hdmap1l6g - Mathbox for Norm Megill

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Theorem hdmap1l6g 41938

Description: Lemmma for hdmap1l6 41943. Part (6) of [Baer] p. 47 line 39. (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
hdmap1l6g	⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)) = (((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

**Proof of Theorem hdmap1l6g**
Step	Hyp	Ref	Expression
1		hdmap1l6.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1l6.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1l6.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap1l6.p	. . 3 ⊢ + = (+_g‘𝑈)
5		hdmap1l6.s	. . 3 ⊢ − = (-_g‘𝑈)
6		hdmap1l6c.o	. . 3 ⊢ 0 = (0_g‘𝑈)
7		hdmap1l6.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
8		hdmap1l6.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		hdmap1l6.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
10		hdmap1l6.a	. . 3 ⊢ ✚ = (+_g‘𝐶)
11		hdmap1l6.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
12		hdmap1l6.q	. . 3 ⊢ 𝑄 = (0_g‘𝐶)
13		hdmap1l6.l	. . 3 ⊢ 𝐿 = (LSpan‘𝐶)
14		hdmap1l6.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
15		hdmap1l6.i	. . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
16		hdmap1l6.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		hdmap1l6.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
18		hdmap1l6cl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19		hdmap1l6.mn	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
20		hdmap1l6d.xn	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
21		hdmap1l6d.yz	. . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
22		hdmap1l6d.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
23		hdmap1l6d.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
24		hdmap1l6d.w	. . 3 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
25		hdmap1l6d.wn	. . 3 ⊢ (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	hdmap1l6d 41935	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)))
27	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	hdmap1l6e 41936	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
28	1, 2, 16	dvhlmod 41232	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
29	24	eldifad 3910	. . . . . 6 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
30	22	eldifad 3910	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
31	23	eldifad 3910	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
32	3, 4	lmodass 20813	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑤 + 𝑌) + 𝑍) = (𝑤 + (𝑌 + 𝑍)))
33	28, 29, 30, 31, 32	syl13anc 1374	. . . . 5 ⊢ (𝜑 → ((𝑤 + 𝑌) + 𝑍) = (𝑤 + (𝑌 + 𝑍)))
34	33	oteq3d 4840	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)⟩ = ⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩)
35	34	fveq2d 6834	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)⟩) = (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩))
36	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	hdmap1l6f 41937	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)))
37	36	oveq1d 7369	. . 3 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = (((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
38	27, 35, 37	3eqtr3d 2776	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = (((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
39	26, 38	eqtr3d 2770	1 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)) = (((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∖ cdif 3895 {csn 4577 {cpr 4579 ⟨cotp 4585 ‘cfv 6488 (class class class)co 7354 Basecbs 17124 +_gcplusg 17165 0_gc0g 17347 -_gcsg 18852 LModclmod 20797 LSpanclspn 20908 HLchlt 39472 LHypclh 40106 DVecHcdvh 41200 LCDualclcd 41708 mapdcmpd 41746 HDMap1chdma1 41913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092 ax-riotaBAD 39075

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7618 df-om 7805 df-1st 7929 df-2nd 7930 df-tpos 8164 df-undef 8211 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-2o 8394 df-er 8630 df-map 8760 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-nn 12135 df-2 12197 df-3 12198 df-4 12199 df-5 12200 df-6 12201 df-n0 12391 df-z 12478 df-uz 12741 df-fz 13412 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17125 df-ress 17146 df-plusg 17178 df-mulr 17179 df-sca 17181 df-vsca 17182 df-0g 17349 df-mre 17492 df-mrc 17493 df-acs 17495 df-proset 18204 df-poset 18223 df-plt 18238 df-lub 18254 df-glb 18255 df-join 18256 df-meet 18257 df-p0 18333 df-p1 18334 df-lat 18342 df-clat 18409 df-mgm 18552 df-sgrp 18631 df-mnd 18647 df-submnd 18696 df-grp 18853 df-minusg 18854 df-sbg 18855 df-subg 19040 df-cntz 19233 df-oppg 19262 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20063 df-rng 20075 df-ur 20104 df-ring 20157 df-oppr 20259 df-dvdsr 20279 df-unit 20280 df-invr 20310 df-dvr 20323 df-nzr 20432 df-rlreg 20613 df-domn 20614 df-drng 20650 df-lmod 20799 df-lss 20869 df-lsp 20909 df-lvec 21041 df-lsatoms 39098 df-lshyp 39099 df-lcv 39141 df-lfl 39180 df-lkr 39208 df-ldual 39246 df-oposet 39298 df-ol 39300 df-oml 39301 df-covers 39388 df-ats 39389 df-atl 39420 df-cvlat 39444 df-hlat 39473 df-llines 39620 df-lplanes 39621 df-lvols 39622 df-lines 39623 df-psubsp 39625 df-pmap 39626 df-padd 39918 df-lhyp 40110 df-laut 40111 df-ldil 40226 df-ltrn 40227 df-trl 40281 df-tgrp 40865 df-tendo 40877 df-edring 40879 df-dveca 41125 df-disoa 41151 df-dvech 41201 df-dib 41261 df-dic 41295 df-dih 41351 df-doch 41470 df-djh 41517 df-lcdual 41709 df-mapd 41747 df-hdmap1 41915

This theorem is referenced by: hdmap1l6h 41939

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