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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdindp0 | Structured version Visualization version GIF version | ||
| Description: Vector independence lemma. (Contributed by NM, 29-Apr-2015.) |
| Ref | Expression |
|---|---|
| mapdindp1.v | ⊢ 𝑉 = (Base‘𝑊) |
| mapdindp1.p | ⊢ + = (+g‘𝑊) |
| mapdindp1.o | ⊢ 0 = (0g‘𝑊) |
| mapdindp1.n | ⊢ 𝑁 = (LSpan‘𝑊) |
| mapdindp1.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
| mapdindp1.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
| mapdindp1.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
| mapdindp1.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
| mapdindp1.W | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
| mapdindp1.e | ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
| mapdindp1.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
| mapdindp1.f | ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
| mapdindp1.yz | ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 ) |
| Ref | Expression |
|---|---|
| mapdindp0 | ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
| 2 | mapdindp1.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 3 | mapdindp1.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
| 4 | lveclmod 21105 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
| 5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
| 6 | mapdindp1.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
| 7 | 6 | eldifad 3963 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
| 8 | mapdindp1.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
| 9 | 8, 1, 2 | lspsncl 20975 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
| 10 | 5, 7, 9 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
| 11 | mapdindp1.e | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
| 12 | 11, 10 | eqeltrrd 2842 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) |
| 13 | eqid 2737 | . . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
| 14 | 1, 13 | lsmcl 21082 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑊)) |
| 15 | 5, 10, 12, 14 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑊)) |
| 16 | 1 | lsssssubg 20956 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
| 17 | 5, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
| 18 | 17, 10 | sseldd 3984 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
| 19 | 11, 18 | eqeltrrd 2842 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) |
| 20 | 8, 2 | lspsnid 20991 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌})) |
| 21 | 5, 7, 20 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌})) |
| 22 | mapdindp1.z | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
| 23 | 22 | eldifad 3963 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
| 24 | 8, 2 | lspsnid 20991 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ (𝑁‘{𝑍})) |
| 25 | 5, 23, 24 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑍})) |
| 26 | mapdindp1.p | . . . . . 6 ⊢ + = (+g‘𝑊) | |
| 27 | 26, 13 | lsmelvali 19668 | . . . . 5 ⊢ ((((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) ∧ (𝑌 ∈ (𝑁‘{𝑌}) ∧ 𝑍 ∈ (𝑁‘{𝑍}))) → (𝑌 + 𝑍) ∈ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍}))) |
| 28 | 18, 19, 21, 25, 27 | syl22anc 839 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍}))) |
| 29 | 1, 2, 5, 15, 28 | ellspsn5 20994 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍}))) |
| 30 | 11 | oveq2d 7447 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍}))) |
| 31 | 13 | lsmidm 19681 | . . . . 5 ⊢ ((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑌})) |
| 32 | 18, 31 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑌})) |
| 33 | 30, 32 | eqtr3d 2779 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) = (𝑁‘{𝑌})) |
| 34 | 29, 33 | sseqtrd 4020 | . 2 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌})) |
| 35 | mapdindp1.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
| 36 | 8, 26 | lmodvacl 20873 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉) |
| 37 | 5, 7, 23, 36 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉) |
| 38 | mapdindp1.yz | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 ) | |
| 39 | eldifsn 4786 | . . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 )) | |
| 40 | 37, 38, 39 | sylanbrc 583 | . . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 })) |
| 41 | 8, 35, 2, 3, 40, 7 | lspsncmp 21118 | . 2 ⊢ (𝜑 → ((𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))) |
| 42 | 34, 41 | mpbid 232 | 1 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 {cpr 4628 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 SubGrpcsubg 19138 LSSumclsm 19652 LModclmod 20858 LSubSpclss 20929 LSpanclspn 20969 LVecclvec 21101 |
| 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-op 4633 df-uni 4908 df-int 4947 df-iun 4993 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-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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 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-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 |
| This theorem is referenced by: mapdindp1 41722 mapdindp2 41723 |
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