hdmap1l6h - Mathbox for Norm Megill

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Theorem hdmap1l6h 41201

Description: Lemmma for hdmap1l6 41205. 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 41200	. . 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 40976	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
28	1, 2, 16	dvhlvec 40493	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
29	24	eldifad 3955	. . . . . . . 8 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
30	18	eldifad 3955	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
31	22	eldifad 3955	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
32	3, 7, 28, 29, 30, 31, 25	lspindpi 20983	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})))
33	32	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}))
34	33	necomd 2990	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤}))
35	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 34, 18, 29	hdmap1cl 41188	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ∈ 𝐷)
36	23	eldifad 3955	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
37	3, 7, 28, 30, 31, 36, 20	lspindpi 20983	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
38	37	simpld 494	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
39	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 38, 18, 31	hdmap1cl 41188	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
40	37	simprd 495	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
41	1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 40, 18, 36	hdmap1cl 41188	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
42	9, 10	lmodass 20722	. . . 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 2766	. 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 41104	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)}))
46	1, 2, 16	dvhlmod 40494	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
47	3, 4	lmodvacl 20721	. . . . 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 41188	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) ∈ 𝐷)
50	9, 10	lmodvacl 20721	. . . 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 20723	. . 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 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∖ cdif 3940 {csn 4623 {cpr 4625 ⟨cotp 4631 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 +_gcplusg 17206 0_gc0g 17394 -_gcsg 18865 LModclmod 20706 LSpanclspn 20818 HLchlt 38733 LHypclh 39368 DVecHcdvh 40462 LCDualclcd 40970 mapdcmpd 41008 HDMap1chdma1 41175

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38336

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-undef 8259 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-0g 17396 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cntz 19233 df-oppg 19262 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-drng 20589 df-lmod 20708 df-lss 20779 df-lsp 20819 df-lvec 20951 df-lsatoms 38359 df-lshyp 38360 df-lcv 38402 df-lfl 38441 df-lkr 38469 df-ldual 38507 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tgrp 40127 df-tendo 40139 df-edring 40141 df-dveca 40387 df-disoa 40413 df-dvech 40463 df-dib 40523 df-dic 40557 df-dih 40613 df-doch 40732 df-djh 40779 df-lcdual 40971 df-mapd 41009 df-hdmap1 41177

This theorem is referenced by: hdmap1l6i 41202

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