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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 2737 | . . . . 5 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
| 5 | 3, 4 | pwsval 17531 | . . . 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 7446 | . 2 ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (((Scalar‘𝑅)Xs(𝐼 × {𝑅})) Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)))) |
| 8 | fconstmpt 5747 | . . . 4 ⊢ (𝐼 × {𝑅}) = (𝑥 ∈ 𝐼 ↦ 𝑅) | |
| 9 | 8 | oveq2i 7442 | . . 3 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) |
| 10 | pwsgsum.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 11 | eqid 2737 | . . 3 ⊢ (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) | |
| 12 | pwsgsum.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑊) | |
| 13 | fvexd 6921 | . . 3 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V) | |
| 14 | 1 | adantr 480 | . . 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 6910 | . . . . 5 ⊢ (𝜑 → (0g‘𝑌) = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
| 19 | 17, 18 | eqtrid 2789 | . . . 4 ⊢ (𝜑 → 0 = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
| 20 | 16, 19 | breqtrd 5169 | . . 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 19999 | . 2 ⊢ (𝜑 → (((Scalar‘𝑅)Xs(𝐼 × {𝑅})) Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) |
| 22 | 7, 21 | eqtrd 2777 | 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 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 (class class class)co 7431 finSupp cfsupp 9401 Basecbs 17247 Scalarcsca 17300 0gc0g 17484 Σg cgsu 17485 Xscprds 17490 ↑s cpws 17491 CMndccmn 19798 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-cntz 19335 df-cmn 19800 |
| This theorem is referenced by: frlmgsum 21792 evls1fpws 22373 plypf1 26251 evlsvvval 42573 |
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