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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 2731 | . . . . 5 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
6 | islindf5.v | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑇) | |
7 | eqid 2731 | . . . . 5 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
8 | eqid 2731 | . . . . 5 ⊢ (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑇)) | |
9 | eqid 2731 | . . . . 5 ⊢ (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = (Base‘((Scalar‘𝑇) freeLMod 𝐼)) | |
10 | 4, 5, 6, 7, 8, 9 | islindf4 21703 | . . . 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 1370 | . . 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 7419 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∘f · 𝐴) = (𝑦 ∘f · 𝐴)) | |
13 | 12 | oveq2d 7428 | . . . . . . . . 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 7445 | . . . . . . . . 9 ⊢ (𝑇 Σg (𝑦 ∘f · 𝐴)) ∈ V | |
16 | 13, 14, 15 | fvmpt 6998 | . . . . . . . 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 2733 | . . . . . 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 20710 | . . . . . . . . . . . 12 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring) |
21 | 1, 20 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring) |
22 | 19, 21 | eqeltrd 2832 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring) |
23 | islindf5.f | . . . . . . . . . . 11 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
24 | eqid 2731 | . . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
25 | 23, 24 | frlm0 21619 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → (𝐼 × {(0g‘𝑅)}) = (0g‘𝐹)) |
26 | 22, 2, 25 | syl2anc 583 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 × {(0g‘𝑅)}) = (0g‘𝐹)) |
27 | 19 | fveq2d 6895 | . . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘𝑇))) |
28 | 27 | sneqd 4640 | . . . . . . . . . 10 ⊢ (𝜑 → {(0g‘𝑅)} = {(0g‘(Scalar‘𝑇))}) |
29 | 28 | xpeq2d 5706 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 × {(0g‘𝑅)}) = (𝐼 × {(0g‘(Scalar‘𝑇))})) |
30 | 26, 29 | eqtr3d 2773 | . . . . . . . 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 2742 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 = (0g‘𝐹) ↔ 𝑦 = (𝐼 × {(0g‘(Scalar‘𝑇))}))) |
33 | 18, 32 | imbi12d 344 | . . . . 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 3174 | . . . 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 2737 | . . . . . . . . 9 ⊢ (𝜑 → (Scalar‘𝑇) = 𝑅) |
36 | 35 | oveq1d 7427 | . . . . . . . 8 ⊢ (𝜑 → ((Scalar‘𝑇) freeLMod 𝐼) = (𝑅 freeLMod 𝐼)) |
37 | 36, 23 | eqtr4di 2789 | . . . . . . 7 ⊢ (𝜑 → ((Scalar‘𝑇) freeLMod 𝐼) = 𝐹) |
38 | 37 | fveq2d 6895 | . . . . . 6 ⊢ (𝜑 → (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = (Base‘𝐹)) |
39 | islindf5.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐹) | |
40 | 38, 39 | eqtr4di 2789 | . . . . 5 ⊢ (𝜑 → (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = 𝐵) |
41 | 40 | raleqdv 3324 | . . . 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 282 | . . 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 282 | . 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 21663 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
45 | lmghm 20875 | . . 3 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸 ∈ (𝐹 GrpHom 𝑇)) | |
46 | eqid 2731 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
47 | 39, 4, 46, 7 | ghmf1 19167 | . . 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 282 | 1 ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ⟶wf 6539 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7412 ∘f cof 7672 Basecbs 17151 Scalarcsca 17207 ·𝑠 cvsca 17208 0gc0g 17392 Σg cgsu 17393 GrpHom cghm 19134 Ringcrg 20134 LModclmod 20702 LMHom clmhm 20863 freeLMod cfrlm 21611 LIndF clindf 21669 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-nzr 20411 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lmhm 20866 df-lbs 20919 df-sra 21019 df-rgmod 21020 df-dsmm 21597 df-frlm 21612 df-uvc 21648 df-lindf 21671 |
This theorem is referenced by: indlcim 21705 |
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