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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 6789 | . . . 4 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V) | |
4 | pwsplusgval.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | pwsplusgval.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
6 | fnconstg 6662 | . . . . 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 17197 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
13 | 5, 4, 12 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
14 | 13 | fveq2d 6778 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑌) = (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
15 | 9, 14 | eqtrid 2790 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
16 | 8, 15 | eleqtrd 2841 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
17 | pwsplusgval.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
18 | 17, 15 | eleqtrd 2841 | . . . 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 17186 | . . 3 ⊢ (𝜑 → (𝐹(.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(.r‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥)))) |
21 | fvconst2g 7077 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) | |
22 | 5, 21 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) |
23 | 22 | fveq2d 6778 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (.r‘((𝐼 × {𝑅})‘𝑥)) = (.r‘𝑅)) |
24 | pwsmulrval.a | . . . . . 6 ⊢ · = (.r‘𝑅) | |
25 | 23, 24 | eqtr4di 2796 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (.r‘((𝐼 × {𝑅})‘𝑥)) = · ) |
26 | 25 | oveqd 7292 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(.r‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥)) = ((𝐹‘𝑥) · (𝐺‘𝑥))) |
27 | 26 | mpteq2dva 5174 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(.r‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))) |
28 | 20, 27 | eqtrd 2778 | . 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 6778 | . . . 4 ⊢ (𝜑 → (.r‘𝑌) = (.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
31 | 29, 30 | eqtrid 2790 | . . 3 ⊢ (𝜑 → ∙ = (.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
32 | 31 | oveqd 7292 | . 2 ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝐹(.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))𝐺)) |
33 | fvexd 6789 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ V) | |
34 | fvexd 6789 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ V) | |
35 | eqid 2738 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
36 | 10, 35, 9, 5, 4, 8 | pwselbas 17200 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝑅)) |
37 | 36 | feqmptd 6837 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
38 | 10, 35, 9, 5, 4, 17 | pwselbas 17200 | . . . 4 ⊢ (𝜑 → 𝐺:𝐼⟶(Base‘𝑅)) |
39 | 38 | feqmptd 6837 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐼 ↦ (𝐺‘𝑥))) |
40 | 4, 33, 34, 37, 39 | offval2 7553 | . 2 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))) |
41 | 28, 32, 40 | 3eqtr4d 2788 | 1 ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝐹 ∘f · 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 {csn 4561 ↦ cmpt 5157 × cxp 5587 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 ∘f cof 7531 Basecbs 16912 .rcmulr 16963 Scalarcsca 16965 Xscprds 17156 ↑s cpws 17157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-prds 17158 df-pws 17160 |
This theorem is referenced by: mpfmulcl 21316 mpfind 21317 evl1muld 21509 pf1mulcl 21520 ply1rem 25328 fta1glem2 25331 fta1blem 25333 plypf1 25373 pwspjmhmmgpd 40267 evlsmulval 40278 |
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