mapdindp0 - Mathbox for Norm Megill

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Theorem mapdindp0 37008

Description: Vector independence lemma. (Contributed by NM, 29-Apr-2015.)

**Hypotheses**
Ref	Expression
mapdindp1.v	⊢ 𝑉 = (Base‘𝑊)
mapdindp1.p	⊢ + = (+_g‘𝑊)
mapdindp1.o	⊢ 0 = (0_g‘𝑊)
mapdindp1.n	⊢ 𝑁 = (LSpan‘𝑊)
mapdindp1.w	⊢ (𝜑 → 𝑊 ∈ LVec)
mapdindp1.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdindp1.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdindp1.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
mapdindp1.W	⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
mapdindp1.e	⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
mapdindp1.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdindp1.f	⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
mapdindp1.yz	⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 )

**Assertion**
Ref	Expression
mapdindp0	⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))

**Proof of Theorem mapdindp0**
Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
2		mapdindp1.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
3		mapdindp1.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
4		lveclmod 19106	. . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
6		mapdindp1.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
7	6	eldifad 3586	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
8		mapdindp1.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
9	8, 1, 2	lspsncl 18977	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
10	5, 7, 9	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
11		mapdindp1.e	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
12	11, 10	eqeltrrd 2702	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
13		eqid 2622	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
14	1, 13	lsmcl 19083	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑊))
15	5, 10, 12, 14	syl3anc 1326	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑊))
16	1	lsssssubg 18958	. . . . . . 7 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
17	5, 16	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
18	17, 10	sseldd 3604	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
19	11, 18	eqeltrrd 2702	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊))
20	8, 2	lspsnid 18993	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
21	5, 7, 20	syl2anc 693	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌}))
22		mapdindp1.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
23	22	eldifad 3586	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
24	8, 2	lspsnid 18993	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ (𝑁‘{𝑍}))
25	5, 23, 24	syl2anc 693	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑍}))
26		mapdindp1.p	. . . . . 6 ⊢ + = (+_g‘𝑊)
27	26, 13	lsmelvali 18065	. . . . 5 ⊢ ((((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) ∧ (𝑌 ∈ (𝑁‘{𝑌}) ∧ 𝑍 ∈ (𝑁‘{𝑍}))) → (𝑌 + 𝑍) ∈ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
28	18, 19, 21, 25, 27	syl22anc 1327	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
29	1, 2, 5, 15, 28	lspsnel5a 18996	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
30	11	oveq2d 6666	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
31	13	lsmidm 18077	. . . . 5 ⊢ ((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑌}))
32	18, 31	syl 17	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑌}))
33	30, 32	eqtr3d 2658	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) = (𝑁‘{𝑌}))
34	29, 33	sseqtrd 3641	. 2 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌}))
35		mapdindp1.o	. . 3 ⊢ 0 = (0_g‘𝑊)
36	8, 26	lmodvacl 18877	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
37	5, 7, 23, 36	syl3anc 1326	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
38		mapdindp1.yz	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 )
39		eldifsn 4317	. . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 ))
40	37, 38, 39	sylanbrc 698	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }))
41	8, 35, 2, 3, 40, 7	lspsncmp 19116	. 2 ⊢ (𝜑 → ((𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})))
42	34, 41	mpbid 222	1 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 {cpr 4179 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 SubGrpcsubg 17588 LSSumclsm 18049 LModclmod 18863 LSubSpclss 18932 LSpanclspn 18971 LVecclvec 19102

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103

This theorem is referenced by: mapdindp1 37009 mapdindp2 37010

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