hdmap1l6c - Mathbox for Norm Megill

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Theorem hdmap1l6c 42304

Description: Lemmma for hdmap1l6 42313. (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 42084	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		lmodgrp 20857	. . . 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 41601	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
18		hdmap1l6cl.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19	18	eldifad 3895	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
20		hdmap1l6c.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
21		hdmap1l6c.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 = 0 )
22	1, 7, 3	dvhlmod 41602	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
23	8, 9	lmod0vcl 20881	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉)
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑉)
25	21, 24	eqeltrd 2839	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
26		hdmap1l6c.ne	. . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
27	8, 10, 17, 19, 20, 25, 26	lspindpi 21125	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
28	27	simpld 495	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
29	1, 7, 8, 9, 10, 2, 11, 12, 13, 14, 3, 15, 16, 28, 18, 20	hdmap1cl 42296	. . 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 18935	. . 3 ⊢ ((𝐶 ∈ Grp ∧ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷) → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ 𝑄) = (𝐼‘⟨𝑋, 𝐹, 𝑌⟩))
33	6, 29, 32	syl2anc 590	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ 𝑄) = (𝐼‘⟨𝑋, 𝐹, 𝑌⟩))
34	21	oteq3d 4818	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝐹, 𝑍⟩ = ⟨𝑋, 𝐹, 0 ⟩)
35	34	fveq2d 6831	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = (𝐼‘⟨𝑋, 𝐹, 0 ⟩))
36	1, 7, 8, 9, 2, 11, 31, 14, 3, 15, 19	hdmap1val0 42291	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
37	35, 36	eqtrd 2774	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝑄)
38	37	oveq2d 7372	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ 𝑄))
39	21	oveq2d 7372	. . . . 5 ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑌 + 0 ))
40		lmodgrp 20857	. . . . . . 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 18935	. . . . . 6 ⊢ ((𝑈 ∈ Grp ∧ 𝑌 ∈ 𝑉) → (𝑌 + 0 ) = 𝑌)
44	41, 20, 43	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝑌 + 0 ) = 𝑌)
45	39, 44	eqtrd 2774	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = 𝑌)
46	45	oteq3d 4818	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩ = ⟨𝑋, 𝐹, 𝑌⟩)
47	46	fveq2d 6831	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑌⟩))
48	33, 38, 47	3eqtr4rd 2785	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 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∖ cdif 3880 {csn 4555 {cpr 4557 ⟨cotp 4563 ‘cfv 6485 (class class class)co 7356 Basecbs 17170 +_gcplusg 17211 0_gc0g 17393 Grpcgrp 18900 -_gcsg 18902 LModclmod 20850 LSpanclspn 20961 HLchlt 39842 LHypclh 40476 DVecHcdvh 41570 LCDualclcd 42078 mapdcmpd 42116 HDMap1chdma1 42283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-riotaBAD 39445

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-ot 4564 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-0g 17395 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18251 df-poset 18270 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18389 df-clat 18456 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-cntz 19283 df-oppg 19312 df-lsm 19602 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-nzr 20485 df-rlreg 20666 df-domn 20667 df-drng 20703 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lvec 21093 df-lsatoms 39468 df-lshyp 39469 df-lcv 39511 df-lfl 39550 df-lkr 39578 df-ldual 39616 df-oposet 39668 df-ol 39670 df-oml 39671 df-covers 39758 df-ats 39759 df-atl 39790 df-cvlat 39814 df-hlat 39843 df-llines 39990 df-lplanes 39991 df-lvols 39992 df-lines 39993 df-psubsp 39995 df-pmap 39996 df-padd 40288 df-lhyp 40480 df-laut 40481 df-ldil 40596 df-ltrn 40597 df-trl 40651 df-tgrp 41235 df-tendo 41247 df-edring 41249 df-dveca 41495 df-disoa 41521 df-dvech 41571 df-dib 41631 df-dic 41665 df-dih 41721 df-doch 41840 df-djh 41887 df-lcdual 42079 df-mapd 42117 df-hdmap1 42285

This theorem is referenced by: hdmap1l6k 42312

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