hdmap1l6e - Mathbox for Norm Megill

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Theorem hdmap1l6e 41797

Description: Lemmma for hdmap1l6 41804. Part (6) in [Baer] p. 47 line 38. (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
hdmap1l6e	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

**Proof of Theorem hdmap1l6e**
Step	Hyp	Ref	Expression
1		hdmap1l6.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1l6.u	. 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1l6.v	. 2 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap1l6.p	. 2 ⊢ + = (+_g‘𝑈)
5		hdmap1l6.s	. 2 ⊢ − = (-_g‘𝑈)
6		hdmap1l6c.o	. 2 ⊢ 0 = (0_g‘𝑈)
7		hdmap1l6.n	. 2 ⊢ 𝑁 = (LSpan‘𝑈)
8		hdmap1l6.c	. 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		hdmap1l6.d	. 2 ⊢ 𝐷 = (Base‘𝐶)
10		hdmap1l6.a	. 2 ⊢ ✚ = (+_g‘𝐶)
11		hdmap1l6.r	. 2 ⊢ 𝑅 = (-_g‘𝐶)
12		hdmap1l6.q	. 2 ⊢ 𝑄 = (0_g‘𝐶)
13		hdmap1l6.l	. 2 ⊢ 𝐿 = (LSpan‘𝐶)
14		hdmap1l6.m	. 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
15		hdmap1l6.i	. 2 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
16		hdmap1l6.k	. 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		hdmap1l6.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
18		hdmap1l6cl.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19		hdmap1l6.mn	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
20	1, 2, 16	dvhlmod 41093	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
21		hdmap1l6d.w	. . . . 5 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
22	21	eldifad 3975	. . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
23		hdmap1l6d.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
24	23	eldifad 3975	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
25	3, 4	lmodvacl 20890	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) ∈ 𝑉)
26	20, 22, 24, 25	syl3anc 1370	. . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ 𝑉)
27	1, 2, 16	dvhlvec 41092	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
28	18	eldifad 3975	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
29		hdmap1l6d.wn	. . . . . 6 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
30	3, 7, 27, 22, 28, 24, 29	lspindpi 21152	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})))
31	30	simprd 495	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))
32	3, 4, 6, 7, 20, 22, 24, 31	lmodindp1 21030	. . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ≠ 0 )
33		eldifsn 4791	. . 3 ⊢ ((𝑤 + 𝑌) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑤 + 𝑌) ∈ 𝑉 ∧ (𝑤 + 𝑌) ≠ 0 ))
34	26, 32, 33	sylanbrc 583	. 2 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ (𝑉 ∖ { 0 }))
35		hdmap1l6d.z	. 2 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
36	35	eldifad 3975	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
37		hdmap1l6d.yz	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
38		hdmap1l6d.xn	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
39	3, 7, 27, 28, 24, 36, 38	lspindpi 21152	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
40	39	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
41	3, 4, 6, 7, 27, 18, 23, 35, 21, 37, 40, 29	mapdindp3 41705	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)}))
42	3, 4, 6, 7, 27, 18, 23, 35, 21, 37, 40, 29	mapdindp4 41706	. . . . 5 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)}))
43	3, 6, 7, 27, 18, 26, 36, 41, 42	lspindp1 21153	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)})))
44	43	simprd 495	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)}))
45		prcom 4737	. . . . 5 ⊢ {(𝑤 + 𝑌), 𝑍} = {𝑍, (𝑤 + 𝑌)}
46	45	fveq2i 6910	. . . 4 ⊢ (𝑁‘{(𝑤 + 𝑌), 𝑍}) = (𝑁‘{𝑍, (𝑤 + 𝑌)})
47	46	eleq2i 2831	. . 3 ⊢ (𝑋 ∈ (𝑁‘{(𝑤 + 𝑌), 𝑍}) ↔ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)}))
48	44, 47	sylnibr 329	. 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{(𝑤 + 𝑌), 𝑍}))
49	3, 7, 27, 36, 28, 26, 42	lspindpi 21152	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)})))
50	49	simprd 495	. . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)}))
51	50	necomd 2994	. 2 ⊢ (𝜑 → (𝑁‘{(𝑤 + 𝑌)}) ≠ (𝑁‘{𝑍}))
52		eqidd 2736	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) = (𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩))
53		eqidd 2736	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))
54	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 34, 35, 48, 51, 52, 53	hdmap1l6a 41792	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 {cpr 4633 ⟨cotp 4639 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +_gcplusg 17298 0_gc0g 17486 -_gcsg 18966 LModclmod 20875 LSpanclspn 20987 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 LCDualclcd 41569 mapdcmpd 41607 HDMap1chdma1 41774

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lcv 39001 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378 df-lcdual 41570 df-mapd 41608 df-hdmap1 41776

This theorem is referenced by: hdmap1l6g 41799

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