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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 6424 | . . . . . 6 ⊢ (𝑁‘𝑌) ∈ V | |
2 | fvex 6424 | . . . . . 6 ⊢ (𝐺‘𝑧) ∈ V | |
3 | 1, 2 | ifex 4325 | . . . . 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 6234 | . . . 4 ⊢ dom 𝐹 = 𝑆 |
6 | 5 | difeq1i 3922 | . . 3 ⊢ (dom 𝐹 ∖ (𝑆 ∖ {𝑋})) = (𝑆 ∖ (𝑆 ∖ {𝑋})) |
7 | snssi 4527 | . . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → {𝑋} ⊆ 𝑆) | |
8 | 7 | 3ad2ant2 1165 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) → {𝑋} ⊆ 𝑆) |
9 | 8 | 3ad2ant2 1165 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → {𝑋} ⊆ 𝑆) |
10 | dfss4 4059 | . . . . 5 ⊢ ({𝑋} ⊆ 𝑆 ↔ (𝑆 ∖ (𝑆 ∖ {𝑋})) = {𝑋}) | |
11 | 9, 10 | sylib 210 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → (𝑆 ∖ (𝑆 ∖ {𝑋})) = {𝑋}) |
12 | snfi 8280 | . . . 4 ⊢ {𝑋} ∈ Fin | |
13 | 11, 12 | syl6eqel 2886 | . . 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 | syl5eqel 2882 | . 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 43042 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋})))) → 𝐹 ∈ (𝐸 ↑𝑚 𝑆)) |
22 | 21 | 3adant3 1163 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 ∈ (𝐸 ↑𝑚 𝑆)) |
23 | elmapfun 8119 | . . 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 8117 | . . . . 5 ⊢ (𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋})) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸) | |
26 | 4 | fdmdifeqresdif 42919 | . . . . 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 1166 | . . 3 ⊢ ((𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) → 𝐺 = (𝐹 ↾ (𝑆 ∖ {𝑋}))) |
29 | 28 | 3ad2ant2 1165 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐺 = (𝐹 ↾ (𝑆 ∖ {𝑋}))) |
30 | simp3 1169 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐺 finSupp 0 ) | |
31 | 18 | fvexi 6425 | . . 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 8544 | 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 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 Vcvv 3385 ∖ cdif 3766 ⊆ wss 3769 ifcif 4277 𝒫 cpw 4349 {csn 4368 class class class wbr 4843 ↦ cmpt 4922 dom cdm 5312 ↾ cres 5314 Fun wfun 6095 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 ↑𝑚 cmap 8095 Fincfn 8195 finSupp cfsupp 8517 Basecbs 16184 Scalarcsca 16270 0gc0g 16415 invgcminusg 17739 LModclmod 19181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-en 8196 df-fin 8199 df-fsupp 8518 df-0g 16417 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-grp 17741 df-minusg 17742 df-ring 18865 df-lmod 19183 |
This theorem is referenced by: lincext3 43044 lindslinindsimp1 43045 islindeps2 43071 |
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