Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2821 | . . . 4 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
2 | eqid 2821 | . . . 4 ⊢ (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) = (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) | |
3 | fvexd 6671 | . . . 4 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V) | |
4 | pwsplusgval.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | pwsplusgval.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
6 | fnconstg 6553 | . . . . 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 2821 | . . . . . . . . 9 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
12 | 10, 11 | pwsval 16742 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
13 | 5, 4, 12 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
14 | 13 | fveq2d 6660 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑌) = (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
15 | 9, 14 | syl5eq 2868 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
16 | 8, 15 | eleqtrd 2915 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
17 | pwsplusgval.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
18 | 17, 15 | eleqtrd 2915 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
19 | eqid 2821 | . . . 4 ⊢ (.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) = (.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) | |
20 | 1, 2, 3, 4, 7, 16, 18, 19 | prdsmulrval 16731 | . . 3 ⊢ (𝜑 → (𝐹(.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(.r‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥)))) |
21 | fvconst2g 6950 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) | |
22 | 5, 21 | sylan 582 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) |
23 | 22 | fveq2d 6660 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (.r‘((𝐼 × {𝑅})‘𝑥)) = (.r‘𝑅)) |
24 | pwsmulrval.a | . . . . . 6 ⊢ · = (.r‘𝑅) | |
25 | 23, 24 | syl6eqr 2874 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (.r‘((𝐼 × {𝑅})‘𝑥)) = · ) |
26 | 25 | oveqd 7159 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(.r‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥)) = ((𝐹‘𝑥) · (𝐺‘𝑥))) |
27 | 26 | mpteq2dva 5147 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(.r‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))) |
28 | 20, 27 | eqtrd 2856 | . 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 6660 | . . . 4 ⊢ (𝜑 → (.r‘𝑌) = (.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
31 | 29, 30 | syl5eq 2868 | . . 3 ⊢ (𝜑 → ∙ = (.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
32 | 31 | oveqd 7159 | . 2 ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝐹(.r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))𝐺)) |
33 | fvexd 6671 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ V) | |
34 | fvexd 6671 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ V) | |
35 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
36 | 10, 35, 9, 5, 4, 8 | pwselbas 16745 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝑅)) |
37 | 36 | feqmptd 6719 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
38 | 10, 35, 9, 5, 4, 17 | pwselbas 16745 | . . . 4 ⊢ (𝜑 → 𝐺:𝐼⟶(Base‘𝑅)) |
39 | 38 | feqmptd 6719 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐼 ↦ (𝐺‘𝑥))) |
40 | 4, 33, 34, 37, 39 | offval2 7412 | . 2 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))) |
41 | 28, 32, 40 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝐹 ∘f · 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3486 {csn 4553 ↦ cmpt 5132 × cxp 5539 Fn wfn 6336 ‘cfv 6341 (class class class)co 7142 ∘f cof 7393 Basecbs 16466 .rcmulr 16549 Scalarcsca 16551 Xscprds 16702 ↑s cpws 16703 |
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 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-sup 8892 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-fz 12883 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-hom 16572 df-cco 16573 df-prds 16704 df-pws 16706 |
This theorem is referenced by: mpfmulcl 20302 mpfind 20303 evl1muld 20489 pf1mulcl 20500 ply1rem 24743 fta1glem2 24746 fta1blem 24748 plypf1 24788 |
Copyright terms: Public domain | W3C validator |