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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 16761 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
7 | 2, 3, 6 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
8 | 7 | fveq2d 6676 | . . . 4 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))) |
9 | 1, 8 | syl5eq 2870 | . . 3 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))) |
10 | 9 | oveqd 7175 | . 2 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))𝑋)) |
11 | eqid 2823 | . . 3 ⊢ (𝐹Xs(𝐼 × {𝑅})) = (𝐹Xs(𝐼 × {𝑅})) | |
12 | eqid 2823 | . . 3 ⊢ (Base‘(𝐹Xs(𝐼 × {𝑅}))) = (Base‘(𝐹Xs(𝐼 × {𝑅}))) | |
13 | eqid 2823 | . . 3 ⊢ ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅}))) = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅}))) | |
14 | pwsvscaval.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
15 | 5 | fvexi 6686 | . . . 4 ⊢ 𝐹 ∈ V |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
17 | fnconstg 6569 | . . . 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 6676 | . . . . 5 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
23 | 21, 22 | syl5eq 2870 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
24 | 20, 23 | eleqtrd 2917 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
25 | 11, 12, 13, 14, 16, 3, 18, 19, 24 | prdsvscaval 16754 | . 2 ⊢ (𝜑 → (𝐴( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))𝑋) = (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥)))) |
26 | fvconst2g 6966 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) | |
27 | 2, 26 | sylan 582 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) |
28 | 27 | fveq2d 6676 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥)) = ( ·𝑠 ‘𝑅)) |
29 | pwsvscaval.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑅) | |
30 | 28, 29 | syl6eqr 2876 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥)) = · ) |
31 | 30 | oveqd 7175 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥)) = (𝐴 · (𝑋‘𝑥))) |
32 | 31 | mpteq2dva 5163 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝐴 · (𝑋‘𝑥)))) |
33 | 19 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ 𝐾) |
34 | fvexd 6687 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑋‘𝑥) ∈ V) | |
35 | fconstmpt 5616 | . . . . 5 ⊢ (𝐼 × {𝐴}) = (𝑥 ∈ 𝐼 ↦ 𝐴) | |
36 | 35 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐼 × {𝐴}) = (𝑥 ∈ 𝐼 ↦ 𝐴)) |
37 | eqid 2823 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
38 | 4, 37, 21, 2, 3, 20 | pwselbas 16764 | . . . . 5 ⊢ (𝜑 → 𝑋:𝐼⟶(Base‘𝑅)) |
39 | 38 | feqmptd 6735 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐼 ↦ (𝑋‘𝑥))) |
40 | 3, 33, 34, 36, 39 | offval2 7428 | . . 3 ⊢ (𝜑 → ((𝐼 × {𝐴}) ∘f · 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝐴 · (𝑋‘𝑥)))) |
41 | 32, 40 | eqtr4d 2861 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥))) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
42 | 10, 25, 41 | 3eqtrd 2862 | 1 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 ↦ cmpt 5148 × cxp 5555 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 Xscprds 16721 ↑s cpws 16722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-prds 16723 df-pws 16725 |
This theorem is referenced by: pwsvscaval 16770 pwsdiaglmhm 19831 pwssplit3 19835 frlmvscafval 20912 |
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