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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 2735 | . . . 4 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
5 | lvecindp2.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lveclmod 21123 | . . . . . 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 3975 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
10 | lvecindp2.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
11 | lvecindp2.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊) | |
12 | 10, 11 | lspsnsubg 20996 | . . . . 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 3975 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
16 | 10, 11 | lspsnsubg 20996 | . . . . 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 21147 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
20 | lmodabl 20924 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
21 | 7, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
22 | 4, 21, 13, 17 | ablcntzd 19890 | . . . 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 | ellspsni 21017 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) |
28 | lvecindp2.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
29 | 10, 23, 24, 25, 11, 7, 28, 9 | ellspsni 21017 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝑋) ∈ (𝑁‘{𝑋})) |
30 | lvecindp2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
31 | 10, 23, 24, 25, 11, 7, 30, 15 | ellspsni 21017 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑌) ∈ (𝑁‘{𝑌})) |
32 | lvecindp2.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝐾) | |
33 | 10, 23, 24, 25, 11, 7, 32, 15 | ellspsni 21017 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝑌) ∈ (𝑁‘{𝑌})) |
34 | 2, 3, 4, 13, 17, 19, 22, 27, 29, 31, 33 | subgdisjb 19726 | . . 3 ⊢ (𝜑 → (((𝐴 · 𝑋) + (𝐵 · 𝑌)) = ((𝐶 · 𝑋) + (𝐷 · 𝑌)) ↔ ((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌)))) |
35 | 1, 34 | mpbid 232 | . 2 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌))) |
36 | eldifsni 4795 | . . . . 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 21132 | . . 3 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐶 · 𝑋) ↔ 𝐴 = 𝐶)) |
39 | eldifsni 4795 | . . . . 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 21132 | . . 3 ⊢ (𝜑 → ((𝐵 · 𝑌) = (𝐷 · 𝑌) ↔ 𝐵 = 𝐷)) |
42 | 38, 41 | anbi12d 632 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
43 | 35, 42 | mpbid 232 | 1 ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 SubGrpcsubg 19151 Cntzccntz 19346 Abelcabl 19814 LModclmod 20875 LSpanclspn 20987 LVecclvec 21119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 |
This theorem is referenced by: mapdpglem30 41685 baerlem3lem1 41690 baerlem5alem1 41691 hdmap14lem9 41859 |
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