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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 2610 | . . . . 5 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
6 | islindf5.v | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑇) | |
7 | eqid 2610 | . . . . 5 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
8 | eqid 2610 | . . . . 5 ⊢ (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑇)) | |
9 | eqid 2610 | . . . . 5 ⊢ (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = (Base‘((Scalar‘𝑇) freeLMod 𝐼)) | |
10 | 4, 5, 6, 7, 8, 9 | islindf4 19996 | . . . 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 1318 | . . 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 6556 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∘𝑓 · 𝐴) = (𝑦 ∘𝑓 · 𝐴)) | |
13 | 12 | oveq2d 6565 | . . . . . . . . 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 6577 | . . . . . . . . 9 ⊢ (𝑇 Σg (𝑦 ∘𝑓 · 𝐴)) ∈ V | |
16 | 13, 14, 15 | fvmpt 6191 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (𝐸‘𝑦) = (𝑇 Σg (𝑦 ∘𝑓 · 𝐴))) |
17 | 16 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐸‘𝑦) = (𝑇 Σg (𝑦 ∘𝑓 · 𝐴))) |
18 | 17 | eqeq1d 2612 | . . . . . 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 18694 | . . . . . . . . . . . 12 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring) |
21 | 1, 20 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring) |
22 | 19, 21 | eqeltrd 2688 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring) |
23 | islindf5.f | . . . . . . . . . . 11 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
24 | eqid 2610 | . . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
25 | 23, 24 | frlm0 19917 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → (𝐼 × {(0g‘𝑅)}) = (0g‘𝐹)) |
26 | 22, 2, 25 | syl2anc 691 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 × {(0g‘𝑅)}) = (0g‘𝐹)) |
27 | 19 | fveq2d 6107 | . . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘𝑇))) |
28 | 27 | sneqd 4137 | . . . . . . . . . 10 ⊢ (𝜑 → {(0g‘𝑅)} = {(0g‘(Scalar‘𝑇))}) |
29 | 28 | xpeq2d 5063 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 × {(0g‘𝑅)}) = (𝐼 × {(0g‘(Scalar‘𝑇))})) |
30 | 26, 29 | eqtr3d 2646 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝐹) = (𝐼 × {(0g‘(Scalar‘𝑇))})) |
31 | 30 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (0g‘𝐹) = (𝐼 × {(0g‘(Scalar‘𝑇))})) |
32 | 31 | eqeq2d 2620 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 = (0g‘𝐹) ↔ 𝑦 = (𝐼 × {(0g‘(Scalar‘𝑇))}))) |
33 | 18, 32 | imbi12d 333 | . . . . 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 2968 | . . . 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 2616 | . . . . . . . . 9 ⊢ (𝜑 → (Scalar‘𝑇) = 𝑅) |
36 | 35 | oveq1d 6564 | . . . . . . . 8 ⊢ (𝜑 → ((Scalar‘𝑇) freeLMod 𝐼) = (𝑅 freeLMod 𝐼)) |
37 | 36, 23 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝜑 → ((Scalar‘𝑇) freeLMod 𝐼) = 𝐹) |
38 | 37 | fveq2d 6107 | . . . . . 6 ⊢ (𝜑 → (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = (Base‘𝐹)) |
39 | islindf5.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐹) | |
40 | 38, 39 | syl6eqr 2662 | . . . . 5 ⊢ (𝜑 → (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = 𝐵) |
41 | 40 | raleqdv 3121 | . . . 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 270 | . . 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 270 | . 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 19956 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
45 | lmghm 18852 | . . 3 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸 ∈ (𝐹 GrpHom 𝑇)) | |
46 | eqid 2610 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
47 | 39, 4, 46, 7 | ghmf1 17512 | . . 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 270 | 1 ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 Σg cgsu 15924 GrpHom cghm 17480 Ringcrg 18370 LModclmod 18686 LMHom clmhm 18840 freeLMod cfrlm 19909 LIndF clindf 19962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lmhm 18843 df-lbs 18896 df-sra 18993 df-rgmod 18994 df-nzr 19079 df-dsmm 19895 df-frlm 19910 df-uvc 19941 df-lindf 19964 |
This theorem is referenced by: indlcim 19998 |
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