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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 2793 | . . . 4 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
5 | lvecindp2.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lveclmod 19556 | . . . . . 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 3866 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
10 | lvecindp2.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
11 | lvecindp2.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊) | |
12 | 10, 11 | lspsnsubg 19430 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
13 | 7, 9, 12 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
14 | lvecindp2.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
15 | 14 | eldifad 3866 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
16 | 10, 11 | lspsnsubg 19430 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
17 | 7, 15, 16 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
18 | lvecindp2.q | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
19 | 10, 3, 11, 5, 9, 15, 18 | lspdisj2 19577 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
20 | lmodabl 19359 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
21 | 7, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
22 | 4, 21, 13, 17 | ablcntzd 18688 | . . . 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 19451 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) |
28 | lvecindp2.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
29 | 10, 23, 24, 25, 11, 7, 28, 9 | lspsneli 19451 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝑋) ∈ (𝑁‘{𝑋})) |
30 | lvecindp2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
31 | 10, 23, 24, 25, 11, 7, 30, 15 | lspsneli 19451 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑌) ∈ (𝑁‘{𝑌})) |
32 | lvecindp2.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝐾) | |
33 | 10, 23, 24, 25, 11, 7, 32, 15 | lspsneli 19451 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝑌) ∈ (𝑁‘{𝑌})) |
34 | 2, 3, 4, 13, 17, 19, 22, 27, 29, 31, 33 | subgdisjb 18534 | . . 3 ⊢ (𝜑 → (((𝐴 · 𝑋) + (𝐵 · 𝑌)) = ((𝐶 · 𝑋) + (𝐷 · 𝑌)) ↔ ((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌)))) |
35 | 1, 34 | mpbid 233 | . 2 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌))) |
36 | eldifsni 4623 | . . . . 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 19562 | . . 3 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐶 · 𝑋) ↔ 𝐴 = 𝐶)) |
39 | eldifsni 4623 | . . . . 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 19562 | . . 3 ⊢ (𝜑 → ((𝐵 · 𝑌) = (𝐷 · 𝑌) ↔ 𝐵 = 𝐷)) |
42 | 38, 41 | anbi12d 630 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
43 | 35, 42 | mpbid 233 | 1 ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ≠ wne 2982 ∖ cdif 3851 {csn 4466 ‘cfv 6217 (class class class)co 7007 Basecbs 16300 +gcplusg 16382 Scalarcsca 16385 ·𝑠 cvsca 16386 0gc0g 16530 SubGrpcsubg 18015 Cntzccntz 18174 Abelcabl 18622 LModclmod 19312 LSpanclspn 19421 LVecclvec 19552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-tpos 7734 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-3 11538 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-0g 16532 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-grp 17852 df-minusg 17853 df-sbg 17854 df-subg 18018 df-cntz 18176 df-cmn 18623 df-abl 18624 df-mgp 18918 df-ur 18930 df-ring 18977 df-oppr 19051 df-dvdsr 19069 df-unit 19070 df-invr 19100 df-drng 19182 df-lmod 19314 df-lss 19382 df-lsp 19422 df-lvec 19553 |
This theorem is referenced by: mapdpglem30 38319 baerlem3lem1 38324 baerlem5alem1 38325 hdmap14lem9 38493 |
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