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Theorem lspindpi 21049

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 21020	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LMod)
5		eqid 2730	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
6	5	lsssssubg 20871	. . . . . . . . . 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 20890	. . . . . . . . . 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 3950	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
14		lspindpi.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
15	9, 5, 10	lspsncl 20890	. . . . . . . . . 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 3950	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊))
18		eqid 2730	. . . . . . . . 9 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
19	18	lsmub1 19594	. . . . . . . 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 21003	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
22	20, 21	sseqtrrd 3987	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑍}))
23		sseq1 3975	. . . . . 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 20891	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑊))
26		lspindpi.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
27	9, 5, 10, 4, 25, 26	ellspsn5b 20908	. . . . 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 2940	. . 3 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
30	1, 29	mpd 15	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
31	18	lsmub2 19595	. . . . . . . 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 3987	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑍}) ⊆ (𝑁‘{𝑌, 𝑍}))
34		sseq1 3975	. . . . . 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 2940	. . 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 1540 ∈ wcel 2109 ≠ wne 2926 ⊆ wss 3917 {csn 4592 {cpr 4594 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 SubGrpcsubg 19059 LSSumclsm 19571 LModclmod 20773 LSubSpclss 20844 LSpanclspn 20884 LVecclvec 21016

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-0g 17411 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-lsm 19573 df-cmn 19719 df-abl 19720 df-mgp 20057 df-ur 20098 df-ring 20151 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017

This theorem is referenced by: lspindp1 21050 baerlem5amN 41717 baerlem5bmN 41718 baerlem5abmN 41719 mapdindp4 41724 mapdh6bN 41738 mapdh6cN 41739 mapdh6dN 41740 mapdh6eN 41741 mapdh6fN 41742 mapdh6hN 41744 mapdh7eN 41749 mapdh7dN 41751 mapdh7fN 41752 mapdh75fN 41756 mapdh8aa 41777 mapdh8ab 41778 mapdh8ad 41780 mapdh8c 41782 mapdh8d0N 41783 mapdh8d 41784 mapdh8e 41785 mapdh9a 41790 mapdh9aOLDN 41791 hdmap1eq4N 41807 hdmap1l6b 41812 hdmap1l6c 41813 hdmap1l6d 41814 hdmap1l6e 41815 hdmap1l6f 41816 hdmap1l6h 41818 hdmap1eulemOLDN 41824 hdmapval0 41834 hdmapval3lemN 41838 hdmap10lem 41840 hdmap11lem1 41842 hdmap14lem11 41879

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