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Theorem lspindpi 19907

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 19881	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LMod)
5		eqid 2824	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
6	5	lsssssubg 19733	. . . . . . . . . 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 19752	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
12	4, 8, 11	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
13	7, 12	sseldd 3971	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
14		lspindpi.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
15	9, 5, 10	lspsncl 19752	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
16	4, 14, 15	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
17	7, 16	sseldd 3971	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊))
18		eqid 2824	. . . . . . . . 9 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
19	18	lsmub1 18785	. . . . . . . 8 ⊢ (((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
20	13, 17, 19	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
21	9, 10, 18, 4, 8, 14	lsmpr 19864	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
22	20, 21	sseqtrrd 4011	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑍}))
23		sseq1 3995	. . . . . 6 ⊢ ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑍})))
24	22, 23	syl5ibrcom 249	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
25	9, 5, 10, 4, 8, 14	lspprcl 19753	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑊))
26		lspindpi.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
27	9, 5, 10, 4, 25, 26	lspsnel5 19770	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
28	24, 27	sylibrd 261	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) → 𝑋 ∈ (𝑁‘{𝑌, 𝑍})))
29	28	necon3bd 3033	. . 3 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
30	1, 29	mpd 15	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
31	18	lsmub2 18786	. . . . . . . 8 ⊢ (((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑍}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
32	13, 17, 31	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑍}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
33	32, 21	sseqtrrd 4011	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑍}) ⊆ (𝑁‘{𝑌, 𝑍}))
34		sseq1 3995	. . . . . 6 ⊢ ((𝑁‘{𝑋}) = (𝑁‘{𝑍}) → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑍}) ⊆ (𝑁‘{𝑌, 𝑍})))
35	33, 34	syl5ibrcom 249	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑍}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
36	35, 27	sylibrd 261	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑍}) → 𝑋 ∈ (𝑁‘{𝑌, 𝑍})))
37	36	necon3bd 3033	. . 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 514	1 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ⊆ wss 3939 {csn 4570 {cpr 4572 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 SubGrpcsubg 18276 LSSumclsm 18762 LModclmod 19637 LSubSpclss 19706 LSpanclspn 19746 LVecclvec 19877

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-cntz 18450 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878

This theorem is referenced by: lspindp1 19908 baerlem5amN 38856 baerlem5bmN 38857 baerlem5abmN 38858 mapdindp4 38863 mapdh6bN 38877 mapdh6cN 38878 mapdh6dN 38879 mapdh6eN 38880 mapdh6fN 38881 mapdh6hN 38883 mapdh7eN 38888 mapdh7dN 38890 mapdh7fN 38891 mapdh75fN 38895 mapdh8aa 38916 mapdh8ab 38917 mapdh8ad 38919 mapdh8c 38921 mapdh8d0N 38922 mapdh8d 38923 mapdh8e 38924 mapdh9a 38929 mapdh9aOLDN 38930 hdmap1eq4N 38946 hdmap1l6b 38951 hdmap1l6c 38952 hdmap1l6d 38953 hdmap1l6e 38954 hdmap1l6f 38955 hdmap1l6h 38957 hdmap1eulemOLDN 38963 hdmapval0 38973 hdmapval3lemN 38977 hdmap10lem 38979 hdmap11lem1 38981 hdmap14lem11 39018

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