mapdh6iN - Mathbox for Norm Megill

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Theorem mapdh6iN 39754

Description: Lemmma for mapdh6N 39757. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
mapdh.q	⊢ 𝑄 = (0_g‘𝐶)
mapdh.i	⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
mapdh.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdh.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdh.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdh.v	⊢ 𝑉 = (Base‘𝑈)
mapdh.s	⊢ − = (-_g‘𝑈)
mapdhc.o	⊢ 0 = (0_g‘𝑈)
mapdh.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdh.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdh.d	⊢ 𝐷 = (Base‘𝐶)
mapdh.r	⊢ 𝑅 = (-_g‘𝐶)
mapdh.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdh.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdhc.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdhcl.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdh.p	⊢ + = (+_g‘𝑈)
mapdh.a	⊢ ✚ = (+_g‘𝐶)
mapdh6i.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
mapdh6i.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh6i.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
mapdh6i.yz	⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))

**Assertion**
Ref	Expression
mapdh6iN	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Distinct variable groups: 𝑥,𝐷,ℎ ℎ,𝐹,𝑥 𝑥,𝐽 𝑥,𝑀 𝑥,𝑁 𝑥, 0 𝑥,𝑄 𝑥,𝑅 𝑥, − ℎ,𝑋,𝑥 ℎ,𝑌,𝑥 𝜑,ℎ 0 ,ℎ 𝐶,ℎ 𝐷,ℎ ℎ,𝐽 ℎ,𝑀 ℎ,𝑁 𝑅,ℎ 𝑈,ℎ − ,ℎ ℎ,𝑍,𝑥 ✚ ,ℎ ℎ,𝐼,𝑥 + ,ℎ,𝑥 ℎ,𝑉 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) ✚ (𝑥) 𝑄(ℎ) 𝑈(𝑥) 𝐻(𝑥,ℎ) 𝐾(𝑥,ℎ) 𝑉(𝑥) 𝑊(𝑥,ℎ)

Proof of Theorem mapdh6iN Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapdh.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdh.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdh.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		mapdh.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
5		mapdh.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		mapdhcl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
7	6	eldifad 3904	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
8		mapdh6i.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3904	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10	1, 2, 3, 4, 5, 7, 9	dvh3dim 39456	. 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
11		mapdh.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
12		mapdh.i	. . . 4 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
13		mapdh.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
14		mapdh.s	. . . 4 ⊢ − = (-_g‘𝑈)
15		mapdhc.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
16		mapdh.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
17		mapdh.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
18		mapdh.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
19		mapdh.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
20	5	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21		mapdhc.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
22	21	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝐹 ∈ 𝐷)
23		mapdh.mn	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
24	23	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
25	6	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
26		mapdh.p	. . . 4 ⊢ + = (+_g‘𝑈)
27		mapdh.a	. . . 4 ⊢ ✚ = (+_g‘𝐶)
28		mapdh6i.xn	. . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
29	28	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
30		mapdh6i.yz	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
31	30	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
32	8	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
33		mapdh6i.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
34	33	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑍 ∈ (𝑉 ∖ { 0 }))
35		eqid 2740	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
36	1, 2, 5	dvhlmod 39120	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
37	36	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LMod)
38	3, 35, 4, 36, 7, 9	lspprcl 20238	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
39	38	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
40		simp2 1136	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ 𝑉)
41		simp3 1137	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
42	15, 35, 37, 39, 40, 41	lssneln0 20212	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ (𝑉 ∖ { 0 }))
43	11, 12, 1, 13, 2, 3, 14, 15, 4, 16, 17, 18, 19, 20, 22, 24, 25, 26, 27, 29, 31, 32, 34, 42, 41	mapdh6hN 39753	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
44	43	rexlimdv3a 3217	. 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))))
45	10, 44	mpd 15	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 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ∃wrex 3067 Vcvv 3431 ∖ cdif 3889 ifcif 4465 {csn 4567 {cpr 4569 ⟨cotp 4575 ↦ cmpt 5162 ‘cfv 6432 ℩crio 7227 (class class class)co 7271 1^st c1st 7822 2^nd c2nd 7823 Basecbs 16910 +_gcplusg 16960 0_gc0g 17148 -_gcsg 18577 LModclmod 20121 LSubSpclss 20191 LSpanclspn 20231 HLchlt 37360 LHypclh 37994 DVecHcdvh 39088 LCDualclcd 39596 mapdcmpd 39634

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-riotaBAD 36963

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-tpos 8033 df-undef 8080 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-0g 17150 df-mre 17293 df-mrc 17294 df-acs 17296 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-oppg 18948 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 df-lsatoms 36986 df-lshyp 36987 df-lcv 37029 df-lfl 37068 df-lkr 37096 df-ldual 37134 df-oposet 37186 df-ol 37188 df-oml 37189 df-covers 37276 df-ats 37277 df-atl 37308 df-cvlat 37332 df-hlat 37361 df-llines 37508 df-lplanes 37509 df-lvols 37510 df-lines 37511 df-psubsp 37513 df-pmap 37514 df-padd 37806 df-lhyp 37998 df-laut 37999 df-ldil 38114 df-ltrn 38115 df-trl 38169 df-tgrp 38753 df-tendo 38765 df-edring 38767 df-dveca 39013 df-disoa 39039 df-dvech 39089 df-dib 39149 df-dic 39183 df-dih 39239 df-doch 39358 df-djh 39405 df-lcdual 39597 df-mapd 39635

This theorem is referenced by: mapdh6jN 39755

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