hdmap11lem1 - Mathbox for Norm Megill

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Theorem hdmap11lem1 41835

Description: Lemma for hdmapadd 41837. (Contributed by NM, 26-May-2015.)

**Hypotheses**
Ref	Expression
hdmap11.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap11.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap11.v	⊢ 𝑉 = (Base‘𝑈)
hdmap11.p	⊢ + = (+_g‘𝑈)
hdmap11.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap11.a	⊢ ✚ = (+_g‘𝐶)
hdmap11.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap11.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap11.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hdmap11.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
hdmap11.e	⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmap11.o	⊢ 0 = (0_g‘𝑈)
hdmap11.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap11.d	⊢ 𝐷 = (Base‘𝐶)
hdmap11.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap11.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap11.j	⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)
hdmap11.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap11lem0.1a	⊢ (𝜑 → 𝑧 ∈ 𝑉)
hdmap11lem0.6	⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
hdmap11lem0.2a	⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸}))

**Assertion**
Ref	Expression
hdmap11lem1	⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))

**Proof of Theorem hdmap11lem1**
Step	Hyp	Ref	Expression
1		hdmap11.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap11.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap11.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap11.p	. . 3 ⊢ + = (+_g‘𝑈)
5		hdmap11.o	. . 3 ⊢ 0 = (0_g‘𝑈)
6		hdmap11.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
7		hdmap11.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		hdmap11.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
9		hdmap11.a	. . 3 ⊢ ✚ = (+_g‘𝐶)
10		hdmap11.l	. . 3 ⊢ 𝐿 = (LSpan‘𝐶)
11		hdmap11.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
12		hdmap11.i	. . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
13		hdmap11.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14		eqid 2729	. . . . . 6 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
15		hdmap11.j	. . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)
16		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
17		eqid 2729	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
18		hdmap11.e	. . . . . . 7 ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
19	1, 16, 17, 2, 3, 5, 18, 13	dvheveccl 41106	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 }))
20	1, 2, 3, 5, 7, 8, 14, 15, 13, 19	hvmapcl2 41760	. . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0_g‘𝐶)}))
21	20	eldifad 3926	. . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷)
22	1, 2, 3, 5, 6, 7, 10, 11, 15, 13, 19	mapdhvmap 41763	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝐸})) = (𝐿‘{(𝐽‘𝐸)}))
23		hdmap11lem0.2a	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸}))
24	23	necomd 2980	. . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑧}))
25		hdmap11lem0.1a	. . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝑉)
26	1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 21, 22, 24, 19, 25	hdmap1cl 41798	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩) ∈ 𝐷)
27		eqid 2729	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
28	1, 2, 13	dvhlmod 41104	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
29		hdmap11.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
30		hdmap11.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
31	3, 27, 6, 28, 29, 30	lspprcl 20884	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
32		hdmap11lem0.6	. . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
33	5, 27, 28, 31, 25, 32	lssneln0 20859	. . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑉 ∖ { 0 }))
34		eqidd 2730	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩) = (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩))
35		eqid 2729	. . . . . 6 ⊢ (-_g‘𝑈) = (-_g‘𝑈)
36		eqid 2729	. . . . . 6 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
37	1, 2, 3, 35, 5, 6, 7, 8, 36, 10, 11, 12, 13, 19, 21, 33, 26, 24, 22	hdmap1eq 41795	. . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩) = (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩) ↔ ((𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩)}) ∧ (𝑀‘(𝑁‘{(𝐸(-_g‘𝑈)𝑧)})) = (𝐿‘{((𝐽‘𝐸)(-_g‘𝐶)(𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩))}))))
38	34, 37	mpbid 232	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩)}) ∧ (𝑀‘(𝑁‘{(𝐸(-_g‘𝑈)𝑧)})) = (𝐿‘{((𝐽‘𝐸)(-_g‘𝐶)(𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩))})))
39	38	simpld 494	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩)}))
40	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 26, 33, 29, 30, 32, 39	hdmap1l6 41815	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), (𝑋 + 𝑌)⟩) = ((𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑋⟩) ✚ (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑌⟩)))
41		hdmap11.s	. . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
42	3, 4	lmodvacl 20781	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
43	28, 29, 30, 42	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
44	1, 2, 13	dvhlvec 41103	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec)
45	19	eldifad 3926	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
46	3, 4, 6, 28, 29, 30	lspprvacl 20905	. . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌}))
47	27, 6, 28, 31, 46	ellspsn5 20902	. . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋, 𝑌}))
48	47, 32	ssneldd 3949	. . . . 5 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{(𝑋 + 𝑌)}))
49	3, 6, 28, 25, 43, 48	lspsnne2 21028	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{(𝑋 + 𝑌)}))
50	3, 6, 5, 44, 45, 43, 33, 23, 49	hdmaplem4 41768	. . 3 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{(𝑋 + 𝑌)})))
51	1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 43, 25, 50	hdmapval2 41826	. 2 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), (𝑋 + 𝑌)⟩))
52	3, 6, 44, 25, 29, 30, 32	lspindpi 21042	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑌})))
53	52	simpld 494	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}))
54	3, 6, 5, 44, 45, 29, 33, 23, 53	hdmaplem4 41768	. . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑋})))
55	1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 29, 25, 54	hdmapval2 41826	. . 3 ⊢ (𝜑 → (𝑆‘𝑋) = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑋⟩))
56	52	simprd 495	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑌}))
57	3, 6, 5, 44, 45, 30, 33, 23, 56	hdmaplem4 41768	. . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑌})))
58	1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 30, 25, 57	hdmapval2 41826	. . 3 ⊢ (𝜑 → (𝑆‘𝑌) = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑌⟩))
59	55, 58	oveq12d 7405	. 2 ⊢ (𝜑 → ((𝑆‘𝑋) ✚ (𝑆‘𝑌)) = ((𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑋⟩) ✚ (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑌⟩)))
60	40, 51, 59	3eqtr4d 2774	1 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {csn 4589 {cpr 4591 ⟨cop 4595 ⟨cotp 4597 I cid 5532 ↾ cres 5640 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 +_gcplusg 17220 0_gc0g 17402 -_gcsg 18867 LModclmod 20766 LSubSpclss 20837 LSpanclspn 20877 HLchlt 39343 LHypclh 39978 LTrncltrn 40095 DVecHcdvh 41072 LCDualclcd 41580 mapdcmpd 41618 HVMapchvm 41750 HDMap1chdma1 41785 HDMapchdma 41786

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-riotaBAD 38946

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-ot 4598 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-undef 8252 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-mre 17547 df-mrc 17548 df-acs 17550 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18391 df-clat 18458 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cntz 19249 df-oppg 19278 df-lsm 19566 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-nzr 20422 df-rlreg 20603 df-domn 20604 df-drng 20640 df-lmod 20768 df-lss 20838 df-lsp 20878 df-lvec 21010 df-lsatoms 38969 df-lshyp 38970 df-lcv 39012 df-lfl 39051 df-lkr 39079 df-ldual 39117 df-oposet 39169 df-ol 39171 df-oml 39172 df-covers 39259 df-ats 39260 df-atl 39291 df-cvlat 39315 df-hlat 39344 df-llines 39492 df-lplanes 39493 df-lvols 39494 df-lines 39495 df-psubsp 39497 df-pmap 39498 df-padd 39790 df-lhyp 39982 df-laut 39983 df-ldil 40098 df-ltrn 40099 df-trl 40153 df-tgrp 40737 df-tendo 40749 df-edring 40751 df-dveca 40997 df-disoa 41023 df-dvech 41073 df-dib 41133 df-dic 41167 df-dih 41223 df-doch 41342 df-djh 41389 df-lcdual 41581 df-mapd 41619 df-hvmap 41751 df-hdmap1 41787 df-hdmap 41788

This theorem is referenced by: hdmap11lem2 41836

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