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Mirrors > Home > MPE Home > Th. List > lvecindp2 | Structured version Visualization version GIF version |
Description: Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.) |
Ref | Expression |
---|---|
lvecindp2.v | ⊢ 𝑉 = (Base‘𝑊) |
lvecindp2.p | ⊢ + = (+g‘𝑊) |
lvecindp2.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lvecindp2.k | ⊢ 𝐾 = (Base‘𝐹) |
lvecindp2.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lvecindp2.o | ⊢ 0 = (0g‘𝑊) |
lvecindp2.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lvecindp2.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lvecindp2.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lvecindp2.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
lvecindp2.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
lvecindp2.b | ⊢ (𝜑 → 𝐵 ∈ 𝐾) |
lvecindp2.c | ⊢ (𝜑 → 𝐶 ∈ 𝐾) |
lvecindp2.d | ⊢ (𝜑 → 𝐷 ∈ 𝐾) |
lvecindp2.q | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
lvecindp2.e | ⊢ (𝜑 → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = ((𝐶 · 𝑋) + (𝐷 · 𝑌))) |
Ref | Expression |
---|---|
lvecindp2 | ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecindp2.e | . . 3 ⊢ (𝜑 → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = ((𝐶 · 𝑋) + (𝐷 · 𝑌))) | |
2 | lvecindp2.p | . . . 4 ⊢ + = (+g‘𝑊) | |
3 | lvecindp2.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
4 | eqid 2622 | . . . 4 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
5 | lvecindp2.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lveclmod 19106 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
8 | lvecindp2.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
9 | 8 | eldifad 3586 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
10 | lvecindp2.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
11 | lvecindp2.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊) | |
12 | 10, 11 | lspsnsubg 18980 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
13 | 7, 9, 12 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
14 | lvecindp2.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
15 | 14 | eldifad 3586 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
16 | 10, 11 | lspsnsubg 18980 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
17 | 7, 15, 16 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
18 | lvecindp2.q | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
19 | 10, 3, 11, 5, 9, 15, 18 | lspdisj2 19127 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
20 | lmodabl 18910 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
21 | 7, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
22 | 4, 21, 13, 17 | ablcntzd 18260 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ ((Cntz‘𝑊)‘(𝑁‘{𝑌}))) |
23 | lvecindp2.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
24 | lvecindp2.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
25 | lvecindp2.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
26 | lvecindp2.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
27 | 10, 23, 24, 25, 11, 7, 26, 9 | lspsneli 19001 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) |
28 | lvecindp2.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
29 | 10, 23, 24, 25, 11, 7, 28, 9 | lspsneli 19001 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝑋) ∈ (𝑁‘{𝑋})) |
30 | lvecindp2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
31 | 10, 23, 24, 25, 11, 7, 30, 15 | lspsneli 19001 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑌) ∈ (𝑁‘{𝑌})) |
32 | lvecindp2.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝐾) | |
33 | 10, 23, 24, 25, 11, 7, 32, 15 | lspsneli 19001 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝑌) ∈ (𝑁‘{𝑌})) |
34 | 2, 3, 4, 13, 17, 19, 22, 27, 29, 31, 33 | subgdisjb 18106 | . . 3 ⊢ (𝜑 → (((𝐴 · 𝑋) + (𝐵 · 𝑌)) = ((𝐶 · 𝑋) + (𝐷 · 𝑌)) ↔ ((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌)))) |
35 | 1, 34 | mpbid 222 | . 2 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌))) |
36 | eldifsni 4320 | . . . . 5 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
37 | 8, 36 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
38 | 10, 23, 24, 25, 3, 5, 26, 28, 9, 37 | lvecvscan2 19112 | . . 3 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐶 · 𝑋) ↔ 𝐴 = 𝐶)) |
39 | eldifsni 4320 | . . . . 5 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
40 | 14, 39 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 0 ) |
41 | 10, 23, 24, 25, 3, 5, 30, 32, 15, 40 | lvecvscan2 19112 | . . 3 ⊢ (𝜑 → ((𝐵 · 𝑌) = (𝐷 · 𝑌) ↔ 𝐵 = 𝐷)) |
42 | 38, 41 | anbi12d 747 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
43 | 35, 42 | mpbid 222 | 1 ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 {csn 4177 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 SubGrpcsubg 17588 Cntzccntz 17748 Abelcabl 18194 LModclmod 18863 LSpanclspn 18971 LVecclvec 19102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 |
This theorem is referenced by: mapdpglem30 36991 baerlem3lem1 36996 baerlem5alem1 36997 hdmap14lem9 37168 |
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