Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19487. (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 | fvex 6113 | . . 3 ⊢ (𝐴‘𝑘) ∈ V | |
7 | 6 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ V) |
8 | mptscmfsuppd.z | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵) | |
9 | eqid 2610 | . 2 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
10 | eqid 2610 | . 2 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
11 | mptscmfsuppd.n | . 2 ⊢ · = ( ·𝑠 ‘𝑃) | |
12 | mptscmfsuppd.a | . . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶𝑌) | |
13 | 12 | feqmptd 6159 | . . 3 ⊢ (𝜑 → 𝐴 = (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘))) |
14 | mptscmfsuppd.f | . . 3 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆)) | |
15 | 13, 14 | eqbrtrrd 4607 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)) finSupp (0g‘𝑆)) |
16 | 1, 2, 4, 5, 7, 8, 9, 10, 11, 15 | mptscmfsupp0 18751 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 finSupp cfsupp 8158 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 LModclmod 18686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-supp 7183 df-er 7629 df-en 7842 df-fin 7845 df-fsupp 8159 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-ring 18372 df-lmod 18688 |
This theorem is referenced by: ply1coefsupp 19486 |
Copyright terms: Public domain | W3C validator |