hdmap1l6a - Mathbox for Norm Megill

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Theorem hdmap1l6a 42561

Description: Lemma for hdmap1l6 42573. 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 42560	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{(𝐺 ✚ 𝐸)}))
27	24, 25	oveq12d 7430	. . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = (𝐺 ✚ 𝐸))
28	27	sneqd 4602	. . . 4 ⊢ (𝜑 → {((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))} = {(𝐺 ✚ 𝐸)})
29	28	fveq2d 6887	. . 3 ⊢ (𝜑 → (𝐿‘{((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))}) = (𝐿‘{(𝐺 ✚ 𝐸)}))
30	26, 29	eqtr4d 2801	. 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 42559	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))
32	27	oveq2d 7428	. . . . 5 ⊢ (𝜑 → (𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))) = (𝐹𝑅(𝐺 ✚ 𝐸)))
33	32	sneqd 4602	. . . 4 ⊢ (𝜑 → {(𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))} = {(𝐹𝑅(𝐺 ✚ 𝐸))})
34	33	fveq2d 6887	. . 3 ⊢ (𝜑 → (𝐿‘{(𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))}) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))
35	31, 34	eqtr4d 2801	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))}))
36	1, 2, 16	dvhlmod 41862	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
37	20	eldifad 3918	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
38	21	eldifad 3918	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
39	3, 4	lmodvacl 20977	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
40	36, 37, 38, 39	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
41	3, 4, 6, 7, 36, 37, 38, 23	lmodindp1 21116	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 )
42		eldifsn 4754	. . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 ))
43	40, 41, 42	sylanbrc 594	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }))
44	1, 8, 16	lcdlmod 42344	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
45	1, 2, 16	dvhlvec 41861	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
46	18	eldifad 3918	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
47	3, 6, 7, 45, 37, 21, 46, 23, 22	lspindp2 21240	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})))
48	47	simpld 499	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
49	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 48, 18, 37	hdmap1cl 42556	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
50	3, 6, 7, 45, 20, 38, 46, 23, 22	lspindp1 21238	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
51	50	simpld 499	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
52	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 51, 18, 38	hdmap1cl 42556	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
53	9, 10	lmodvacl 20977	. . . 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 1398	. . 3 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) ∈ 𝐷)
55		eqid 2763	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
56	3, 55, 7, 36, 37, 38	lspprcl 21080	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈))
57	3, 4, 7, 36, 37, 38	lspprvacl 21101	. . . . . 6 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑁‘{𝑌, 𝑍}))
58	55, 7, 36, 56, 57	ellspsn5 21098	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌, 𝑍}))
59	3, 55, 7, 36, 56, 46	ellspsn5b 21097	. . . . . 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 4001	. . . . 5 ⊢ (((𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌, 𝑍}) ∧ ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})) → (𝑁‘{(𝑌 + 𝑍)}) ≠ (𝑁‘{𝑋}))
62	58, 60, 61	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ≠ (𝑁‘{𝑋}))
63	62	necomd 3013	. . 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 42553	. 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 725	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 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3903 ⊆ wss 3906 {csn 4590 {cpr 4592 ⟨cotp 4598 ‘cfv 6538 (class class class)co 7412 Basecbs 17270 +_gcplusg 17311 0_gc0g 17493 -_gcsg 19003 LModclmod 20962 LSubSpclss 21033 LSpanclspn 21073 HLchlt 40102 LHypclh 40736 DVecHcdvh 41830 LCDualclcd 42338 mapdcmpd 42376 HDMap1chdma1 42543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-riotaBAD 39705

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-ot 4599 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 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 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-tpos 8223 df-undef 8270 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-n0 12506 df-z 12593 df-uz 12864 df-fz 13537 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-0g 17495 df-mre 17639 df-mrc 17640 df-acs 17642 df-proset 18351 df-poset 18370 df-plt 18385 df-lub 18401 df-glb 18402 df-join 18403 df-meet 18404 df-p0 18480 df-p1 18481 df-lat 18489 df-clat 18556 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-grp 19004 df-minusg 19005 df-sbg 19006 df-subg 19190 df-cntz 19388 df-oppg 19417 df-lsm 19707 df-cmn 19853 df-abl 19854 df-mgp 20218 df-rng 20232 df-ur 20265 df-ring 20318 df-oppr 20420 df-dvdsr 20440 df-unit 20441 df-invr 20471 df-dvr 20484 df-nzr 20597 df-rlreg 20780 df-domn 20781 df-drng 20816 df-lmod 20964 df-lss 21034 df-lsp 21074 df-lvec 21205 df-lsatoms 39728 df-lshyp 39729 df-lcv 39771 df-lfl 39810 df-lkr 39838 df-ldual 39876 df-oposet 39928 df-ol 39930 df-oml 39931 df-covers 40018 df-ats 40019 df-atl 40050 df-cvlat 40074 df-hlat 40103 df-llines 40250 df-lplanes 40251 df-lvols 40252 df-lines 40253 df-psubsp 40255 df-pmap 40256 df-padd 40548 df-lhyp 40740 df-laut 40741 df-ldil 40856 df-ltrn 40857 df-trl 40911 df-tgrp 41495 df-tendo 41507 df-edring 41509 df-dveca 41755 df-disoa 41781 df-dvech 41831 df-dib 41891 df-dic 41925 df-dih 41981 df-doch 42100 df-djh 42147 df-lcdual 42339 df-mapd 42377 df-hdmap1 42545

This theorem is referenced by: hdmap1l6d 42565 hdmap1l6e 42566 hdmap1l6f 42567 hdmap1l6j 42571

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