mapdh6iN - Mathbox for Norm Megill

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Theorem mapdh6iN 36513

Description: Lemmma for mapdh6N 36516. 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 3567	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
8		mapdh6i.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3567	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10	1, 2, 3, 4, 5, 7, 9	dvh3dim 36215	. 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 1080	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21		mapdhc.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
22	21	3ad2ant1 1080	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝐹 ∈ 𝐷)
23		mapdh.mn	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
24	23	3ad2ant1 1080	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
25	6	3ad2ant1 1080	. . . 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 1080	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
30		mapdh6i.yz	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
31	30	3ad2ant1 1080	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
32	8	3ad2ant1 1080	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
33		mapdh6i.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
34	33	3ad2ant1 1080	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑍 ∈ (𝑉 ∖ { 0 }))
35		eqid 2621	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
36	1, 2, 5	dvhlmod 35879	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
37	36	3ad2ant1 1080	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LMod)
38	3, 35, 4, 36, 7, 9	lspprcl 18897	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
39	38	3ad2ant1 1080	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
40		simp2 1060	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ 𝑉)
41		simp3 1061	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
42	3, 15, 35, 37, 39, 40, 41	lssneln0 18871	. . . 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 36512	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
44	43	rexlimdv3a 3026	. 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 384 ∧ w3a 1036 = wceq 1480 ∈ wcel 1987 ∃wrex 2908 Vcvv 3186 ∖ cdif 3552 ifcif 4058 {csn 4148 {cpr 4150 ⟨cotp 4156 ↦ cmpt 4673 ‘cfv 5847 ℩crio 6564 (class class class)co 6604 1^st c1st 7111 2^nd c2nd 7112 Basecbs 15781 +_gcplusg 15862 0_gc0g 16021 -_gcsg 17345 LModclmod 18784 LSubSpclss 18851 LSpanclspn 18890 HLchlt 34117 LHypclh 34750 DVecHcdvh 35847 LCDualclcd 36355 mapdcmpd 36393

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1836 ax-6 1885 ax-7 1932 ax-8 1989 ax-9 1996 ax-10 2016 ax-11 2031 ax-12 2044 ax-13 2245 ax-ext 2601 ax-rep 4731 ax-sep 4741 ax-nul 4749 ax-pow 4803 ax-pr 4867 ax-un 6902 ax-cnex 9936 ax-resscn 9937 ax-1cn 9938 ax-icn 9939 ax-addcl 9940 ax-addrcl 9941 ax-mulcl 9942 ax-mulrcl 9943 ax-mulcom 9944 ax-addass 9945 ax-mulass 9946 ax-distr 9947 ax-i2m1 9948 ax-1ne0 9949 ax-1rid 9950 ax-rnegex 9951 ax-rrecex 9952 ax-cnre 9953 ax-pre-lttri 9954 ax-pre-lttrn 9955 ax-pre-ltadd 9956 ax-pre-mulgt0 9957 ax-riotaBAD 33719

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-fal 1486 df-ex 1702 df-nf 1707 df-sb 1878 df-eu 2473 df-mo 2474 df-clab 2608 df-cleq 2614 df-clel 2617 df-nfc 2750 df-ne 2791 df-nel 2894 df-ral 2912 df-rex 2913 df-reu 2914 df-rmo 2915 df-rab 2916 df-v 3188 df-sbc 3418 df-csb 3515 df-dif 3558 df-un 3560 df-in 3562 df-ss 3569 df-pss 3571 df-nul 3892 df-if 4059 df-pw 4132 df-sn 4149 df-pr 4151 df-tp 4153 df-op 4155 df-ot 4157 df-uni 4403 df-int 4441 df-iun 4487 df-iin 4488 df-br 4614 df-opab 4674 df-mpt 4675 df-tr 4713 df-eprel 4985 df-id 4989 df-po 4995 df-so 4996 df-fr 5033 df-we 5035 df-xp 5080 df-rel 5081 df-cnv 5082 df-co 5083 df-dm 5084 df-rn 5085 df-res 5086 df-ima 5087 df-pred 5639 df-ord 5685 df-on 5686 df-lim 5687 df-suc 5688 df-iota 5810 df-fun 5849 df-fn 5850 df-f 5851 df-f1 5852 df-fo 5853 df-f1o 5854 df-fv 5855 df-riota 6565 df-ov 6607 df-oprab 6608 df-mpt2 6609 df-of 6850 df-om 7013 df-1st 7113 df-2nd 7114 df-tpos 7297 df-undef 7344 df-wrecs 7352 df-recs 7413 df-rdg 7451 df-1o 7505 df-oadd 7509 df-er 7687 df-map 7804 df-en 7900 df-dom 7901 df-sdom 7902 df-fin 7903 df-pnf 10020 df-mnf 10021 df-xr 10022 df-ltxr 10023 df-le 10024 df-sub 10212 df-neg 10213 df-nn 10965 df-2 11023 df-3 11024 df-4 11025 df-5 11026 df-6 11027 df-n0 11237 df-z 11322 df-uz 11632 df-fz 12269 df-struct 15783 df-ndx 15784 df-slot 15785 df-base 15786 df-sets 15787 df-ress 15788 df-plusg 15875 df-mulr 15876 df-sca 15878 df-vsca 15879 df-0g 16023 df-mre 16167 df-mrc 16168 df-acs 16170 df-preset 16849 df-poset 16867 df-plt 16879 df-lub 16895 df-glb 16896 df-join 16897 df-meet 16898 df-p0 16960 df-p1 16961 df-lat 16967 df-clat 17029 df-mgm 17163 df-sgrp 17205 df-mnd 17216 df-submnd 17257 df-grp 17346 df-minusg 17347 df-sbg 17348 df-subg 17512 df-cntz 17671 df-oppg 17697 df-lsm 17972 df-cmn 18116 df-abl 18117 df-mgp 18411 df-ur 18423 df-ring 18470 df-oppr 18544 df-dvdsr 18562 df-unit 18563 df-invr 18593 df-dvr 18604 df-drng 18670 df-lmod 18786 df-lss 18852 df-lsp 18891 df-lvec 19022 df-lsatoms 33743 df-lshyp 33744 df-lcv 33786 df-lfl 33825 df-lkr 33853 df-ldual 33891 df-oposet 33943 df-ol 33945 df-oml 33946 df-covers 34033 df-ats 34034 df-atl 34065 df-cvlat 34089 df-hlat 34118 df-llines 34264 df-lplanes 34265 df-lvols 34266 df-lines 34267 df-psubsp 34269 df-pmap 34270 df-padd 34562 df-lhyp 34754 df-laut 34755 df-ldil 34870 df-ltrn 34871 df-trl 34926 df-tgrp 35511 df-tendo 35523 df-edring 35525 df-dveca 35771 df-disoa 35798 df-dvech 35848 df-dib 35908 df-dic 35942 df-dih 35998 df-doch 36117 df-djh 36164 df-lcdual 36356 df-mapd 36394

This theorem is referenced by: mapdh6jN 36514

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