hdmap1l6k - Mathbox for Norm Megill

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Theorem hdmap1l6k 39813

Description: Lemmma for hdmap1l6 39814. Eliminate nonzero vector requirement. (Contributed by NM, 1-May-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}))
hdmap1l6k.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
hdmap1l6k.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
hdmap1l6k.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

**Assertion**
Ref	Expression
hdmap1l6k	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

**Proof of Theorem hdmap1l6k**
Step	Hyp	Ref	Expression
1		hdmap1l6.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1l6.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1l6.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap1l6.p	. . 3 ⊢ + = (+_g‘𝑈)
5		hdmap1l6.s	. . 3 ⊢ − = (-_g‘𝑈)
6		hdmap1l6c.o	. . 3 ⊢ 0 = (0_g‘𝑈)
7		hdmap1l6.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
8		hdmap1l6.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		hdmap1l6.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
10		hdmap1l6.a	. . 3 ⊢ ✚ = (+_g‘𝐶)
11		hdmap1l6.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
12		hdmap1l6.q	. . 3 ⊢ 𝑄 = (0_g‘𝐶)
13		hdmap1l6.l	. . 3 ⊢ 𝐿 = (LSpan‘𝐶)
14		hdmap1l6.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
15		hdmap1l6.i	. . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
16		hdmap1l6.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
18		hdmap1l6.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
19	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝐹 ∈ 𝐷)
20		hdmap1l6cl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
22		hdmap1l6.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
23	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
24		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑌 = 0 )
25		hdmap1l6k.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
26	25	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑍 ∈ 𝑉)
27		hdmap1l6k.xn	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
28	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
29	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 23, 24, 26, 28	hdmap1l6b 39804	. 2 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
30	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝐹 ∈ 𝐷)
32	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
33	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
34		hdmap1l6k.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
35	34	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝑌 ∈ 𝑉)
36		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝑍 = 0 )
37	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
38	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 30, 31, 32, 33, 35, 36, 37	hdmap1l6c 39805	. 2 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
39	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
40	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝐹 ∈ 𝐷)
41	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑋 ∈ (𝑉 ∖ { 0 }))
42	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
43	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
44	34	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑌 ∈ 𝑉)
45		simprl 767	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑌 ≠ 0 )
46		eldifsn 4725	. . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
47	44, 45, 46	sylanbrc 582	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑌 ∈ (𝑉 ∖ { 0 }))
48	25	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑍 ∈ 𝑉)
49		simprr 769	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑍 ≠ 0 )
50		eldifsn 4725	. . . 4 ⊢ (𝑍 ∈ (𝑉 ∖ { 0 }) ↔ (𝑍 ∈ 𝑉 ∧ 𝑍 ≠ 0 ))
51	48, 49, 50	sylanbrc 582	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑍 ∈ (𝑉 ∖ { 0 }))
52	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 40, 41, 42, 43, 47, 51	hdmap1l6j 39812	. 2 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
53	29, 38, 52	pm2.61da2ne 3034	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 1541 ∈ wcel 2109 ≠ wne 2944 ∖ cdif 3888 {csn 4566 {cpr 4568 ⟨cotp 4574 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 +_gcplusg 16943 0_gc0g 17131 -_gcsg 18560 LSpanclspn 20214 HLchlt 37343 LHypclh 37977 DVecHcdvh 39071 LCDualclcd 39579 mapdcmpd 39617 HDMap1chdma1 39784

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-riotaBAD 36946

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-tpos 8026 df-undef 8073 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-0g 17133 df-mre 17276 df-mrc 17277 df-acs 17279 df-proset 17994 df-poset 18012 df-plt 18029 df-lub 18045 df-glb 18046 df-join 18047 df-meet 18048 df-p0 18124 df-p1 18125 df-lat 18131 df-clat 18198 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-grp 18561 df-minusg 18562 df-sbg 18563 df-subg 18733 df-cntz 18904 df-oppg 18931 df-lsm 19222 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-ring 19766 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-dvr 19906 df-drng 19974 df-lmod 20106 df-lss 20175 df-lsp 20215 df-lvec 20346 df-lsatoms 36969 df-lshyp 36970 df-lcv 37012 df-lfl 37051 df-lkr 37079 df-ldual 37117 df-oposet 37169 df-ol 37171 df-oml 37172 df-covers 37259 df-ats 37260 df-atl 37291 df-cvlat 37315 df-hlat 37344 df-llines 37491 df-lplanes 37492 df-lvols 37493 df-lines 37494 df-psubsp 37496 df-pmap 37497 df-padd 37789 df-lhyp 37981 df-laut 37982 df-ldil 38097 df-ltrn 38098 df-trl 38152 df-tgrp 38736 df-tendo 38748 df-edring 38750 df-dveca 38996 df-disoa 39022 df-dvech 39072 df-dib 39132 df-dic 39166 df-dih 39222 df-doch 39341 df-djh 39388 df-lcdual 39580 df-mapd 39618 df-hdmap1 39786

This theorem is referenced by: hdmap1l6 39814

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