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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 2798 | . . . . . . . . 9 ⊢ (invg‘𝑊) = (invg‘𝑊) | |
6 | lmodindp1.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊) | |
7 | 4, 5, 6 | lspsnneg 19771 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑋})) |
8 | 2, 3, 7 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑋})) |
9 | 8 | eqcomd 2804 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{((invg‘𝑊)‘𝑋)})) |
10 | 9 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{((invg‘𝑊)‘𝑋)})) |
11 | lmodgrp 19634 | . . . . . . . . . 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 18146 | . . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (((invg‘𝑊)‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 )) |
17 | 12, 3, 13, 16 | syl3anc 1368 | . . . . . . . 8 ⊢ (𝜑 → (((invg‘𝑊)‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 )) |
18 | 17 | biimpar 481 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → ((invg‘𝑊)‘𝑋) = 𝑌) |
19 | 18 | sneqd 4537 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → {((invg‘𝑊)‘𝑋)} = {𝑌}) |
20 | 19 | fveq2d 6649 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑌})) |
21 | 10, 20 | eqtrd 2833 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
22 | 21 | ex 416 | . . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
23 | 22 | necon3d 3008 | . 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {csn 4525 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Grpcgrp 18095 invgcminusg 18096 LModclmod 19627 LSpanclspn 19736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mgp 19233 df-ur 19245 df-ring 19292 df-lmod 19629 df-lss 19697 df-lsp 19737 |
This theorem is referenced by: lcfrlem17 38855 mapdh6aN 39031 mapdh6eN 39036 hdmap1l6a 39105 hdmap1l6e 39110 hdmaprnlem3eN 39154 |
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