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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 2733 | . . . 4 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
5 | lvecindp2.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lveclmod 20710 | . . . . . 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 3960 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
10 | lvecindp2.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
11 | lvecindp2.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊) | |
12 | 10, 11 | lspsnsubg 20584 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
13 | 7, 9, 12 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
14 | lvecindp2.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
15 | 14 | eldifad 3960 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
16 | 10, 11 | lspsnsubg 20584 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
17 | 7, 15, 16 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
18 | lvecindp2.q | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
19 | 10, 3, 11, 5, 9, 15, 18 | lspdisj2 20733 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
20 | lmodabl 20512 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
21 | 7, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
22 | 4, 21, 13, 17 | ablcntzd 19720 | . . . 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 20605 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) |
28 | lvecindp2.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
29 | 10, 23, 24, 25, 11, 7, 28, 9 | lspsneli 20605 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝑋) ∈ (𝑁‘{𝑋})) |
30 | lvecindp2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
31 | 10, 23, 24, 25, 11, 7, 30, 15 | lspsneli 20605 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑌) ∈ (𝑁‘{𝑌})) |
32 | lvecindp2.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝐾) | |
33 | 10, 23, 24, 25, 11, 7, 32, 15 | lspsneli 20605 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝑌) ∈ (𝑁‘{𝑌})) |
34 | 2, 3, 4, 13, 17, 19, 22, 27, 29, 31, 33 | subgdisjb 19556 | . . 3 ⊢ (𝜑 → (((𝐴 · 𝑋) + (𝐵 · 𝑌)) = ((𝐶 · 𝑋) + (𝐷 · 𝑌)) ↔ ((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌)))) |
35 | 1, 34 | mpbid 231 | . 2 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌))) |
36 | eldifsni 4793 | . . . . 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 20718 | . . 3 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐶 · 𝑋) ↔ 𝐴 = 𝐶)) |
39 | eldifsni 4793 | . . . . 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 20718 | . . 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 231 | 1 ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3945 {csn 4628 ‘cfv 6541 (class class class)co 7406 Basecbs 17141 +gcplusg 17194 Scalarcsca 17197 ·𝑠 cvsca 17198 0gc0g 17382 SubGrpcsubg 18995 Cntzccntz 19174 Abelcabl 19644 LModclmod 20464 LSpanclspn 20575 LVecclvec 20706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-0g 17384 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-grp 18819 df-minusg 18820 df-sbg 18821 df-subg 18998 df-cntz 19176 df-cmn 19645 df-abl 19646 df-mgp 19983 df-ur 20000 df-ring 20052 df-oppr 20143 df-dvdsr 20164 df-unit 20165 df-invr 20195 df-drng 20310 df-lmod 20466 df-lss 20536 df-lsp 20576 df-lvec 20707 |
This theorem is referenced by: mapdpglem30 40562 baerlem3lem1 40567 baerlem5alem1 40568 hdmap14lem9 40736 |
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