hdmap1l6e - Mathbox for Norm Megill

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Theorem hdmap1l6e 43063

Description: Lemmma for hdmap1l6 43070. Part (6) in [Baer] p. 47 line 38. (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}))
hdmap1l6d.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
hdmap1l6d.yz	⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
hdmap1l6d.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
hdmap1l6d.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
hdmap1l6d.w	⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
hdmap1l6d.wn	⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))

**Assertion**
Ref	Expression
hdmap1l6e	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) + (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

**Proof of Theorem hdmap1l6e**
Step	Hyp	Ref	Expression
1		hdmap1l6.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1l6.u	. 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1l6.v	. 2 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap1l6.p	. 2 ⊢ + = (+_g‘𝑈)
5		hdmap1l6.s	. 2 ⊢ − = (-_g‘𝑈)
6		hdmap1l6c.o	. 2 ⊢ 0 = (0_g‘𝑈)
7		hdmap1l6.n	. 2 ⊢ 𝑁 = (LSpan‘𝑈)
8		hdmap1l6.c	. 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		hdmap1l6.d	. 2 ⊢ 𝐷 = (Base‘𝐶)
10		hdmap1l6.a	. 2 ⊢ + = (+_g‘𝐶)
11		hdmap1l6.r	. 2 ⊢ 𝑅 = (-_g‘𝐶)
12		hdmap1l6.q	. 2 ⊢ 𝑄 = (0_g‘𝐶)
13		hdmap1l6.l	. 2 ⊢ 𝐿 = (LSpan‘𝐶)
14		hdmap1l6.m	. 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
15		hdmap1l6.i	. 2 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
16		hdmap1l6.k	. 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		hdmap1l6.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
18		hdmap1l6cl.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19		hdmap1l6.mn	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
20	1, 2, 16	dvhlmod 42359	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
21		hdmap1l6d.w	. . . . 5 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
22	21	eldifad 4054	. . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
23		hdmap1l6d.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
24	23	eldifad 4054	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
25	3, 4	lmodvacl 21180	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) ∈ 𝑉)
26	20, 22, 24, 25	syl3anc 1399	. . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ 𝑉)
27	1, 2, 16	dvhlvec 42358	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
28	18	eldifad 4054	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
29		hdmap1l6d.wn	. . . . . 6 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
30	3, 7, 27, 22, 28, 24, 29	lspindpi 21440	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})))
31	30	simprd 501	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))
32	3, 4, 6, 7, 20, 22, 24, 31	lmodindp1 21319	. . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ≠ 0 )
33		eldifsn 4748	. . 3 ⊢ ((𝑤 + 𝑌) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑤 + 𝑌) ∈ 𝑉 ∧ (𝑤 + 𝑌) ≠ 0 ))
34	26, 32, 33	sylanbrc 595	. 2 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ (𝑉 ∖ { 0 }))
35		hdmap1l6d.z	. 2 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
36	35	eldifad 4054	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
37		hdmap1l6d.yz	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
38		hdmap1l6d.xn	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
39	3, 7, 27, 28, 24, 36, 38	lspindpi 21440	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
40	39	simpld 500	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
41	3, 4, 6, 7, 27, 18, 23, 35, 21, 37, 40, 29	mapdindp3 42971	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)}))
42	3, 4, 6, 7, 27, 18, 23, 35, 21, 37, 40, 29	mapdindp4 42972	. . . . 5 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)}))
43	3, 6, 7, 27, 18, 26, 36, 41, 42	lspindp1 21441	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)})))
44	43	simprd 501	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)}))
45		prcom 4693	. . . . 5 ⊢ {(𝑤 + 𝑌), 𝑍} = {𝑍, (𝑤 + 𝑌)}
46	45	fveq2i 6911	. . . 4 ⊢ (𝑁‘{(𝑤 + 𝑌), 𝑍}) = (𝑁‘{𝑍, (𝑤 + 𝑌)})
47	46	eleq2i 2856	. . 3 ⊢ (𝑋 ∈ (𝑁‘{(𝑤 + 𝑌), 𝑍}) ↔ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)}))
48	44, 47	sylnibr 332	. 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{(𝑤 + 𝑌), 𝑍}))
49	3, 7, 27, 36, 28, 26, 42	lspindpi 21440	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)})))
50	49	simprd 501	. . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)}))
51	50	necomd 3014	. 2 ⊢ (𝜑 → (𝑁‘{(𝑤 + 𝑌)}) ≠ (𝑁‘{𝑍}))
52		eqidd 2764	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) = (𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩))
53		eqidd 2764	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))
54	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 34, 35, 48, 51, 52, 53	hdmap1l6a 43058	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, (𝑤 + 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 6558 (class class class)co 7443 Basecbs 17416 +_gcplusg 17457 0_gc0g 17639 -_gcsg 19175 LModclmod 21165 LSpanclspn 21276 HLchlt 40599 LHypclh 41233 DVecHcdvh 42327 LCDualclcd 42835 mapdcmpd 42873 HDMap1chdma1 43040

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 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-riotaBAD 40202

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 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-tpos 8271 df-undef 8318 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-n0 12634 df-z 12721 df-uz 12993 df-fz 13667 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-sca 17473 df-vsca 17474 df-0g 17641 df-mre 17785 df-mrc 17786 df-acs 17788 df-proset 18497 df-poset 18516 df-plt 18531 df-lub 18547 df-glb 18548 df-join 18549 df-meet 18550 df-p0 18626 df-p1 18627 df-lat 18635 df-clat 18702 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-subg 19362 df-cntz 19561 df-oppg 19590 df-lsm 19880 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-ring 20491 df-oppr 20597 df-dvdsr 20617 df-unit 20618 df-invr 20648 df-dvr 20661 df-nzr 20793 df-rlreg 20976 df-domn 20977 df-drng 21012 df-lmod 21167 df-lss 21237 df-lsp 21277 df-lvec 21408 df-lsatoms 40225 df-lshyp 40226 df-lcv 40268 df-lfl 40307 df-lkr 40335 df-ldual 40373 df-oposet 40425 df-ol 40427 df-oml 40428 df-covers 40515 df-ats 40516 df-atl 40547 df-cvlat 40571 df-hlat 40600 df-llines 40747 df-lplanes 40748 df-lvols 40749 df-lines 40750 df-psubsp 40752 df-pmap 40753 df-padd 41045 df-lhyp 41237 df-laut 41238 df-ldil 41353 df-ltrn 41354 df-trl 41408 df-tgrp 41992 df-tendo 42004 df-edring 42006 df-dveca 42252 df-disoa 42278 df-dvech 42328 df-dib 42388 df-dic 42422 df-dih 42478 df-doch 42597 df-djh 42644 df-lcdual 42836 df-mapd 42874 df-hdmap1 43042

This theorem is used by: hdmap1l6g 43065

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