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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdindp3 | Structured version Visualization version GIF version |
Description: Vector independence lemma. (Contributed by NM, 29-Apr-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 |
---|---|
mapdindp3 | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdindp1.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | lveclmod 19612 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | mapdindp1.W | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
5 | 4 | eldifad 3835 | . . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
6 | mapdindp1.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
7 | 6 | eldifad 3835 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
8 | mapdindp1.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | mapdindp1.p | . . . . 5 ⊢ + = (+g‘𝑊) | |
10 | mapdindp1.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
11 | 8, 9, 10 | lspvadd 19602 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑤 + 𝑌)}) ⊆ (𝑁‘{𝑤, 𝑌})) |
12 | 3, 5, 7, 11 | syl3anc 1351 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑤 + 𝑌)}) ⊆ (𝑁‘{𝑤, 𝑌})) |
13 | mapdindp1.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
14 | mapdindp1.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
15 | mapdindp1.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
16 | mapdindp1.f | . . . . 5 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
17 | 8, 13, 10, 1, 14, 7, 5, 15, 16 | lspindp1 19639 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌}))) |
18 | 17 | simprd 488 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌})) |
19 | 12, 18 | ssneldd 3855 | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{(𝑤 + 𝑌)})) |
20 | 14 | eldifad 3835 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
21 | 8, 10 | lspsnid 19499 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
22 | 3, 20, 21 | syl2anc 576 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋})) |
23 | eleq2 2848 | . . . 4 ⊢ ((𝑁‘{𝑋}) = (𝑁‘{(𝑤 + 𝑌)}) → (𝑋 ∈ (𝑁‘{𝑋}) ↔ 𝑋 ∈ (𝑁‘{(𝑤 + 𝑌)}))) | |
24 | 22, 23 | syl5ibcom 237 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{(𝑤 + 𝑌)}) → 𝑋 ∈ (𝑁‘{(𝑤 + 𝑌)}))) |
25 | 24 | necon3bd 2975 | . 2 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{(𝑤 + 𝑌)}) → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)}))) |
26 | 19, 25 | mpd 15 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 ∖ cdif 3820 ⊆ wss 3823 {csn 4435 {cpr 4437 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 +gcplusg 16419 0gc0g 16567 LModclmod 19368 LSpanclspn 19477 LVecclvec 19608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-tpos 7693 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-0g 16569 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-subg 18072 df-cntz 18230 df-lsm 18534 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-oppr 19108 df-dvdsr 19126 df-unit 19127 df-invr 19157 df-drng 19239 df-lmod 19370 df-lss 19438 df-lsp 19478 df-lvec 19609 |
This theorem is referenced by: mapdh6eN 38350 hdmap1l6e 38424 |
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