hdmap1l6b - Mathbox for Norm Megill

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Theorem hdmap1l6b 41805

Description: Lemmma for hdmap1l6 41815. (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}))
hdmap1l6b.y	⊢ (𝜑 → 𝑌 = 0 )
hdmap1l6b.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
hdmap1l6b.ne	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

**Assertion**
Ref	Expression
hdmap1l6b	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

**Proof of Theorem hdmap1l6b**
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 41586	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		lmodgrp 20773	. . . 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 41103	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
18		hdmap1l6cl.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19	18	eldifad 3926	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
20		hdmap1l6b.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 = 0 )
21	1, 7, 3	dvhlmod 41104	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
22	8, 9	lmod0vcl 20797	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉)
23	21, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑉)
24	20, 23	eqeltrd 2828	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
25		hdmap1l6b.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
26		hdmap1l6b.ne	. . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
27	8, 10, 17, 19, 24, 25, 26	lspindpi 21042	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
28	27	simprd 495	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
29	1, 7, 8, 9, 10, 2, 11, 12, 13, 14, 3, 15, 16, 28, 18, 25	hdmap1cl 41798	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
30		hdmap1l6.a	. . . 4 ⊢ ✚ = (+_g‘𝐶)
31		hdmap1l6.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
32	11, 30, 31	grplid 18899	. . 3 ⊢ ((𝐶 ∈ Grp ∧ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷) → (𝑄 ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))
33	6, 29, 32	syl2anc 584	. 2 ⊢ (𝜑 → (𝑄 ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))
34	20	oteq3d 4851	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝐹, 𝑌⟩ = ⟨𝑋, 𝐹, 0 ⟩)
35	34	fveq2d 6862	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐼‘⟨𝑋, 𝐹, 0 ⟩))
36	1, 7, 8, 9, 2, 11, 31, 14, 3, 15, 19	hdmap1val0 41793	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
37	35, 36	eqtrd 2764	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑄)
38	37	oveq1d 7402	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = (𝑄 ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
39	20	oveq1d 7402	. . . . 5 ⊢ (𝜑 → (𝑌 + 𝑍) = ( 0 + 𝑍))
40		lmodgrp 20773	. . . . . . 7 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp)
41	21, 40	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Grp)
42		hdmap1l6.p	. . . . . . 7 ⊢ + = (+_g‘𝑈)
43	8, 42, 9	grplid 18899	. . . . . 6 ⊢ ((𝑈 ∈ Grp ∧ 𝑍 ∈ 𝑉) → ( 0 + 𝑍) = 𝑍)
44	41, 25, 43	syl2anc 584	. . . . 5 ⊢ (𝜑 → ( 0 + 𝑍) = 𝑍)
45	39, 44	eqtrd 2764	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = 𝑍)
46	45	oteq3d 4851	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩ = ⟨𝑋, 𝐹, 𝑍⟩)
47	46	fveq2d 6862	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))
48	33, 38, 47	3eqtr4rd 2775	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 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3911 {csn 4589 {cpr 4591 ⟨cotp 4597 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 +_gcplusg 17220 0_gc0g 17402 Grpcgrp 18865 -_gcsg 18867 LModclmod 20766 LSpanclspn 20877 HLchlt 39343 LHypclh 39978 DVecHcdvh 41072 LCDualclcd 41580 mapdcmpd 41618 HDMap1chdma1 41785

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 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-riotaBAD 38946

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-ot 4598 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-undef 8252 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-mre 17547 df-mrc 17548 df-acs 17550 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18391 df-clat 18458 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cntz 19249 df-oppg 19278 df-lsm 19566 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-nzr 20422 df-rlreg 20603 df-domn 20604 df-drng 20640 df-lmod 20768 df-lss 20838 df-lsp 20878 df-lvec 21010 df-lsatoms 38969 df-lshyp 38970 df-lcv 39012 df-lfl 39051 df-lkr 39079 df-ldual 39117 df-oposet 39169 df-ol 39171 df-oml 39172 df-covers 39259 df-ats 39260 df-atl 39291 df-cvlat 39315 df-hlat 39344 df-llines 39492 df-lplanes 39493 df-lvols 39494 df-lines 39495 df-psubsp 39497 df-pmap 39498 df-padd 39790 df-lhyp 39982 df-laut 39983 df-ldil 40098 df-ltrn 40099 df-trl 40153 df-tgrp 40737 df-tendo 40749 df-edring 40751 df-dveca 40997 df-disoa 41023 df-dvech 41073 df-dib 41133 df-dic 41167 df-dih 41223 df-doch 41342 df-djh 41389 df-lcdual 41581 df-mapd 41619 df-hdmap1 41787

This theorem is referenced by: hdmap1l6k 41814

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