lspindpi - Metamath Proof Explorer

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Theorem lspindpi 21415

Description: Partial independence property. (Contributed by NM, 23-Apr-2015.)

**Hypotheses**
Ref	Expression
lspindpi.v	⊢ 𝑉 = (Base‘𝑊)
lspindpi.n	⊢ 𝑁 = (LSpan‘𝑊)
lspindpi.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lspindpi.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lspindpi.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lspindpi.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
lspindpi.e	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

**Assertion**
Ref	Expression
lspindpi	⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))

**Proof of Theorem lspindpi**
Step	Hyp	Ref	Expression
1		lspindpi.e	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
2		lspindpi.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ LVec)
3		lveclmod 21386	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
4	2, 3	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LMod)
5		eqid 2760	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
6	5	lsssssubg 21238	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
7	4, 6	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
8		lspindpi.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
9		lspindpi.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Base‘𝑊)
10		lspindpi.n	. . . . . . . . . . 11 ⊢ 𝑁 = (LSpan‘𝑊)
11	9, 5, 10	lspsncl 21257	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
12	4, 8, 11	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
13	7, 12	sseldd 3909	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
14		lspindpi.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
15	9, 5, 10	lspsncl 21257	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
16	4, 14, 15	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
17	7, 16	sseldd 3909	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊))
18		eqid 2760	. . . . . . . . 9 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
19	18	lsmub1 19876	. . . . . . . 8 ⊢ (((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
20	13, 17, 19	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
21	9, 10, 18, 4, 8, 14	lsmpr 21369	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
22	20, 21	sseqtrrd 3945	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑍}))
23		sseq1 3933	. . . . . 6 ⊢ ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑍})))
24	22, 23	syl5ibrcom 250	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
25	9, 5, 10, 4, 8, 14	lspprcl 21258	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑊))
26		lspindpi.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
27	9, 5, 10, 4, 25, 26	ellspsn5b 21275	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
28	24, 27	sylibrd 262	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) → 𝑋 ∈ (𝑁‘{𝑌, 𝑍})))
29	28	necon3bd 2969	. . 3 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
30	1, 29	mpd 16	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
31	18	lsmub2 19877	. . . . . . . 8 ⊢ (((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑍}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
32	13, 17, 31	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑍}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
33	32, 21	sseqtrrd 3945	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑍}) ⊆ (𝑁‘{𝑌, 𝑍}))
34		sseq1 3933	. . . . . 6 ⊢ ((𝑁‘{𝑋}) = (𝑁‘{𝑍}) → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑍}) ⊆ (𝑁‘{𝑌, 𝑍})))
35	33, 34	syl5ibrcom 250	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑍}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
36	35, 27	sylibrd 262	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑍}) → 𝑋 ∈ (𝑁‘{𝑌, 𝑍})))
37	36	necon3bd 2969	. . 3 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
38	1, 37	mpd 16	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
39	30, 38	jca 521	1 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ⊆ wss 3895 {csn 4584 {cpr 4586 ‘cfv 6538 (class class class)co 7420 Basecbs 17392 SubGrpcsubg 19335 LSSumclsm 19853 LModclmod 21140 LSubSpclss 21211 LSpanclspn 21251 LVecclvec 21382

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-er 8720 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-0g 17617 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-submnd 18984 df-grp 19152 df-minusg 19153 df-sbg 19154 df-subg 19338 df-cntz 19536 df-lsm 19855 df-cmn 20001 df-abl 20002 df-mgp 20366 df-ur 20413 df-ring 20466 df-lmod 21142 df-lss 21212 df-lsp 21252 df-lvec 21383

This theorem is used by: lspindp1 21416 baerlem5amN 42866 baerlem5bmN 42867 baerlem5abmN 42868 mapdindp4 42873 mapdh6bN 42887 mapdh6cN 42888 mapdh6dN 42889 mapdh6eN 42890 mapdh6fN 42891 mapdh6hN 42893 mapdh7eN 42898 mapdh7dN 42900 mapdh7fN 42901 mapdh75fN 42905 mapdh8aa 42926 mapdh8ab 42927 mapdh8ad 42929 mapdh8c 42931 mapdh8d0N 42932 mapdh8d 42933 mapdh8e 42934 mapdh9a 42939 mapdh9aOLDN 42940 hdmap1eq4N 42956 hdmap1l6b 42961 hdmap1l6c 42962 hdmap1l6d 42963 hdmap1l6e 42964 hdmap1l6f 42965 hdmap1l6h 42967 hdmap1eulemOLDN 42973 hdmapval0 42983 hdmapval3lemN 42987 hdmap10lem 42989 hdmap11lem1 42991 hdmap14lem11 43028

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