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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 2737 | . . . . 5 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
| 6 | islindf5.v | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑇) | |
| 7 | eqid 2737 | . . . . 5 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
| 8 | eqid 2737 | . . . . 5 ⊢ (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑇)) | |
| 9 | eqid 2737 | . . . . 5 ⊢ (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = (Base‘((Scalar‘𝑇) freeLMod 𝐼)) | |
| 10 | 4, 5, 6, 7, 8, 9 | islindf4 21858 | . . . 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 1373 | . . 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 7438 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∘f · 𝐴) = (𝑦 ∘f · 𝐴)) | |
| 13 | 12 | oveq2d 7447 | . . . . . . . . 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 7464 | . . . . . . . . 9 ⊢ (𝑇 Σg (𝑦 ∘f · 𝐴)) ∈ V | |
| 16 | 13, 14, 15 | fvmpt 7016 | . . . . . . . 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 2739 | . . . . . 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 20866 | . . . . . . . . . . . 12 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring) |
| 21 | 1, 20 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring) |
| 22 | 19, 21 | eqeltrd 2841 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring) |
| 23 | islindf5.f | . . . . . . . . . . 11 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
| 24 | eqid 2737 | . . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 25 | 23, 24 | frlm0 21774 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → (𝐼 × {(0g‘𝑅)}) = (0g‘𝐹)) |
| 26 | 22, 2, 25 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 × {(0g‘𝑅)}) = (0g‘𝐹)) |
| 27 | 19 | fveq2d 6910 | . . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘𝑇))) |
| 28 | 27 | sneqd 4638 | . . . . . . . . . 10 ⊢ (𝜑 → {(0g‘𝑅)} = {(0g‘(Scalar‘𝑇))}) |
| 29 | 28 | xpeq2d 5715 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 × {(0g‘𝑅)}) = (𝐼 × {(0g‘(Scalar‘𝑇))})) |
| 30 | 26, 29 | eqtr3d 2779 | . . . . . . . 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 2748 | . . . . . 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 3176 | . . . 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 2743 | . . . . . . . . 9 ⊢ (𝜑 → (Scalar‘𝑇) = 𝑅) |
| 36 | 35 | oveq1d 7446 | . . . . . . . 8 ⊢ (𝜑 → ((Scalar‘𝑇) freeLMod 𝐼) = (𝑅 freeLMod 𝐼)) |
| 37 | 36, 23 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝜑 → ((Scalar‘𝑇) freeLMod 𝐼) = 𝐹) |
| 38 | 37 | fveq2d 6910 | . . . . . 6 ⊢ (𝜑 → (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = (Base‘𝐹)) |
| 39 | islindf5.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐹) | |
| 40 | 38, 39 | eqtr4di 2795 | . . . . 5 ⊢ (𝜑 → (Base‘((Scalar‘𝑇) freeLMod 𝐼)) = 𝐵) |
| 41 | 40 | raleqdv 3326 | . . . 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 21818 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
| 45 | lmghm 21030 | . . 3 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸 ∈ (𝐹 GrpHom 𝑇)) | |
| 46 | eqid 2737 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
| 47 | 39, 4, 46, 7 | ghmf1 19264 | . . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ⟶wf 6557 –1-1→wf1 6558 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 Σg cgsu 17485 GrpHom cghm 19230 Ringcrg 20230 LModclmod 20858 LMHom clmhm 21018 freeLMod cfrlm 21766 LIndF clindf 21824 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-nzr 20513 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lmhm 21021 df-lbs 21074 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 df-uvc 21803 df-lindf 21826 |
| This theorem is referenced by: indlcim 21860 |
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