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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 22318. (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 6922 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ V) | |
7 | mptscmfsuppd.z | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵) | |
8 | eqid 2735 | . 2 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
9 | eqid 2735 | . 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 5172 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)) finSupp (0g‘𝑆)) |
15 | 1, 2, 4, 5, 6, 7, 8, 9, 10, 14 | mptscmfsupp0 20942 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 finSupp cfsupp 9399 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 LModclmod 20875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8185 df-1o 8505 df-en 8985 df-fin 8988 df-fsupp 9400 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-ring 20253 df-lmod 20877 |
This theorem is referenced by: ply1coefsupp 22317 |
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