hdmap1l6k - Mathbox for Norm Megill

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Theorem hdmap1l6k 43068

Description: Lemmma for hdmap1l6 43069. 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 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
18		hdmap1l6.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
19	18	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝐹 ∈ 𝐷)
20		hdmap1l6cl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
22		hdmap1l6.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
23	22	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
24		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑌 = 0 )
25		hdmap1l6k.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
26	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑍 ∈ 𝑉)
27		hdmap1l6k.xn	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
28	27	adantr 486	. . 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 43059	. 2 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) + (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
30	16	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	18	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝐹 ∈ 𝐷)
32	20	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
33	22	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
34		hdmap1l6k.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
35	34	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝑌 ∈ 𝑉)
36		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝑍 = 0 )
37	27	adantr 486	. . 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 43060	. 2 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) + (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
39	16	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
40	18	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝐹 ∈ 𝐷)
41	20	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑋 ∈ (𝑉 ∖ { 0 }))
42	22	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
43	27	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
44	34	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑌 ∈ 𝑉)
45		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑌 ≠ 0 )
46		eldifsn 4748	. . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
47	44, 45, 46	sylanbrc 595	. . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑌 ∈ (𝑉 ∖ { 0 }))
48	25	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑍 ∈ 𝑉)
49		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑍 ≠ 0 )
50		eldifsn 4748	. . . 4 ⊢ (𝑍 ∈ (𝑉 ∖ { 0 }) ↔ (𝑍 ∈ 𝑉 ∧ 𝑍 ≠ 0 ))
51	48, 49, 50	sylanbrc 595	. . 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 43067	. 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 3047	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) + (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∖ cdif 4048 {csn 4588 {cpr 4590 ⟨cotp 4861 ‘cfv 6555 (class class class)co 7440 Basecbs 17413 +_gcplusg 17454 0_gc0g 17636 -_gcsg 19172 LSpanclspn 21273 HLchlt 40598 LHypclh 41232 DVecHcdvh 42326 LCDualclcd 42834 mapdcmpd 42872 HDMap1chdma1 43039

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-riotaBAD 40201

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-ot 4862 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-tpos 8268 df-undef 8315 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-n0 12631 df-z 12718 df-uz 12990 df-fz 13664 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-sca 17470 df-vsca 17471 df-0g 17638 df-mre 17782 df-mrc 17783 df-acs 17785 df-proset 18494 df-poset 18513 df-plt 18528 df-lub 18544 df-glb 18545 df-join 18546 df-meet 18547 df-p0 18623 df-p1 18624 df-lat 18632 df-clat 18699 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-submnd 19005 df-grp 19173 df-minusg 19174 df-sbg 19175 df-subg 19359 df-cntz 19558 df-oppg 19587 df-lsm 19877 df-cmn 20023 df-abl 20024 df-mgp 20388 df-rng 20402 df-ur 20435 df-ring 20488 df-oppr 20594 df-dvdsr 20614 df-unit 20615 df-invr 20645 df-dvr 20658 df-nzr 20790 df-rlreg 20973 df-domn 20974 df-drng 21009 df-lmod 21164 df-lss 21234 df-lsp 21274 df-lvec 21405 df-lsatoms 40224 df-lshyp 40225 df-lcv 40267 df-lfl 40306 df-lkr 40334 df-ldual 40372 df-oposet 40424 df-ol 40426 df-oml 40427 df-covers 40514 df-ats 40515 df-atl 40546 df-cvlat 40570 df-hlat 40599 df-llines 40746 df-lplanes 40747 df-lvols 40748 df-lines 40749 df-psubsp 40751 df-pmap 40752 df-padd 41044 df-lhyp 41236 df-laut 41237 df-ldil 41352 df-ltrn 41353 df-trl 41407 df-tgrp 41991 df-tendo 42003 df-edring 42005 df-dveca 42251 df-disoa 42277 df-dvech 42327 df-dib 42387 df-dic 42421 df-dih 42477 df-doch 42596 df-djh 42643 df-lcdual 42835 df-mapd 42873 df-hdmap1 43041

This theorem is used by: hdmap1l6 43069

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