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Mirrors > Home > MPE Home > Th. List > pwsmulrval | Structured version Visualization version GIF version |
Description: Value of multiplication in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
Ref | Expression |
---|---|
pwsplusgval.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsplusgval.b | ⊢ 𝐵 = (Base‘𝑌) |
pwsplusgval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
pwsplusgval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
pwsplusgval.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
pwsplusgval.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
pwsmulrval.a | ⊢ · = (.r‘𝑅) |
pwsmulrval.p | ⊢ ∙ = (.r‘𝑌) |
Ref | Expression |
---|---|
pwsmulrval | ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝐹 ∘f · 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . 4 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
2 | eqid 2738 | . . . 4 ⊢ (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) = (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) | |
3 | fvexd 6683 | . . . 4 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V) | |
4 | pwsplusgval.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | pwsplusgval.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
6 | fnconstg 6560 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝐼 × {𝑅}) Fn 𝐼) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 × {𝑅}) Fn 𝐼) |
8 | pwsplusgval.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
9 | pwsplusgval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
10 | pwsplusgval.y | . . . . . . . . 9 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
11 | eqid 2738 | . . . . . . . . 9 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
12 | 10, 11 | pwsval 16855 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
13 | 5, 4, 12 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
14 | 13 | fveq2d 6672 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑌) = (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
15 | 9, 14 | syl5eq 2785 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
16 | 8, 15 | eleqtrd 2835 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
17 | pwsplusgval.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
18 | 17, 15 | eleqtrd 2835 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
19 | eqid 2738 | . . . 4 ⊢ (.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) = (.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) | |
20 | 1, 2, 3, 4, 7, 16, 18, 19 | prdsmulrval 16844 | . . 3 ⊢ (𝜑 → (𝐹(.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(.r‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥)))) |
21 | fvconst2g 6968 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) | |
22 | 5, 21 | sylan 583 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) |
23 | 22 | fveq2d 6672 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (.r‘((𝐼 × {𝑅})‘𝑥)) = (.r‘𝑅)) |
24 | pwsmulrval.a | . . . . . 6 ⊢ · = (.r‘𝑅) | |
25 | 23, 24 | eqtr4di 2791 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (.r‘((𝐼 × {𝑅})‘𝑥)) = · ) |
26 | 25 | oveqd 7181 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(.r‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥)) = ((𝐹‘𝑥) · (𝐺‘𝑥))) |
27 | 26 | mpteq2dva 5122 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(.r‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))) |
28 | 20, 27 | eqtrd 2773 | . 2 ⊢ (𝜑 → (𝐹(.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))) |
29 | pwsmulrval.p | . . . 4 ⊢ ∙ = (.r‘𝑌) | |
30 | 13 | fveq2d 6672 | . . . 4 ⊢ (𝜑 → (.r‘𝑌) = (.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
31 | 29, 30 | syl5eq 2785 | . . 3 ⊢ (𝜑 → ∙ = (.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
32 | 31 | oveqd 7181 | . 2 ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝐹(.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))𝐺)) |
33 | fvexd 6683 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ V) | |
34 | fvexd 6683 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ V) | |
35 | eqid 2738 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
36 | 10, 35, 9, 5, 4, 8 | pwselbas 16858 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝑅)) |
37 | 36 | feqmptd 6731 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
38 | 10, 35, 9, 5, 4, 17 | pwselbas 16858 | . . . 4 ⊢ (𝜑 → 𝐺:𝐼⟶(Base‘𝑅)) |
39 | 38 | feqmptd 6731 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐼 ↦ (𝐺‘𝑥))) |
40 | 4, 33, 34, 37, 39 | offval2 7438 | . 2 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))) |
41 | 28, 32, 40 | 3eqtr4d 2783 | 1 ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝐹 ∘f · 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 Vcvv 3397 {csn 4513 ↦ cmpt 5107 × cxp 5517 Fn wfn 6328 ‘cfv 6333 (class class class)co 7164 ∘f cof 7417 Basecbs 16579 .rcmulr 16662 Scalarcsca 16664 Xscprds 16815 ↑s cpws 16816 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-hom 16685 df-cco 16686 df-prds 16817 df-pws 16819 |
This theorem is referenced by: mpfmulcl 20913 mpfind 20914 evl1muld 21106 pf1mulcl 21117 ply1rem 24908 fta1glem2 24911 fta1blem 24913 plypf1 24953 pwspjmhmmgpd 39832 evlsmulval 39841 |
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