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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 20925. (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 6660 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ V) | |
7 | mptscmfsuppd.z | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵) | |
8 | eqid 2798 | . 2 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
9 | eqid 2798 | . 2 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
10 | mptscmfsuppd.n | . 2 ⊢ · = ( ·𝑠 ‘𝑃) | |
11 | mptscmfsuppd.a | . . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶𝑌) | |
12 | 11 | feqmptd 6708 | . . 3 ⊢ (𝜑 → 𝐴 = (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘))) |
13 | mptscmfsuppd.f | . . 3 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆)) | |
14 | 12, 13 | eqbrtrrd 5054 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)) finSupp (0g‘𝑆)) |
15 | 1, 2, 4, 5, 6, 7, 8, 9, 10, 14 | mptscmfsupp0 19692 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 ↦ cmpt 5110 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 finSupp cfsupp 8817 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 LModclmod 19627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-supp 7814 df-er 8272 df-en 8493 df-fin 8496 df-fsupp 8818 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-ring 19292 df-lmod 19629 |
This theorem is referenced by: ply1coefsupp 20924 |
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