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Mirrors > Home > MPE Home > Th. List > islindf5 | Structured version Visualization version GIF version |
Description: A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Ref | Expression |
---|---|
islindf5.f | ⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
islindf5.b | ⊢ 𝐵 = (Base‘𝐹) |
islindf5.c | ⊢ 𝐶 = (Base‘𝑇) |
islindf5.v | ⊢ · = ( ·𝑠 ‘𝑇) |
islindf5.e | ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) |
islindf5.t | ⊢ (𝜑 → 𝑇 ∈ LMod) |
islindf5.i | ⊢ (𝜑 → 𝐼 ∈ 𝑋) |
islindf5.r | ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) |
islindf5.a | ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) |
Ref | Expression |
---|---|
islindf5 | ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islindf5.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ LMod) | |
2 | islindf5.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑋) | |
3 | islindf5.a | . . . 4 ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) | |
4 | islindf5.c | . . . . 5 ⊢ 𝐶 = (Base‘𝑇) | |
5 | eqid 2622 | . . . . 5 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
6 | islindf5.v | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑇) | |
7 | eqid 2622 | . . . . 5 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
8 | eqid 2622 | . . . . 5 ⊢ (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑇)) | |
9 | eqid 2622 | . . . . 5 ⊢ (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = (Base‘((Scalar‘𝑇) freeLMod 𝐼)) | |
10 | 4, 5, 6, 7, 8, 9 | islindf4 20177 | . . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐴:𝐼⟶𝐶) → (𝐴 LIndF 𝑇 ↔ ∀𝑦 ∈ (Base‘((Scalar‘𝑇) freeLMod 𝐼))((𝑇 Σg (𝑦 ∘𝑓 · 𝐴)) = (0g‘𝑇) → 𝑦 = (𝐼 × {(0g‘(Scalar‘𝑇))})))) |
11 | 1, 2, 3, 10 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ ∀𝑦 ∈ (Base‘((Scalar‘𝑇) freeLMod 𝐼))((𝑇 Σg (𝑦 ∘𝑓 · 𝐴)) = (0g‘𝑇) → 𝑦 = (𝐼 × {(0g‘(Scalar‘𝑇))})))) |
12 | oveq1 6657 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∘𝑓 · 𝐴) = (𝑦 ∘𝑓 · 𝐴)) | |
13 | 12 | oveq2d 6666 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑇 Σg (𝑥 ∘𝑓 · 𝐴)) = (𝑇 Σg (𝑦 ∘𝑓 · 𝐴))) |
14 | islindf5.e | . . . . . . . . 9 ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) | |
15 | ovex 6678 | . . . . . . . . 9 ⊢ (𝑇 Σg (𝑦 ∘𝑓 · 𝐴)) ∈ V | |
16 | 13, 14, 15 | fvmpt 6282 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (𝐸‘𝑦) = (𝑇 Σg (𝑦 ∘𝑓 · 𝐴))) |
17 | 16 | adantl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐸‘𝑦) = (𝑇 Σg (𝑦 ∘𝑓 · 𝐴))) |
18 | 17 | eqeq1d 2624 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝐸‘𝑦) = (0g‘𝑇) ↔ (𝑇 Σg (𝑦 ∘𝑓 · 𝐴)) = (0g‘𝑇))) |
19 | islindf5.r | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) | |
20 | 5 | lmodring 18871 | . . . . . . . . . . . 12 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring) |
21 | 1, 20 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring) |
22 | 19, 21 | eqeltrd 2701 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring) |
23 | islindf5.f | . . . . . . . . . . 11 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
24 | eqid 2622 | . . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
25 | 23, 24 | frlm0 20098 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → (𝐼 × {(0g‘𝑅)}) = (0g‘𝐹)) |
26 | 22, 2, 25 | syl2anc 693 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 × {(0g‘𝑅)}) = (0g‘𝐹)) |
27 | 19 | fveq2d 6195 | . . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘𝑇))) |
28 | 27 | sneqd 4189 | . . . . . . . . . 10 ⊢ (𝜑 → {(0g‘𝑅)} = {(0g‘(Scalar‘𝑇))}) |
29 | 28 | xpeq2d 5139 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 × {(0g‘𝑅)}) = (𝐼 × {(0g‘(Scalar‘𝑇))})) |
30 | 26, 29 | eqtr3d 2658 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝐹) = (𝐼 × {(0g‘(Scalar‘𝑇))})) |
31 | 30 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (0g‘𝐹) = (𝐼 × {(0g‘(Scalar‘𝑇))})) |
32 | 31 | eqeq2d 2632 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 = (0g‘𝐹) ↔ 𝑦 = (𝐼 × {(0g‘(Scalar‘𝑇))}))) |
33 | 18, 32 | imbi12d 334 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (((𝐸‘𝑦) = (0g‘𝑇) → 𝑦 = (0g‘𝐹)) ↔ ((𝑇 Σg (𝑦 ∘𝑓 · 𝐴)) = (0g‘𝑇) → 𝑦 = (𝐼 × {(0g‘(Scalar‘𝑇))})))) |
34 | 33 | ralbidva 2985 | . . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝐸‘𝑦) = (0g‘𝑇) → 𝑦 = (0g‘𝐹)) ↔ ∀𝑦 ∈ 𝐵 ((𝑇 Σg (𝑦 ∘𝑓 · 𝐴)) = (0g‘𝑇) → 𝑦 = (𝐼 × {(0g‘(Scalar‘𝑇))})))) |
35 | 19 | eqcomd 2628 | . . . . . . . . 9 ⊢ (𝜑 → (Scalar‘𝑇) = 𝑅) |
36 | 35 | oveq1d 6665 | . . . . . . . 8 ⊢ (𝜑 → ((Scalar‘𝑇) freeLMod 𝐼) = (𝑅 freeLMod 𝐼)) |
37 | 36, 23 | syl6eqr 2674 | . . . . . . 7 ⊢ (𝜑 → ((Scalar‘𝑇) freeLMod 𝐼) = 𝐹) |
38 | 37 | fveq2d 6195 | . . . . . 6 ⊢ (𝜑 → (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = (Base‘𝐹)) |
39 | islindf5.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐹) | |
40 | 38, 39 | syl6eqr 2674 | . . . . 5 ⊢ (𝜑 → (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = 𝐵) |
41 | 40 | raleqdv 3144 | . . . 4 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘((Scalar‘𝑇) freeLMod 𝐼))((𝑇 Σg (𝑦 ∘𝑓 · 𝐴)) = (0g‘𝑇) → 𝑦 = (𝐼 × {(0g‘(Scalar‘𝑇))})) ↔ ∀𝑦 ∈ 𝐵 ((𝑇 Σg (𝑦 ∘𝑓 · 𝐴)) = (0g‘𝑇) → 𝑦 = (𝐼 × {(0g‘(Scalar‘𝑇))})))) |
42 | 34, 41 | bitr4d 271 | . . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝐸‘𝑦) = (0g‘𝑇) → 𝑦 = (0g‘𝐹)) ↔ ∀𝑦 ∈ (Base‘((Scalar‘𝑇) freeLMod 𝐼))((𝑇 Σg (𝑦 ∘𝑓 · 𝐴)) = (0g‘𝑇) → 𝑦 = (𝐼 × {(0g‘(Scalar‘𝑇))})))) |
43 | 11, 42 | bitr4d 271 | . 2 ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ ∀𝑦 ∈ 𝐵 ((𝐸‘𝑦) = (0g‘𝑇) → 𝑦 = (0g‘𝐹)))) |
44 | 23, 39, 4, 6, 14, 1, 2, 19, 3 | frlmup1 20137 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
45 | lmghm 19031 | . . 3 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸 ∈ (𝐹 GrpHom 𝑇)) | |
46 | eqid 2622 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
47 | 39, 4, 46, 7 | ghmf1 17689 | . . 3 ⊢ (𝐸 ∈ (𝐹 GrpHom 𝑇) → (𝐸:𝐵–1-1→𝐶 ↔ ∀𝑦 ∈ 𝐵 ((𝐸‘𝑦) = (0g‘𝑇) → 𝑦 = (0g‘𝐹)))) |
48 | 44, 45, 47 | 3syl 18 | . 2 ⊢ (𝜑 → (𝐸:𝐵–1-1→𝐶 ↔ ∀𝑦 ∈ 𝐵 ((𝐸‘𝑦) = (0g‘𝑇) → 𝑦 = (0g‘𝐹)))) |
49 | 43, 48 | bitr4d 271 | 1 ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ⟶wf 5884 –1-1→wf1 5885 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 Basecbs 15857 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 Σg cgsu 16101 GrpHom cghm 17657 Ringcrg 18547 LModclmod 18863 LMHom clmhm 19019 freeLMod cfrlm 20090 LIndF clindf 20143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lmhm 19022 df-lbs 19075 df-sra 19172 df-rgmod 19173 df-nzr 19258 df-dsmm 20076 df-frlm 20091 df-uvc 20122 df-lindf 20145 |
This theorem is referenced by: indlcim 20179 |
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