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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 22302. (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 6921 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ V) | |
| 7 | mptscmfsuppd.z | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵) | |
| 8 | eqid 2737 | . 2 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
| 9 | eqid 2737 | . 2 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
| 10 | mptscmfsuppd.n | . 2 ⊢ · = ( ·𝑠 ‘𝑃) | |
| 11 | mptscmfsuppd.a | . . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶𝑌) | |
| 12 | 11 | feqmptd 6977 | . . 3 ⊢ (𝜑 → 𝐴 = (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘))) | 
| 13 | mptscmfsuppd.f | . . 3 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆)) | |
| 14 | 12, 13 | eqbrtrrd 5167 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)) finSupp (0g‘𝑆)) | 
| 15 | 1, 2, 4, 5, 6, 7, 8, 9, 10, 14 | mptscmfsupp0 20925 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 finSupp cfsupp 9401 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 LModclmod 20858 | 
| 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-pr 5432 ax-un 7755 | 
| 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-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-op 4633 df-uni 4908 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-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-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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8186 df-1o 8506 df-en 8986 df-fin 8989 df-fsupp 9402 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-ring 20232 df-lmod 20860 | 
| This theorem is referenced by: ply1coefsupp 22301 | 
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