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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lincext2 | Structured version Visualization version GIF version |
Description: Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
Ref | Expression |
---|---|
lincext.b | ⊢ 𝐵 = (Base‘𝑀) |
lincext.r | ⊢ 𝑅 = (Scalar‘𝑀) |
lincext.e | ⊢ 𝐸 = (Base‘𝑅) |
lincext.0 | ⊢ 0 = (0g‘𝑅) |
lincext.z | ⊢ 𝑍 = (0g‘𝑀) |
lincext.n | ⊢ 𝑁 = (invg‘𝑅) |
lincext.f | ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧))) |
Ref | Expression |
---|---|
lincext2 | ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6909 | . . . . . 6 ⊢ (𝑁‘𝑌) ∈ V | |
2 | fvex 6909 | . . . . . 6 ⊢ (𝐺‘𝑧) ∈ V | |
3 | 1, 2 | ifex 4580 | . . . . 5 ⊢ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧)) ∈ V |
4 | lincext.f | . . . . 5 ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧))) | |
5 | 3, 4 | dmmpti 6700 | . . . 4 ⊢ dom 𝐹 = 𝑆 |
6 | 5 | difeq1i 4114 | . . 3 ⊢ (dom 𝐹 ∖ (𝑆 ∖ {𝑋})) = (𝑆 ∖ (𝑆 ∖ {𝑋})) |
7 | snssi 4813 | . . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → {𝑋} ⊆ 𝑆) | |
8 | 7 | 3ad2ant2 1131 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) → {𝑋} ⊆ 𝑆) |
9 | 8 | 3ad2ant2 1131 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → {𝑋} ⊆ 𝑆) |
10 | dfss4 4257 | . . . . 5 ⊢ ({𝑋} ⊆ 𝑆 ↔ (𝑆 ∖ (𝑆 ∖ {𝑋})) = {𝑋}) | |
11 | 9, 10 | sylib 217 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → (𝑆 ∖ (𝑆 ∖ {𝑋})) = {𝑋}) |
12 | snfi 9071 | . . . 4 ⊢ {𝑋} ∈ Fin | |
13 | 11, 12 | eqeltrdi 2833 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → (𝑆 ∖ (𝑆 ∖ {𝑋})) ∈ Fin) |
14 | 6, 13 | eqeltrid 2829 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → (dom 𝐹 ∖ (𝑆 ∖ {𝑋})) ∈ Fin) |
15 | lincext.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
16 | lincext.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑀) | |
17 | lincext.e | . . . 4 ⊢ 𝐸 = (Base‘𝑅) | |
18 | lincext.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
19 | lincext.z | . . . 4 ⊢ 𝑍 = (0g‘𝑀) | |
20 | lincext.n | . . . 4 ⊢ 𝑁 = (invg‘𝑅) | |
21 | 15, 16, 17, 18, 19, 20, 4 | lincext1 47713 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})))) → 𝐹 ∈ (𝐸 ↑m 𝑆)) |
22 | 21 | 3adant3 1129 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 ∈ (𝐸 ↑m 𝑆)) |
23 | elmapfun 8885 | . . 3 ⊢ (𝐹 ∈ (𝐸 ↑m 𝑆) → Fun 𝐹) | |
24 | 22, 23 | syl 17 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → Fun 𝐹) |
25 | elmapi 8868 | . . . . 5 ⊢ (𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸) | |
26 | 4 | fdmdifeqresdif 47596 | . . . . 5 ⊢ (𝐺:(𝑆 ∖ {𝑋})⟶𝐸 → 𝐺 = (𝐹 ↾ (𝑆 ∖ {𝑋}))) |
27 | 25, 26 | syl 17 | . . . 4 ⊢ (𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})) → 𝐺 = (𝐹 ↾ (𝑆 ∖ {𝑋}))) |
28 | 27 | 3ad2ant3 1132 | . . 3 ⊢ ((𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) → 𝐺 = (𝐹 ↾ (𝑆 ∖ {𝑋}))) |
29 | 28 | 3ad2ant2 1131 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐺 = (𝐹 ↾ (𝑆 ∖ {𝑋}))) |
30 | simp3 1135 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐺 finSupp 0 ) | |
31 | 18 | fvexi 6910 | . . 3 ⊢ 0 ∈ V |
32 | 31 | a1i 11 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 0 ∈ V) |
33 | 14, 22, 24, 29, 30, 32 | resfsupp 9426 | 1 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∖ cdif 3941 ⊆ wss 3944 ifcif 4530 𝒫 cpw 4604 {csn 4630 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5678 ↾ cres 5680 Fun wfun 6543 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 Fincfn 8964 finSupp cfsupp 9392 Basecbs 17199 Scalarcsca 17255 0gc0g 17440 invgcminusg 18915 LModclmod 20772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-1o 8487 df-map 8847 df-en 8965 df-fin 8968 df-fsupp 9393 df-0g 17442 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-grp 18917 df-minusg 18918 df-ring 20204 df-lmod 20774 |
This theorem is referenced by: lincext3 47715 lindslinindsimp1 47716 islindeps2 47742 |
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