𝑋‘𝑥))) |
40 | 3, 33, 34, 36, 39 | offval2 7717 | . . 3 ⊢ (𝜑 → ((𝐼 × {𝐴}) ∘f · 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝐴 · (𝑋‘𝑥)))) |
41 | 32, 40 | eqtr4d 2778 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥))) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
42 | 10, 25, 41 | 3eqtrd 2779 | 1 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 ↦ cmpt 5231 × cxp 5687 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 Xscprds 17492 ↑s cpws 17493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-prds 17494 df-pws 17496 |
This theorem is referenced by: pwsvscaval 17542 pwsdiaglmhm 21074 pwssplit3 21078 frlmvscafval 21804 mhphf2 42585 |
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