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Mirrors > Home > MPE Home > Th. List > Mathboxes > lincreslvec3 | Structured version Visualization version GIF version |
Description: Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.) |
Ref | Expression |
---|---|
lincresunit.b | ⊢ 𝐵 = (Base‘𝑀) |
lincresunit.r | ⊢ 𝑅 = (Scalar‘𝑀) |
lincresunit.e | ⊢ 𝐸 = (Base‘𝑅) |
lincresunit.u | ⊢ 𝑈 = (Unit‘𝑅) |
lincresunit.0 | ⊢ 0 = (0g‘𝑅) |
lincresunit.z | ⊢ 𝑍 = (0g‘𝑀) |
lincresunit.n | ⊢ 𝑁 = (invg‘𝑅) |
lincresunit.i | ⊢ 𝐼 = (invr‘𝑅) |
lincresunit.t | ⊢ · = (.r‘𝑅) |
lincresunit.g | ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) |
Ref | Expression |
---|---|
lincreslvec3 | ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lveclmod 20349 | . . . 4 ⊢ (𝑀 ∈ LVec → 𝑀 ∈ LMod) | |
2 | 1 | 3anim2i 1151 | . . 3 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆)) |
3 | 2 | 3ad2ant1 1131 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆)) |
4 | simp21 1204 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝐹 ∈ (𝐸 ↑m 𝑆)) | |
5 | elmapi 8611 | . . . . . 6 ⊢ (𝐹 ∈ (𝐸 ↑m 𝑆) → 𝐹:𝑆⟶𝐸) | |
6 | 5 | 3ad2ant1 1131 | . . . . 5 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) → 𝐹:𝑆⟶𝐸) |
7 | simp3 1136 | . . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
8 | ffvelrn 6953 | . . . . 5 ⊢ ((𝐹:𝑆⟶𝐸 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ 𝐸) | |
9 | 6, 7, 8 | syl2anr 596 | . . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝐸) |
10 | simpr2 1193 | . . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ≠ 0 ) | |
11 | lincresunit.r | . . . . . . . 8 ⊢ 𝑅 = (Scalar‘𝑀) | |
12 | 11 | lvecdrng 20348 | . . . . . . 7 ⊢ (𝑀 ∈ LVec → 𝑅 ∈ DivRing) |
13 | 12 | 3ad2ant2 1132 | . . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) → 𝑅 ∈ DivRing) |
14 | 13 | adantr 480 | . . . . 5 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 )) → 𝑅 ∈ DivRing) |
15 | lincresunit.e | . . . . . 6 ⊢ 𝐸 = (Base‘𝑅) | |
16 | lincresunit.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
17 | lincresunit.0 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
18 | 15, 16, 17 | drngunit 19977 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((𝐹‘𝑋) ∈ 𝑈 ↔ ((𝐹‘𝑋) ∈ 𝐸 ∧ (𝐹‘𝑋) ≠ 0 ))) |
19 | 14, 18 | syl 17 | . . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 )) → ((𝐹‘𝑋) ∈ 𝑈 ↔ ((𝐹‘𝑋) ∈ 𝐸 ∧ (𝐹‘𝑋) ≠ 0 ))) |
20 | 9, 10, 19 | mpbir2and 709 | . . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝑈) |
21 | 20 | 3adant3 1130 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐹‘𝑋) ∈ 𝑈) |
22 | simp3 1136 | . . 3 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) → 𝐹 finSupp 0 ) | |
23 | 22 | 3ad2ant2 1132 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝐹 finSupp 0 ) |
24 | simp3 1136 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐹( linC ‘𝑀)𝑆) = 𝑍) | |
25 | lincresunit.b | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
26 | lincresunit.z | . . 3 ⊢ 𝑍 = (0g‘𝑀) | |
27 | lincresunit.n | . . 3 ⊢ 𝑁 = (invg‘𝑅) | |
28 | lincresunit.i | . . 3 ⊢ 𝐼 = (invr‘𝑅) | |
29 | lincresunit.t | . . 3 ⊢ · = (.r‘𝑅) | |
30 | lincresunit.g | . . 3 ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) | |
31 | 25, 11, 15, 16, 17, 26, 27, 28, 29, 30 | lincresunit3 45774 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋) |
32 | 3, 4, 21, 23, 24, 31 | syl131anc 1381 | 1 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 ∖ cdif 3888 𝒫 cpw 4538 {csn 4566 class class class wbr 5078 ↦ cmpt 5161 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 ↑m cmap 8589 finSupp cfsupp 9089 Basecbs 16893 .rcmulr 16944 Scalarcsca 16946 0gc0g 17131 invgcminusg 18559 Unitcui 19862 invrcinvr 19894 DivRingcdr 19972 LModclmod 20104 LVecclvec 20345 linC clinc 45697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-tpos 8026 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-fzo 13365 df-seq 13703 df-hash 14026 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-0g 17133 df-gsum 17134 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-mhm 18411 df-submnd 18412 df-grp 18561 df-minusg 18562 df-mulg 18682 df-ghm 18813 df-cntz 18904 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-ring 19766 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-drng 19974 df-lmod 20106 df-lvec 20346 df-linc 45699 |
This theorem is referenced by: isldepslvec2 45778 |
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