hdmap1l6a - Mathbox for Norm Megill

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Theorem hdmap1l6a 41811

Description: Lemma for hdmap1l6 41823. 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 41810	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{(𝐺 ✚ 𝐸)}))
27	24, 25	oveq12d 7449	. . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = (𝐺 ✚ 𝐸))
28	27	sneqd 4638	. . . 4 ⊢ (𝜑 → {((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))} = {(𝐺 ✚ 𝐸)})
29	28	fveq2d 6910	. . 3 ⊢ (𝜑 → (𝐿‘{((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))}) = (𝐿‘{(𝐺 ✚ 𝐸)}))
30	26, 29	eqtr4d 2780	. 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 41809	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))
32	27	oveq2d 7447	. . . . 5 ⊢ (𝜑 → (𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))) = (𝐹𝑅(𝐺 ✚ 𝐸)))
33	32	sneqd 4638	. . . 4 ⊢ (𝜑 → {(𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))} = {(𝐹𝑅(𝐺 ✚ 𝐸))})
34	33	fveq2d 6910	. . 3 ⊢ (𝜑 → (𝐿‘{(𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))}) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))
35	31, 34	eqtr4d 2780	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))}))
36	1, 2, 16	dvhlmod 41112	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
37	20	eldifad 3963	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
38	21	eldifad 3963	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
39	3, 4	lmodvacl 20873	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
40	36, 37, 38, 39	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
41	3, 4, 6, 7, 36, 37, 38, 23	lmodindp1 21012	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 )
42		eldifsn 4786	. . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 ))
43	40, 41, 42	sylanbrc 583	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }))
44	1, 8, 16	lcdlmod 41594	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
45	1, 2, 16	dvhlvec 41111	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
46	18	eldifad 3963	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
47	3, 6, 7, 45, 37, 21, 46, 23, 22	lspindp2 21137	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})))
48	47	simpld 494	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
49	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 48, 18, 37	hdmap1cl 41806	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
50	3, 6, 7, 45, 20, 38, 46, 23, 22	lspindp1 21135	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
51	50	simpld 494	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
52	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 51, 18, 38	hdmap1cl 41806	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
53	9, 10	lmodvacl 20873	. . . 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 1373	. . 3 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) ∈ 𝐷)
55		eqid 2737	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
56	3, 55, 7, 36, 37, 38	lspprcl 20976	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈))
57	3, 4, 7, 36, 37, 38	lspprvacl 20997	. . . . . 6 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑁‘{𝑌, 𝑍}))
58	55, 7, 36, 56, 57	ellspsn5 20994	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌, 𝑍}))
59	3, 55, 7, 36, 56, 46	ellspsn5b 20993	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
60	22, 59	mtbid 324	. . . . 5 ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}))
61		nssne2 4047	. . . . 5 ⊢ (((𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌, 𝑍}) ∧ ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})) → (𝑁‘{(𝑌 + 𝑍)}) ≠ (𝑁‘{𝑋}))
62	58, 60, 61	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ≠ (𝑁‘{𝑋}))
63	62	necomd 2996	. . 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 41803	. 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 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 {cpr 4628 ⟨cotp 4634 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +_gcplusg 17297 0_gc0g 17484 -_gcsg 18953 LModclmod 20858 LSubSpclss 20929 LSpanclspn 20969 HLchlt 39351 LHypclh 39986 DVecHcdvh 41080 LCDualclcd 41588 mapdcmpd 41626 HDMap1chdma1 41793

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-mre 17629 df-mrc 17630 df-acs 17632 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-nzr 20513 df-rlreg 20694 df-domn 20695 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-lshyp 38978 df-lcv 39020 df-lfl 39059 df-lkr 39087 df-ldual 39125 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tgrp 40745 df-tendo 40757 df-edring 40759 df-dveca 41005 df-disoa 41031 df-dvech 41081 df-dib 41141 df-dic 41175 df-dih 41231 df-doch 41350 df-djh 41397 df-lcdual 41589 df-mapd 41627 df-hdmap1 41795

This theorem is referenced by: hdmap1l6d 41815 hdmap1l6e 41816 hdmap1l6f 41817 hdmap1l6j 41821

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