hdmap1l6h - Mathbox for Norm Megill

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Theorem hdmap1l6h 41346

Description: Lemmma for hdmap1l6 41350. Part (6) of [Baer] p. 48 line 2. (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
hdmap1l6h	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

**Proof of Theorem hdmap1l6h**
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		hdmap1l6d.xn	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
21		hdmap1l6d.yz	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
22		hdmap1l6d.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
23		hdmap1l6d.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
24		hdmap1l6d.w	. . . 4 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
25		hdmap1l6d.wn	. . . 4 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
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	hdmap1l6g 41345	. . 3 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)) = (((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
27	1, 8, 16	lcdlmod 41121	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
28	1, 2, 16	dvhlvec 40638	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
29	24	eldifad 3951	. . . . . . . 8 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
30	18	eldifad 3951	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
31	22	eldifad 3951	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
32	3, 7, 28, 29, 30, 31, 25	lspindpi 21024	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})))
33	32	simpld 493	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}))
34	33	necomd 2986	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤}))
35	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 34, 18, 29	hdmap1cl 41333	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ∈ 𝐷)
36	23	eldifad 3951	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
37	3, 7, 28, 30, 31, 36, 20	lspindpi 21024	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
38	37	simpld 493	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
39	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 38, 18, 31	hdmap1cl 41333	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
40	37	simprd 494	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
41	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 40, 18, 36	hdmap1cl 41333	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
42	9, 10	lmodass 20763	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ∈ 𝐷 ∧ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷 ∧ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)) → (((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))))
43	27, 35, 39, 41, 42	syl13anc 1369	. . 3 ⊢ (𝜑 → (((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))))
44	26, 43	eqtrd 2765	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))))
45	3, 4, 6, 7, 28, 18, 22, 23, 24, 21, 38, 25	mapdindp1 41249	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)}))
46	1, 2, 16	dvhlmod 40639	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
47	3, 4	lmodvacl 20762	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
48	46, 31, 36, 47	syl3anc 1368	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
49	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 45, 18, 48	hdmap1cl 41333	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) ∈ 𝐷)
50	9, 10	lmodvacl 20762	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷 ∧ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷) → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) ∈ 𝐷)
51	27, 39, 41, 50	syl3anc 1368	. . 3 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) ∈ 𝐷)
52	9, 10	lmodlcan 20764	. . 3 ⊢ ((𝐶 ∈ LMod ∧ ((𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) ∈ 𝐷 ∧ ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) ∈ 𝐷 ∧ (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ∈ 𝐷)) → (((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))) ↔ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))))
53	27, 49, 51, 35, 52	syl13anc 1369	. 2 ⊢ (𝜑 → (((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))) ↔ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))))
54	44, 53	mpbid 231	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 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∖ cdif 3936 {csn 4624 {cpr 4626 ⟨cotp 4632 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 +_gcplusg 17232 0_gc0g 17420 -_gcsg 18896 LModclmod 20747 LSpanclspn 20859 HLchlt 38878 LHypclh 39513 DVecHcdvh 40607 LCDualclcd 41115 mapdcmpd 41153 HDMap1chdma1 41320

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-riotaBAD 38481

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-undef 8277 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-0g 17422 df-mre 17565 df-mrc 17566 df-acs 17568 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-cntz 19272 df-oppg 19301 df-lsm 19595 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-dvr 20344 df-drng 20630 df-lmod 20749 df-lss 20820 df-lsp 20860 df-lvec 20992 df-lsatoms 38504 df-lshyp 38505 df-lcv 38547 df-lfl 38586 df-lkr 38614 df-ldual 38652 df-oposet 38704 df-ol 38706 df-oml 38707 df-covers 38794 df-ats 38795 df-atl 38826 df-cvlat 38850 df-hlat 38879 df-llines 39027 df-lplanes 39028 df-lvols 39029 df-lines 39030 df-psubsp 39032 df-pmap 39033 df-padd 39325 df-lhyp 39517 df-laut 39518 df-ldil 39633 df-ltrn 39634 df-trl 39688 df-tgrp 40272 df-tendo 40284 df-edring 40286 df-dveca 40532 df-disoa 40558 df-dvech 40608 df-dib 40668 df-dic 40702 df-dih 40758 df-doch 40877 df-djh 40924 df-lcdual 41116 df-mapd 41154 df-hdmap1 41322

This theorem is referenced by: hdmap1l6i 41347

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