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Mirrors > Home > MPE Home > Th. List > lmodindp1 | Structured version Visualization version GIF version |
Description: Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.) |
Ref | Expression |
---|---|
lmodindp1.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodindp1.p | ⊢ + = (+g‘𝑊) |
lmodindp1.o | ⊢ 0 = (0g‘𝑊) |
lmodindp1.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lmodindp1.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lmodindp1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lmodindp1.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lmodindp1.q | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
lmodindp1 | ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodindp1.q | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
2 | lmodindp1.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | lmodindp1.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | lmodindp1.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊) | |
5 | eqid 2818 | . . . . . . . . 9 ⊢ (invg‘𝑊) = (invg‘𝑊) | |
6 | lmodindp1.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊) | |
7 | 4, 5, 6 | lspsnneg 19707 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑋})) |
8 | 2, 3, 7 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑋})) |
9 | 8 | eqcomd 2824 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{((invg‘𝑊)‘𝑋)})) |
10 | 9 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{((invg‘𝑊)‘𝑋)})) |
11 | lmodgrp 19570 | . . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
12 | 2, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Grp) |
13 | lmodindp1.y | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
14 | lmodindp1.p | . . . . . . . . . 10 ⊢ + = (+g‘𝑊) | |
15 | lmodindp1.o | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑊) | |
16 | 4, 14, 15, 5 | grpinvid1 18092 | . . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (((invg‘𝑊)‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 )) |
17 | 12, 3, 13, 16 | syl3anc 1363 | . . . . . . . 8 ⊢ (𝜑 → (((invg‘𝑊)‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 )) |
18 | 17 | biimpar 478 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → ((invg‘𝑊)‘𝑋) = 𝑌) |
19 | 18 | sneqd 4569 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → {((invg‘𝑊)‘𝑋)} = {𝑌}) |
20 | 19 | fveq2d 6667 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑌})) |
21 | 10, 20 | eqtrd 2853 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
22 | 21 | ex 413 | . . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
23 | 22 | necon3d 3034 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) → (𝑋 + 𝑌) ≠ 0 )) |
24 | 1, 23 | mpd 15 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {csn 4557 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Grpcgrp 18041 invgcminusg 18042 LModclmod 19563 LSpanclspn 19672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mgp 19169 df-ur 19181 df-ring 19228 df-lmod 19565 df-lss 19633 df-lsp 19673 |
This theorem is referenced by: lcfrlem17 38575 mapdh6aN 38751 mapdh6eN 38756 hdmap1l6a 38825 hdmap1l6e 38830 hdmaprnlem3eN 38874 |
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