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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 15969 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
7 | 2, 3, 6 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
8 | 7 | fveq2d 6107 | . . . 4 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))) |
9 | 1, 8 | syl5eq 2656 | . . 3 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))) |
10 | 9 | oveqd 6566 | . 2 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))𝑋)) |
11 | eqid 2610 | . . 3 ⊢ (𝐹Xs(𝐼 × {𝑅})) = (𝐹Xs(𝐼 × {𝑅})) | |
12 | eqid 2610 | . . 3 ⊢ (Base‘(𝐹Xs(𝐼 × {𝑅}))) = (Base‘(𝐹Xs(𝐼 × {𝑅}))) | |
13 | eqid 2610 | . . 3 ⊢ ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅}))) = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅}))) | |
14 | pwsvscaval.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
15 | fvex 6113 | . . . . 5 ⊢ (Scalar‘𝑅) ∈ V | |
16 | 5, 15 | eqeltri 2684 | . . . 4 ⊢ 𝐹 ∈ V |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
18 | fnconstg 6006 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝐼 × {𝑅}) Fn 𝐼) | |
19 | 2, 18 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼 × {𝑅}) Fn 𝐼) |
20 | pwsvscaval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
21 | pwsvscaval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
22 | pwsvscaval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
23 | 7 | fveq2d 6107 | . . . . 5 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
24 | 22, 23 | syl5eq 2656 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
25 | 21, 24 | eleqtrd 2690 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
26 | 11, 12, 13, 14, 17, 3, 19, 20, 25 | prdsvscaval 15962 | . 2 ⊢ (𝜑 → (𝐴( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))𝑋) = (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥)))) |
27 | fvconst2g 6372 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) | |
28 | 2, 27 | sylan 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) |
29 | 28 | fveq2d 6107 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥)) = ( ·𝑠 ‘𝑅)) |
30 | pwsvscaval.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑅) | |
31 | 29, 30 | syl6eqr 2662 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥)) = · ) |
32 | 31 | oveqd 6566 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥)) = (𝐴 · (𝑋‘𝑥))) |
33 | 32 | mpteq2dva 4672 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝐴 · (𝑋‘𝑥)))) |
34 | 20 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ 𝐾) |
35 | fvex 6113 | . . . . 5 ⊢ (𝑋‘𝑥) ∈ V | |
36 | 35 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑋‘𝑥) ∈ V) |
37 | fconstmpt 5085 | . . . . 5 ⊢ (𝐼 × {𝐴}) = (𝑥 ∈ 𝐼 ↦ 𝐴) | |
38 | 37 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐼 × {𝐴}) = (𝑥 ∈ 𝐼 ↦ 𝐴)) |
39 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
40 | 4, 39, 22, 2, 3, 21 | pwselbas 15972 | . . . . 5 ⊢ (𝜑 → 𝑋:𝐼⟶(Base‘𝑅)) |
41 | 40 | feqmptd 6159 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐼 ↦ (𝑋‘𝑥))) |
42 | 3, 34, 36, 38, 41 | offval2 6812 | . . 3 ⊢ (𝜑 → ((𝐼 × {𝐴}) ∘𝑓 · 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝐴 · (𝑋‘𝑥)))) |
43 | 33, 42 | eqtr4d 2647 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥))) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
44 | 10, 26, 43 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 ↦ cmpt 4643 × cxp 5036 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 Xscprds 15929 ↑s cpws 15930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-prds 15931 df-pws 15933 |
This theorem is referenced by: pwsvscaval 15978 pwsdiaglmhm 18878 pwssplit3 18882 frlmvscafval 19928 |
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