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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 2740 | . . . . 5 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
5 | 3, 4 | pwsval 17195 | . . . 4 ⊢ ((𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
6 | 1, 2, 5 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
7 | 6 | oveq1d 7286 | . 2 ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (((Scalar‘𝑅)Xs(𝐼 × {𝑅})) Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)))) |
8 | fconstmpt 5650 | . . . 4 ⊢ (𝐼 × {𝑅}) = (𝑥 ∈ 𝐼 ↦ 𝑅) | |
9 | 8 | oveq2i 7282 | . . 3 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) |
10 | pwsgsum.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
11 | eqid 2740 | . . 3 ⊢ (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) | |
12 | pwsgsum.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑊) | |
13 | fvexd 6786 | . . 3 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V) | |
14 | 1 | adantr 481 | . . 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 6775 | . . . . 5 ⊢ (𝜑 → (0g‘𝑌) = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
19 | 17, 18 | eqtrid 2792 | . . . 4 ⊢ (𝜑 → 0 = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
20 | 16, 19 | breqtrd 5105 | . . 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 19580 | . 2 ⊢ (𝜑 → (((Scalar‘𝑅)Xs(𝐼 × {𝑅})) Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) |
22 | 7, 21 | eqtrd 2780 | 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 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 {csn 4567 class class class wbr 5079 ↦ cmpt 5162 × cxp 5588 ‘cfv 6432 (class class class)co 7271 finSupp cfsupp 9106 Basecbs 16910 Scalarcsca 16963 0gc0g 17148 Σg cgsu 17149 Xscprds 17154 ↑s cpws 17155 CMndccmn 19384 |
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 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-sup 9179 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 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 12437 df-uz 12582 df-fz 13239 df-fzo 13382 df-seq 13720 df-hash 14043 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-hom 16984 df-cco 16985 df-0g 17150 df-gsum 17151 df-prds 17156 df-pws 17158 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-cntz 18921 df-cmn 19386 |
This theorem is referenced by: frlmgsum 20977 plypf1 25371 |
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