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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 2824 | . . . 4 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
5 | lvecindp2.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lveclmod 19881 | . . . . . 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 3951 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
10 | lvecindp2.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
11 | lvecindp2.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊) | |
12 | 10, 11 | lspsnsubg 19755 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
13 | 7, 9, 12 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
14 | lvecindp2.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
15 | 14 | eldifad 3951 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
16 | 10, 11 | lspsnsubg 19755 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
17 | 7, 15, 16 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
18 | lvecindp2.q | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
19 | 10, 3, 11, 5, 9, 15, 18 | lspdisj2 19902 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
20 | lmodabl 19684 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
21 | 7, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
22 | 4, 21, 13, 17 | ablcntzd 18980 | . . . 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 19776 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) |
28 | lvecindp2.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
29 | 10, 23, 24, 25, 11, 7, 28, 9 | lspsneli 19776 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝑋) ∈ (𝑁‘{𝑋})) |
30 | lvecindp2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
31 | 10, 23, 24, 25, 11, 7, 30, 15 | lspsneli 19776 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑌) ∈ (𝑁‘{𝑌})) |
32 | lvecindp2.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝐾) | |
33 | 10, 23, 24, 25, 11, 7, 32, 15 | lspsneli 19776 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝑌) ∈ (𝑁‘{𝑌})) |
34 | 2, 3, 4, 13, 17, 19, 22, 27, 29, 31, 33 | subgdisjb 18822 | . . 3 ⊢ (𝜑 → (((𝐴 · 𝑋) + (𝐵 · 𝑌)) = ((𝐶 · 𝑋) + (𝐷 · 𝑌)) ↔ ((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌)))) |
35 | 1, 34 | mpbid 234 | . 2 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐶 · 𝑋) ∧ (𝐵 · 𝑌) = (𝐷 · 𝑌))) |
36 | eldifsni 4725 | . . . . 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 19887 | . . 3 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐶 · 𝑋) ↔ 𝐴 = 𝐶)) |
39 | eldifsni 4725 | . . . . 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 19887 | . . 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 234 | 1 ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∖ cdif 3936 {csn 4570 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 +gcplusg 16568 Scalarcsca 16571 ·𝑠 cvsca 16572 0gc0g 16716 SubGrpcsubg 18276 Cntzccntz 18448 Abelcabl 18910 LModclmod 19637 LSpanclspn 19746 LVecclvec 19877 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-drng 19507 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878 |
This theorem is referenced by: mapdpglem30 38842 baerlem3lem1 38847 baerlem5alem1 38848 hdmap14lem9 39016 |
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