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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 2821 | . . . . 5 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
6 | islindf5.v | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑇) | |
7 | eqid 2821 | . . . . 5 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
8 | eqid 2821 | . . . . 5 ⊢ (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑇)) | |
9 | eqid 2821 | . . . . 5 ⊢ (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = (Base‘((Scalar‘𝑇) freeLMod 𝐼)) | |
10 | 4, 5, 6, 7, 8, 9 | islindf4 20982 | . . . 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 1367 | . . 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 7163 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∘f · 𝐴) = (𝑦 ∘f · 𝐴)) | |
13 | 12 | oveq2d 7172 | . . . . . . . . 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 7189 | . . . . . . . . 9 ⊢ (𝑇 Σg (𝑦 ∘f · 𝐴)) ∈ V | |
16 | 13, 14, 15 | fvmpt 6768 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (𝐸‘𝑦) = (𝑇 Σg (𝑦 ∘f · 𝐴))) |
17 | 16 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐸‘𝑦) = (𝑇 Σg (𝑦 ∘f · 𝐴))) |
18 | 17 | eqeq1d 2823 | . . . . . 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 19642 | . . . . . . . . . . . 12 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring) |
21 | 1, 20 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring) |
22 | 19, 21 | eqeltrd 2913 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring) |
23 | islindf5.f | . . . . . . . . . . 11 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
24 | eqid 2821 | . . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
25 | 23, 24 | frlm0 20898 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → (𝐼 × {(0g‘𝑅)}) = (0g‘𝐹)) |
26 | 22, 2, 25 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 × {(0g‘𝑅)}) = (0g‘𝐹)) |
27 | 19 | fveq2d 6674 | . . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘𝑇))) |
28 | 27 | sneqd 4579 | . . . . . . . . . 10 ⊢ (𝜑 → {(0g‘𝑅)} = {(0g‘(Scalar‘𝑇))}) |
29 | 28 | xpeq2d 5585 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 × {(0g‘𝑅)}) = (𝐼 × {(0g‘(Scalar‘𝑇))})) |
30 | 26, 29 | eqtr3d 2858 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝐹) = (𝐼 × {(0g‘(Scalar‘𝑇))})) |
31 | 30 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (0g‘𝐹) = (𝐼 × {(0g‘(Scalar‘𝑇))})) |
32 | 31 | eqeq2d 2832 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 = (0g‘𝐹) ↔ 𝑦 = (𝐼 × {(0g‘(Scalar‘𝑇))}))) |
33 | 18, 32 | imbi12d 347 | . . . . 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 3196 | . . . 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 2827 | . . . . . . . . 9 ⊢ (𝜑 → (Scalar‘𝑇) = 𝑅) |
36 | 35 | oveq1d 7171 | . . . . . . . 8 ⊢ (𝜑 → ((Scalar‘𝑇) freeLMod 𝐼) = (𝑅 freeLMod 𝐼)) |
37 | 36, 23 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝜑 → ((Scalar‘𝑇) freeLMod 𝐼) = 𝐹) |
38 | 37 | fveq2d 6674 | . . . . . 6 ⊢ (𝜑 → (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = (Base‘𝐹)) |
39 | islindf5.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐹) | |
40 | 38, 39 | syl6eqr 2874 | . . . . 5 ⊢ (𝜑 → (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = 𝐵) |
41 | 40 | raleqdv 3415 | . . . 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 284 | . . 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 284 | . 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 20942 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
45 | lmghm 19803 | . . 3 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸 ∈ (𝐹 GrpHom 𝑇)) | |
46 | eqid 2821 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
47 | 39, 4, 46, 7 | ghmf1 18387 | . . 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 284 | 1 ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {csn 4567 class class class wbr 5066 ↦ cmpt 5146 × cxp 5553 ⟶wf 6351 –1-1→wf1 6352 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 Σg cgsu 16714 GrpHom cghm 18355 Ringcrg 19297 LModclmod 19634 LMHom clmhm 19791 freeLMod cfrlm 20890 LIndF clindf 20948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-gsum 16716 df-prds 16721 df-pws 16723 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lmhm 19794 df-lbs 19847 df-sra 19944 df-rgmod 19945 df-nzr 20031 df-dsmm 20876 df-frlm 20891 df-uvc 20927 df-lindf 20950 |
This theorem is referenced by: indlcim 20984 |
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