![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2725 | . . . . 5 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
5 | 3, 4 | pwsval 17471 | . . . 4 ⊢ ((𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
6 | 1, 2, 5 | syl2anc 582 | . . 3 ⊢ (𝜑 → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
7 | 6 | oveq1d 7434 | . 2 ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (((Scalar‘𝑅)Xs(𝐼 × {𝑅})) Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)))) |
8 | fconstmpt 5740 | . . . 4 ⊢ (𝐼 × {𝑅}) = (𝑥 ∈ 𝐼 ↦ 𝑅) | |
9 | 8 | oveq2i 7430 | . . 3 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) |
10 | pwsgsum.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
11 | eqid 2725 | . . 3 ⊢ (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) | |
12 | pwsgsum.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑊) | |
13 | fvexd 6911 | . . 3 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V) | |
14 | 1 | adantr 479 | . . 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 6900 | . . . . 5 ⊢ (𝜑 → (0g‘𝑌) = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
19 | 17, 18 | eqtrid 2777 | . . . 4 ⊢ (𝜑 → 0 = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
20 | 16, 19 | breqtrd 5175 | . . 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 19948 | . 2 ⊢ (𝜑 → (((Scalar‘𝑅)Xs(𝐼 × {𝑅})) Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) |
22 | 7, 21 | eqtrd 2765 | 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 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 {csn 4630 class class class wbr 5149 ↦ cmpt 5232 × cxp 5676 ‘cfv 6549 (class class class)co 7419 finSupp cfsupp 9387 Basecbs 17183 Scalarcsca 17239 0gc0g 17424 Σg cgsu 17425 Xscprds 17430 ↑s cpws 17431 CMndccmn 19747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-sup 9467 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-struct 17119 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-hom 17260 df-cco 17261 df-0g 17426 df-gsum 17427 df-prds 17432 df-pws 17434 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-cntz 19280 df-cmn 19749 |
This theorem is referenced by: frlmgsum 21723 evls1fpws 22313 plypf1 26191 evlsvvval 41931 |
Copyright terms: Public domain | W3C validator |