lspindpi - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lspindpi		Structured version Visualization version GIF version

Theorem lspindpi 21087

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 21058	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LMod)
5		eqid 2736	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
6	5	lsssssubg 20909	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
7	4, 6	syl 17	. . . . . . . . 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 20928	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
12	4, 8, 11	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
13	7, 12	sseldd 3934	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
14		lspindpi.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
15	9, 5, 10	lspsncl 20928	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
16	4, 14, 15	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
17	7, 16	sseldd 3934	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊))
18		eqid 2736	. . . . . . . . 9 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
19	18	lsmub1 19586	. . . . . . . 8 ⊢ (((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
20	13, 17, 19	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
21	9, 10, 18, 4, 8, 14	lsmpr 21041	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
22	20, 21	sseqtrrd 3971	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑍}))
23		sseq1 3959	. . . . . 6 ⊢ ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑍})))
24	22, 23	syl5ibrcom 247	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
25	9, 5, 10, 4, 8, 14	lspprcl 20929	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑊))
26		lspindpi.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
27	9, 5, 10, 4, 25, 26	ellspsn5b 20946	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
28	24, 27	sylibrd 259	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) → 𝑋 ∈ (𝑁‘{𝑌, 𝑍})))
29	28	necon3bd 2946	. . 3 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
30	1, 29	mpd 15	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
31	18	lsmub2 19587	. . . . . . . 8 ⊢ (((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑍}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
32	13, 17, 31	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑍}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
33	32, 21	sseqtrrd 3971	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑍}) ⊆ (𝑁‘{𝑌, 𝑍}))
34		sseq1 3959	. . . . . 6 ⊢ ((𝑁‘{𝑋}) = (𝑁‘{𝑍}) → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑍}) ⊆ (𝑁‘{𝑌, 𝑍})))
35	33, 34	syl5ibrcom 247	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑍}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
36	35, 27	sylibrd 259	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑍}) → 𝑋 ∈ (𝑁‘{𝑌, 𝑍})))
37	36	necon3bd 2946	. . 3 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
38	1, 37	mpd 15	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
39	30, 38	jca 511	1 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ⊆ wss 3901 {csn 4580 {cpr 4582 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 SubGrpcsubg 19050 LSSumclsm 19563 LModclmod 20811 LSubSpclss 20882 LSpanclspn 20922 LVecclvec 21054

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-0g 17361 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-cntz 19246 df-lsm 19565 df-cmn 19711 df-abl 19712 df-mgp 20076 df-ur 20117 df-ring 20170 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lvec 21055

This theorem is referenced by: lspindp1 21088 baerlem5amN 41972 baerlem5bmN 41973 baerlem5abmN 41974 mapdindp4 41979 mapdh6bN 41993 mapdh6cN 41994 mapdh6dN 41995 mapdh6eN 41996 mapdh6fN 41997 mapdh6hN 41999 mapdh7eN 42004 mapdh7dN 42006 mapdh7fN 42007 mapdh75fN 42011 mapdh8aa 42032 mapdh8ab 42033 mapdh8ad 42035 mapdh8c 42037 mapdh8d0N 42038 mapdh8d 42039 mapdh8e 42040 mapdh9a 42045 mapdh9aOLDN 42046 hdmap1eq4N 42062 hdmap1l6b 42067 hdmap1l6c 42068 hdmap1l6d 42069 hdmap1l6e 42070 hdmap1l6f 42071 hdmap1l6h 42073 hdmap1eulemOLDN 42079 hdmapval0 42089 hdmapval3lemN 42093 hdmap10lem 42095 hdmap11lem1 42097 hdmap14lem11 42134

W3C validator