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Mirrors > Home > MPE Home > Th. List > pwsgsum | Structured version Visualization version GIF version |
Description: Finite commutative sums in a power structure are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2015.) (Revised by AV, 9-Jun-2019.) |
Ref | Expression |
---|---|
pwsgsum.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsgsum.b | ⊢ 𝐵 = (Base‘𝑅) |
pwsgsum.z | ⊢ 0 = (0g‘𝑌) |
pwsgsum.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
pwsgsum.j | ⊢ (𝜑 → 𝐽 ∈ 𝑊) |
pwsgsum.r | ⊢ (𝜑 → 𝑅 ∈ CMnd) |
pwsgsum.f | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽)) → 𝑈 ∈ 𝐵) |
pwsgsum.w | ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)) finSupp 0 ) |
Ref | Expression |
---|---|
pwsgsum | ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsgsum.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CMnd) | |
2 | pwsgsum.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
3 | pwsgsum.y | . . . . 5 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
4 | eqid 2780 | . . . . 5 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
5 | 3, 4 | pwsval 16621 | . . . 4 ⊢ ((𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
6 | 1, 2, 5 | syl2anc 576 | . . 3 ⊢ (𝜑 → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
7 | 6 | oveq1d 6997 | . 2 ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (((Scalar‘𝑅)Xs(𝐼 × {𝑅})) Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)))) |
8 | fconstmpt 5468 | . . . 4 ⊢ (𝐼 × {𝑅}) = (𝑥 ∈ 𝐼 ↦ 𝑅) | |
9 | 8 | oveq2i 6993 | . . 3 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) |
10 | pwsgsum.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
11 | eqid 2780 | . . 3 ⊢ (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) | |
12 | pwsgsum.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑊) | |
13 | fvexd 6519 | . . 3 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V) | |
14 | 1 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ CMnd) |
15 | pwsgsum.f | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽)) → 𝑈 ∈ 𝐵) | |
16 | pwsgsum.w | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)) finSupp 0 ) | |
17 | pwsgsum.z | . . . . 5 ⊢ 0 = (0g‘𝑌) | |
18 | 6 | fveq2d 6508 | . . . . 5 ⊢ (𝜑 → (0g‘𝑌) = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
19 | 17, 18 | syl5eq 2828 | . . . 4 ⊢ (𝜑 → 0 = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
20 | 16, 19 | breqtrd 4960 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)) finSupp (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
21 | 9, 10, 11, 2, 12, 13, 14, 15, 20 | prdsgsum 18863 | . 2 ⊢ (𝜑 → (((Scalar‘𝑅)Xs(𝐼 × {𝑅})) Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) |
22 | 7, 21 | eqtrd 2816 | 1 ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 Vcvv 3417 {csn 4444 class class class wbr 4934 ↦ cmpt 5013 × cxp 5409 ‘cfv 6193 (class class class)co 6982 finSupp cfsupp 8634 Basecbs 16345 Scalarcsca 16430 0gc0g 16575 Σg cgsu 16576 Xscprds 16581 ↑s cpws 16582 CMndccmn 18678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-se 5371 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-isom 6202 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-1st 7507 df-2nd 7508 df-supp 7640 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-oadd 7915 df-er 8095 df-map 8214 df-ixp 8266 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-fsupp 8635 df-sup 8707 df-oi 8775 df-card 9168 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-nn 11446 df-2 11509 df-3 11510 df-4 11511 df-5 11512 df-6 11513 df-7 11514 df-8 11515 df-9 11516 df-n0 11714 df-z 11800 df-dec 11918 df-uz 12065 df-fz 12715 df-fzo 12856 df-seq 13191 df-hash 13512 df-struct 16347 df-ndx 16348 df-slot 16349 df-base 16351 df-plusg 16440 df-mulr 16441 df-sca 16443 df-vsca 16444 df-ip 16445 df-tset 16446 df-ple 16447 df-ds 16449 df-hom 16451 df-cco 16452 df-0g 16577 df-gsum 16578 df-prds 16583 df-pws 16585 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-cntz 18230 df-cmn 18680 |
This theorem is referenced by: frlmgsum 20633 plypf1 24520 |
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