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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdindp1 | Structured version Visualization version GIF version |
Description: Vector independence lemma. (Contributed by NM, 1-May-2015.) |
Ref | Expression |
---|---|
mapdindp1.v | ⊢ 𝑉 = (Base‘𝑊) |
mapdindp1.p | ⊢ + = (+g‘𝑊) |
mapdindp1.o | ⊢ 0 = (0g‘𝑊) |
mapdindp1.n | ⊢ 𝑁 = (LSpan‘𝑊) |
mapdindp1.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
mapdindp1.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.W | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.e | ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
mapdindp1.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdindp1.f | ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
Ref | Expression |
---|---|
mapdindp1 | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdindp1.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
2 | eldifsni 4693 | . . . . . 6 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
4 | mapdindp1.w | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lveclmod 20115 | . . . . . . . . . 10 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
6 | 4, 5 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ LMod) |
7 | mapdindp1.o | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑊) | |
8 | mapdindp1.n | . . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑊) | |
9 | 7, 8 | lspsn0 20017 | . . . . . . . . 9 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
10 | 6, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{ 0 }) = { 0 }) |
11 | 10 | eqeq2d 2745 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{ 0 }) ↔ (𝑁‘{𝑋}) = { 0 })) |
12 | 1 | eldifad 3869 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
13 | mapdindp1.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊) | |
14 | 13, 7, 8 | lspsneq0 20021 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 )) |
15 | 6, 12, 14 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 )) |
16 | 11, 15 | bitrd 282 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{ 0 }) ↔ 𝑋 = 0 )) |
17 | 16 | necon3bid 2979 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{ 0 }) ↔ 𝑋 ≠ 0 )) |
18 | 3, 17 | mpbird 260 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{ 0 })) |
19 | 18 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{ 0 })) |
20 | sneq 4541 | . . . . 5 ⊢ ((𝑌 + 𝑍) = 0 → {(𝑌 + 𝑍)} = { 0 }) | |
21 | 20 | fveq2d 6710 | . . . 4 ⊢ ((𝑌 + 𝑍) = 0 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{ 0 })) |
22 | 21 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{ 0 })) |
23 | 19, 22 | neeqtrrd 3009 | . 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
24 | mapdindp1.ne | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
25 | 24 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
26 | mapdindp1.p | . . . 4 ⊢ + = (+g‘𝑊) | |
27 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑊 ∈ LVec) |
28 | 1 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
29 | mapdindp1.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
30 | 29 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
31 | mapdindp1.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
32 | 31 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑍 ∈ (𝑉 ∖ { 0 })) |
33 | mapdindp1.W | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
34 | 33 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑤 ∈ (𝑉 ∖ { 0 })) |
35 | mapdindp1.e | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
36 | 35 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
37 | mapdindp1.f | . . . . 5 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
38 | 37 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
39 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑌 + 𝑍) ≠ 0 ) | |
40 | 13, 26, 7, 8, 27, 28, 30, 32, 34, 36, 25, 38, 39 | mapdindp0 39427 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})) |
41 | 25, 40 | neeqtrrd 3009 | . 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
42 | 23, 41 | pm2.61dane 3022 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∖ cdif 3854 {csn 4531 {cpr 4533 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 +gcplusg 16767 0gc0g 16916 LModclmod 19871 LSpanclspn 19980 LVecclvec 20111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-tpos 7957 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-grp 18340 df-minusg 18341 df-sbg 18342 df-subg 18512 df-cntz 18683 df-lsm 18997 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-oppr 19613 df-dvdsr 19631 df-unit 19632 df-invr 19662 df-drng 19741 df-lmod 19873 df-lss 19941 df-lsp 19981 df-lvec 20112 |
This theorem is referenced by: mapdh6dN 39447 mapdh6hN 39451 hdmap1l6d 39521 hdmap1l6h 39525 |
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