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Theorem lspindpi 20733

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 20705	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LMod)
5		eqid 2733	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
6	5	lsssssubg 20557	. . . . . . . . . 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 20576	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
12	4, 8, 11	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
13	7, 12	sseldd 3982	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
14		lspindpi.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
15	9, 5, 10	lspsncl 20576	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
16	4, 14, 15	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
17	7, 16	sseldd 3982	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊))
18		eqid 2733	. . . . . . . . 9 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
19	18	lsmub1 19518	. . . . . . . 8 ⊢ (((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
20	13, 17, 19	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
21	9, 10, 18, 4, 8, 14	lsmpr 20688	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
22	20, 21	sseqtrrd 4022	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑍}))
23		sseq1 4006	. . . . . 6 ⊢ ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑍})))
24	22, 23	syl5ibrcom 246	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
25	9, 5, 10, 4, 8, 14	lspprcl 20577	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑊))
26		lspindpi.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
27	9, 5, 10, 4, 25, 26	lspsnel5 20594	. . . . 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 2955	. . 3 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
30	1, 29	mpd 15	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
31	18	lsmub2 19519	. . . . . . . 8 ⊢ (((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑍}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
32	13, 17, 31	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑍}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
33	32, 21	sseqtrrd 4022	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑍}) ⊆ (𝑁‘{𝑌, 𝑍}))
34		sseq1 4006	. . . . . 6 ⊢ ((𝑁‘{𝑋}) = (𝑁‘{𝑍}) → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑍}) ⊆ (𝑁‘{𝑌, 𝑍})))
35	33, 34	syl5ibrcom 246	. . . . 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 2955	. . 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 513	1 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ⊆ wss 3947 {csn 4627 {cpr 4629 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 SubGrpcsubg 18994 LSSumclsm 19495 LModclmod 20459 LSubSpclss 20530 LSpanclspn 20570 LVecclvec 20701

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-lsm 19497 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-lmod 20461 df-lss 20531 df-lsp 20571 df-lvec 20702

This theorem is referenced by: lspindp1 20734 baerlem5amN 40525 baerlem5bmN 40526 baerlem5abmN 40527 mapdindp4 40532 mapdh6bN 40546 mapdh6cN 40547 mapdh6dN 40548 mapdh6eN 40549 mapdh6fN 40550 mapdh6hN 40552 mapdh7eN 40557 mapdh7dN 40559 mapdh7fN 40560 mapdh75fN 40564 mapdh8aa 40585 mapdh8ab 40586 mapdh8ad 40588 mapdh8c 40590 mapdh8d0N 40591 mapdh8d 40592 mapdh8e 40593 mapdh9a 40598 mapdh9aOLDN 40599 hdmap1eq4N 40615 hdmap1l6b 40620 hdmap1l6c 40621 hdmap1l6d 40622 hdmap1l6e 40623 hdmap1l6f 40624 hdmap1l6h 40626 hdmap1eulemOLDN 40632 hdmapval0 40642 hdmapval3lemN 40646 hdmap10lem 40648 hdmap11lem1 40650 hdmap14lem11 40687

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