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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 2610 | . . . . . . . . 9 ⊢ (invg‘𝑊) = (invg‘𝑊) | |
6 | lmodindp1.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊) | |
7 | 4, 5, 6 | lspsnneg 18827 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑋})) |
8 | 2, 3, 7 | syl2anc 691 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑋})) |
9 | 8 | eqcomd 2616 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{((invg‘𝑊)‘𝑋)})) |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{((invg‘𝑊)‘𝑋)})) |
11 | lmodgrp 18693 | . . . . . . . . . 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 17293 | . . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (((invg‘𝑊)‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 )) |
17 | 12, 3, 13, 16 | syl3anc 1318 | . . . . . . . 8 ⊢ (𝜑 → (((invg‘𝑊)‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 )) |
18 | 17 | biimpar 501 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → ((invg‘𝑊)‘𝑋) = 𝑌) |
19 | 18 | sneqd 4137 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → {((invg‘𝑊)‘𝑋)} = {𝑌}) |
20 | 19 | fveq2d 6107 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑌})) |
21 | 10, 20 | eqtrd 2644 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
22 | 21 | ex 449 | . . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
23 | 22 | necon3d 2803 | . 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {csn 4125 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 invgcminusg 17246 LModclmod 18686 LSpanclspn 18792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-lss 18754 df-lsp 18793 |
This theorem is referenced by: lcfrlem17 35866 mapdh6aN 36042 mapdh6eN 36047 hdmap1l6a 36117 hdmap1l6e 36122 hdmaprnlem3eN 36168 |
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