hdmap1l6k - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1l6k		Structured version Visualization version GIF version

Theorem hdmap1l6k 42657

Description: Lemmma for hdmap1l6 42658. 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 42648	. 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 42649	. 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 4755	. . . 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 4755	. . . 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 42656	. 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 3048	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 2146 ≠ wne 2960 ∖ cdif 3903 {csn 4591 {cpr 4593 ⟨cotp 4599 ‘cfv 6541 (class class class)co 7420 Basecbs 17296 +_gcplusg 17337 0_gc0g 17519 -_gcsg 19051 LSpanclspn 21147 HLchlt 40187 LHypclh 40821 DVecHcdvh 41915 LCDualclcd 42423 mapdcmpd 42461 HDMap1chdma1 42628

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7743 ax-cnex 11176 ax-resscn 11177 ax-1cn 11178 ax-icn 11179 ax-addcl 11180 ax-addrcl 11181 ax-mulcl 11182 ax-mulrcl 11183 ax-mulcom 11184 ax-addass 11185 ax-mulass 11186 ax-distr 11187 ax-i2m1 11188 ax-1ne0 11189 ax-1rid 11190 ax-rnegex 11191 ax-rrecex 11192 ax-cnre 11193 ax-pre-lttri 11194 ax-pre-lttrn 11195 ax-pre-ltadd 11196 ax-pre-mulgt0 11197 ax-riotaBAD 39790

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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7870 df-1st 7993 df-2nd 7994 df-tpos 8229 df-undef 8276 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-en 8951 df-dom 8952 df-sdom 8953 df-fin 8954 df-pnf 11265 df-mnf 11266 df-xr 11267 df-ltxr 11268 df-le 11269 df-sub 11463 df-neg 11464 df-nn 12254 df-2 12323 df-3 12324 df-4 12325 df-5 12326 df-6 12327 df-n0 12525 df-z 12612 df-uz 12884 df-fz 13557 df-struct 17234 df-sets 17251 df-slot 17269 df-ndx 17281 df-base 17297 df-ress 17318 df-plusg 17350 df-mulr 17351 df-sca 17353 df-vsca 17354 df-0g 17521 df-mre 17665 df-mrc 17666 df-acs 17668 df-proset 18377 df-poset 18396 df-plt 18411 df-lub 18427 df-glb 18428 df-join 18429 df-meet 18430 df-p0 18506 df-p1 18507 df-lat 18515 df-clat 18582 df-mgm 18725 df-sgrp 18814 df-mnd 18830 df-submnd 18884 df-grp 19052 df-minusg 19053 df-sbg 19054 df-subg 19238 df-cntz 19436 df-oppg 19465 df-lsm 19755 df-cmn 19901 df-abl 19902 df-mgp 20266 df-rng 20280 df-ur 20313 df-ring 20366 df-oppr 20470 df-dvdsr 20490 df-unit 20491 df-invr 20521 df-dvr 20534 df-nzr 20665 df-rlreg 20848 df-domn 20849 df-drng 20884 df-lmod 21038 df-lss 21108 df-lsp 21148 df-lvec 21279 df-lsatoms 39813 df-lshyp 39814 df-lcv 39856 df-lfl 39895 df-lkr 39923 df-ldual 39961 df-oposet 40013 df-ol 40015 df-oml 40016 df-covers 40103 df-ats 40104 df-atl 40135 df-cvlat 40159 df-hlat 40188 df-llines 40335 df-lplanes 40336 df-lvols 40337 df-lines 40338 df-psubsp 40340 df-pmap 40341 df-padd 40633 df-lhyp 40825 df-laut 40826 df-ldil 40941 df-ltrn 40942 df-trl 40996 df-tgrp 41580 df-tendo 41592 df-edring 41594 df-dveca 41840 df-disoa 41866 df-dvech 41916 df-dib 41976 df-dic 42010 df-dih 42066 df-doch 42185 df-djh 42232 df-lcdual 42424 df-mapd 42462 df-hdmap1 42630

This theorem is used by: hdmap1l6 42658

W3C validator