hdmap1l6c - Mathbox for Norm Megill

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Theorem hdmap1l6c 37102

Description: Lemmma for hdmap1l6 37111. (Contributed by NM, 24-Apr-2015.)

**Hypotheses**
Ref	Expression
hdmap1l6.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap1l6.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1l6.v	⊢ 𝑉 = (Base‘𝑈)
hdmap1l6.p	⊢ + = (+_g‘𝑈)
hdmap1l6.s	⊢ − = (-_g‘𝑈)
hdmap1l6c.o	⊢ 0 = (0_g‘𝑈)
hdmap1l6.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap1l6.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1l6.d	⊢ 𝐷 = (Base‘𝐶)
hdmap1l6.a	⊢ ✚ = (+_g‘𝐶)
hdmap1l6.r	⊢ 𝑅 = (-_g‘𝐶)
hdmap1l6.q	⊢ 𝑄 = (0_g‘𝐶)
hdmap1l6.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap1l6.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1l6.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1l6.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap1l6.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
hdmap1l6cl.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap1l6.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
hdmap1l6c.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
hdmap1l6c.z	⊢ (𝜑 → 𝑍 = 0 )
hdmap1l6c.ne	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

**Assertion**
Ref	Expression
hdmap1l6c	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

**Proof of Theorem hdmap1l6c**
Step	Hyp	Ref	Expression
1		hdmap1l6.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1l6.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
3		hdmap1l6.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	lcdlmod 36881	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		lmodgrp 18870	. . . 4 ⊢ (𝐶 ∈ LMod → 𝐶 ∈ Grp)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Grp)
7		hdmap1l6.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		hdmap1l6.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
9		hdmap1l6c.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
10		hdmap1l6.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
11		hdmap1l6.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
12		hdmap1l6.l	. . . 4 ⊢ 𝐿 = (LSpan‘𝐶)
13		hdmap1l6.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
14		hdmap1l6.i	. . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
15		hdmap1l6.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
16		hdmap1l6.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
17	1, 7, 3	dvhlvec 36398	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
18		hdmap1l6cl.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19	18	eldifad 3586	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
20		hdmap1l6c.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
21		hdmap1l6c.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 = 0 )
22	1, 7, 3	dvhlmod 36399	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
23	8, 9	lmod0vcl 18892	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉)
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑉)
25	21, 24	eqeltrd 2701	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
26		hdmap1l6c.ne	. . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
27	8, 10, 17, 19, 20, 25, 26	lspindpi 19132	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
28	27	simpld 475	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
29	1, 7, 8, 9, 10, 2, 11, 12, 13, 14, 3, 15, 16, 28, 18, 20	hdmap1cl 37094	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
30		hdmap1l6.a	. . . 4 ⊢ ✚ = (+_g‘𝐶)
31		hdmap1l6.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
32	11, 30, 31	grprid 17453	. . 3 ⊢ ((𝐶 ∈ Grp ∧ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷) → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ 𝑄) = (𝐼‘⟨𝑋, 𝐹, 𝑌⟩))
33	6, 29, 32	syl2anc 693	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ 𝑄) = (𝐼‘⟨𝑋, 𝐹, 𝑌⟩))
34	21	oteq3d 4416	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝐹, 𝑍⟩ = ⟨𝑋, 𝐹, 0 ⟩)
35	34	fveq2d 6195	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = (𝐼‘⟨𝑋, 𝐹, 0 ⟩))
36	1, 7, 8, 9, 2, 11, 31, 14, 3, 15, 19	hdmap1val0 37089	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
37	35, 36	eqtrd 2656	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝑄)
38	37	oveq2d 6666	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ 𝑄))
39	21	oveq2d 6666	. . . . 5 ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑌 + 0 ))
40		lmodgrp 18870	. . . . . . 7 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp)
41	22, 40	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Grp)
42		hdmap1l6.p	. . . . . . 7 ⊢ + = (+_g‘𝑈)
43	8, 42, 9	grprid 17453	. . . . . 6 ⊢ ((𝑈 ∈ Grp ∧ 𝑌 ∈ 𝑉) → (𝑌 + 0 ) = 𝑌)
44	41, 20, 43	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝑌 + 0 ) = 𝑌)
45	39, 44	eqtrd 2656	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = 𝑌)
46	45	oteq3d 4416	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩ = ⟨𝑋, 𝐹, 𝑌⟩)
47	46	fveq2d 6195	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑌⟩))
48	33, 38, 47	3eqtr4rd 2667	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 {csn 4177 {cpr 4179 ⟨cotp 4185 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 Grpcgrp 17422 -_gcsg 17424 LModclmod 18863 LSpanclspn 18971 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 LCDualclcd 36875 mapdcmpd 36913 HDMap1chdma1 37081

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 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 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-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 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-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 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-oppg 17776 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-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914 df-hdmap1 37083

This theorem is referenced by: hdmap1l6k 37110

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