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Mirrors > Home > MPE Home > Th. List > mptscmfsuppd | Structured version Visualization version GIF version |
Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 20392. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.) (Proof shortened by AV, 18-Oct-2019.) |
Ref | Expression |
---|---|
mptscmfsuppd.b | ⊢ 𝐵 = (Base‘𝑃) |
mptscmfsuppd.s | ⊢ 𝑆 = (Scalar‘𝑃) |
mptscmfsuppd.n | ⊢ · = ( ·𝑠 ‘𝑃) |
mptscmfsuppd.p | ⊢ (𝜑 → 𝑃 ∈ LMod) |
mptscmfsuppd.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
mptscmfsuppd.z | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵) |
mptscmfsuppd.a | ⊢ (𝜑 → 𝐴:𝑋⟶𝑌) |
mptscmfsuppd.f | ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆)) |
Ref | Expression |
---|---|
mptscmfsuppd | ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptscmfsuppd.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | mptscmfsuppd.p | . 2 ⊢ (𝜑 → 𝑃 ∈ LMod) | |
3 | mptscmfsuppd.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑃) | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑆 = (Scalar‘𝑃)) |
5 | mptscmfsuppd.b | . 2 ⊢ 𝐵 = (Base‘𝑃) | |
6 | fvexd 6678 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ V) | |
7 | mptscmfsuppd.z | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵) | |
8 | eqid 2818 | . 2 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
9 | eqid 2818 | . 2 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
10 | mptscmfsuppd.n | . 2 ⊢ · = ( ·𝑠 ‘𝑃) | |
11 | mptscmfsuppd.a | . . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶𝑌) | |
12 | 11 | feqmptd 6726 | . . 3 ⊢ (𝜑 → 𝐴 = (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘))) |
13 | mptscmfsuppd.f | . . 3 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆)) | |
14 | 12, 13 | eqbrtrrd 5081 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)) finSupp (0g‘𝑆)) |
15 | 1, 2, 4, 5, 6, 7, 8, 9, 10, 14 | mptscmfsupp0 19628 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 class class class wbr 5057 ↦ cmpt 5137 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 finSupp cfsupp 8821 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 0gc0g 16701 LModclmod 19563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-supp 7820 df-er 8278 df-en 8498 df-fin 8501 df-fsupp 8822 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-ring 19228 df-lmod 19565 |
This theorem is referenced by: ply1coefsupp 20391 |
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