hdmap1l6a - Mathbox for Norm Megill

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Theorem hdmap1l6a 39435

Description: Lemma for hdmap1l6 39447. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-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}))
hdmap1l6e.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
hdmap1l6e.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
hdmap1l6e.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
hdmap1l6.yz	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
hdmap1l6.fg	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
hdmap1l6.fe	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)

**Assertion**
Ref	Expression
hdmap1l6a	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

**Proof of Theorem hdmap1l6a**
Step	Hyp	Ref	Expression
1		hdmap1l6.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1l6.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1l6.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap1l6.p	. . . 4 ⊢ + = (+_g‘𝑈)
5		hdmap1l6.s	. . . 4 ⊢ − = (-_g‘𝑈)
6		hdmap1l6c.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
7		hdmap1l6.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
8		hdmap1l6.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		hdmap1l6.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
10		hdmap1l6.a	. . . 4 ⊢ ✚ = (+_g‘𝐶)
11		hdmap1l6.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
12		hdmap1l6.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
13		hdmap1l6.l	. . . 4 ⊢ 𝐿 = (LSpan‘𝐶)
14		hdmap1l6.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
15		hdmap1l6.i	. . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
16		hdmap1l6.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		hdmap1l6.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
18		hdmap1l6cl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19		hdmap1l6.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
20		hdmap1l6e.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
21		hdmap1l6e.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
22		hdmap1l6e.xn	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
23		hdmap1l6.yz	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
24		hdmap1l6.fg	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
25		hdmap1l6.fe	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)
26	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25	hdmap1l6lem2 39434	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{(𝐺 ✚ 𝐸)}))
27	24, 25	oveq12d 7182	. . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = (𝐺 ✚ 𝐸))
28	27	sneqd 4525	. . . 4 ⊢ (𝜑 → {((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))} = {(𝐺 ✚ 𝐸)})
29	28	fveq2d 6672	. . 3 ⊢ (𝜑 → (𝐿‘{((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))}) = (𝐿‘{(𝐺 ✚ 𝐸)}))
30	26, 29	eqtr4d 2776	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))}))
31	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25	hdmap1l6lem1 39433	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))
32	27	oveq2d 7180	. . . . 5 ⊢ (𝜑 → (𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))) = (𝐹𝑅(𝐺 ✚ 𝐸)))
33	32	sneqd 4525	. . . 4 ⊢ (𝜑 → {(𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))} = {(𝐹𝑅(𝐺 ✚ 𝐸))})
34	33	fveq2d 6672	. . 3 ⊢ (𝜑 → (𝐿‘{(𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))}) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))
35	31, 34	eqtr4d 2776	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))}))
36	1, 2, 16	dvhlmod 38736	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
37	20	eldifad 3853	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
38	21	eldifad 3853	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
39	3, 4	lmodvacl 19760	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
40	36, 37, 38, 39	syl3anc 1372	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
41	3, 4, 6, 7, 36, 37, 38, 23	lmodindp1 19898	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 )
42		eldifsn 4672	. . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 ))
43	40, 41, 42	sylanbrc 586	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }))
44	1, 8, 16	lcdlmod 39218	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
45	1, 2, 16	dvhlvec 38735	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
46	18	eldifad 3853	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
47	3, 6, 7, 45, 37, 21, 46, 23, 22	lspindp2 20019	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})))
48	47	simpld 498	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
49	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 48, 18, 37	hdmap1cl 39430	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
50	3, 6, 7, 45, 20, 38, 46, 23, 22	lspindp1 20017	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
51	50	simpld 498	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
52	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 51, 18, 38	hdmap1cl 39430	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
53	9, 10	lmodvacl 19760	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷 ∧ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷) → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) ∈ 𝐷)
54	44, 49, 52, 53	syl3anc 1372	. . 3 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) ∈ 𝐷)
55		eqid 2738	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
56	3, 55, 7, 36, 37, 38	lspprcl 19862	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈))
57	3, 4, 7, 36, 37, 38	lspprvacl 19883	. . . . . 6 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑁‘{𝑌, 𝑍}))
58	55, 7, 36, 56, 57	lspsnel5a 19880	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌, 𝑍}))
59	3, 55, 7, 36, 56, 46	lspsnel5 19879	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
60	22, 59	mtbid 327	. . . . 5 ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}))
61		nssne2 3936	. . . . 5 ⊢ (((𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌, 𝑍}) ∧ ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})) → (𝑁‘{(𝑌 + 𝑍)}) ≠ (𝑁‘{𝑋}))
62	58, 60, 61	syl2anc 587	. . . 4 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ≠ (𝑁‘{𝑋}))
63	62	necomd 2989	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)}))
64	1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 18, 17, 43, 54, 63, 19	hdmap1eq 39427	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) ↔ ((𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))}) ∧ (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))}))))
65	30, 35, 64	mpbir2and 713	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 399 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∖ cdif 3838 ⊆ wss 3841 {csn 4513 {cpr 4515 ⟨cotp 4521 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 +_gcplusg 16661 0_gc0g 16809 -_gcsg 18214 LModclmod 19746 LSubSpclss 19815 LSpanclspn 19855 HLchlt 36976 LHypclh 37610 DVecHcdvh 38704 LCDualclcd 39212 mapdcmpd 39250 HDMap1chdma1 39417

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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-riotaBAD 36579

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-ot 4522 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-undef 7961 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-0g 16811 df-mre 16953 df-mrc 16954 df-acs 16956 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-p1 17759 df-lat 17765 df-clat 17827 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-grp 18215 df-minusg 18216 df-sbg 18217 df-subg 18387 df-cntz 18558 df-oppg 18585 df-lsm 18872 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-dvr 19548 df-drng 19616 df-lmod 19748 df-lss 19816 df-lsp 19856 df-lvec 19987 df-lsatoms 36602 df-lshyp 36603 df-lcv 36645 df-lfl 36684 df-lkr 36712 df-ldual 36750 df-oposet 36802 df-ol 36804 df-oml 36805 df-covers 36892 df-ats 36893 df-atl 36924 df-cvlat 36948 df-hlat 36977 df-llines 37124 df-lplanes 37125 df-lvols 37126 df-lines 37127 df-psubsp 37129 df-pmap 37130 df-padd 37422 df-lhyp 37614 df-laut 37615 df-ldil 37730 df-ltrn 37731 df-trl 37785 df-tgrp 38369 df-tendo 38381 df-edring 38383 df-dveca 38629 df-disoa 38655 df-dvech 38705 df-dib 38765 df-dic 38799 df-dih 38855 df-doch 38974 df-djh 39021 df-lcdual 39213 df-mapd 39251 df-hdmap1 39419

This theorem is referenced by: hdmap1l6d 39439 hdmap1l6e 39440 hdmap1l6f 39441 hdmap1l6j 39445

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