mapdhcl - Mathbox for Norm Megill

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Theorem mapdhcl 37016

Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)

**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 }))
mapdhc.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mapdh.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))

**Assertion**
Ref	Expression
mapdhcl	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)

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 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) 𝑄(ℎ) 𝑈(𝑥) 𝐻(𝑥,ℎ) 𝐼(𝑥,ℎ) 𝐾(𝑥,ℎ) 𝑉(𝑥,ℎ) 𝑊(𝑥,ℎ)

**Proof of Theorem mapdhcl**
Step	Hyp	Ref	Expression
1		oteq3 4413	. . . 4 ⊢ (𝑌 = 0 → ⟨𝑋, 𝐹, 𝑌⟩ = ⟨𝑋, 𝐹, 0 ⟩)
2	1	fveq2d 6195	. . 3 ⊢ (𝑌 = 0 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐼‘⟨𝑋, 𝐹, 0 ⟩))
3	2	eleq1d 2686	. 2 ⊢ (𝑌 = 0 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷 ↔ (𝐼‘⟨𝑋, 𝐹, 0 ⟩) ∈ 𝐷))
4		mapdh.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
5		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)})))))
6		mapdhcl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
8		mapdhc.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
9	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → 𝐹 ∈ 𝐷)
10		mapdhc.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11	10	anim1i 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
12		eldifsn 4317	. . . . 5 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
13	11, 12	sylibr 224	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	4, 5, 7, 9, 13	mapdhval2 37015	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))
15		mapdh.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
16		mapdh.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
17		mapdh.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
18		mapdh.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
19		mapdh.s	. . . . 5 ⊢ − = (-_g‘𝑈)
20		mapdhc.o	. . . . 5 ⊢ 0 = (0_g‘𝑈)
21		mapdh.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
22		mapdh.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
23		mapdh.d	. . . . 5 ⊢ 𝐷 = (Base‘𝐶)
24		mapdh.r	. . . . 5 ⊢ 𝑅 = (-_g‘𝐶)
25		mapdh.j	. . . . 5 ⊢ 𝐽 = (LSpan‘𝐶)
26		mapdh.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
27	26	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28		mapdh.ne	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
29	28	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30		mapdh.mn	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
31	30	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
32	15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 7, 13, 9, 29, 31	mapdpg 36995	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → ∃!ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))
33		riotacl 6625	. . . 4 ⊢ (∃!ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})) → (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))) ∈ 𝐷)
34	32, 33	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))) ∈ 𝐷)
35	14, 34	eqeltrd 2701	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
36	4, 5, 20, 6, 8	mapdhval0 37014	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
37	15, 22, 23, 4, 26	lcd0vcl 36903	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐷)
38	36, 37	eqeltrd 2701	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) ∈ 𝐷)
39	3, 35, 38	pm2.61ne 2879	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃!wreu 2914 Vcvv 3200 ∖ cdif 3571 ifcif 4086 {csn 4177 ⟨cotp 4185 ↦ cmpt 4729 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 Basecbs 15857 0_gc0g 16100 -_gcsg 17424 LSpanclspn 18971 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 LCDualclcd 36875 mapdcmpd 36913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914

This theorem is referenced by: mapdheq4lem 37020 mapdheq4 37021 mapdh6lem1N 37022 mapdh6lem2N 37023 mapdh6aN 37024 mapdh6bN 37026 mapdh6cN 37027 mapdh6dN 37028 mapdh6hN 37032 mapdh7eN 37037 mapdh7cN 37038 mapdh7fN 37040 mapdh75e 37041 mapdh75fN 37044 mapdh8aa 37065 mapdh8d0N 37071 mapdh8d 37072 mapdh9a 37079 mapdh9aOLDN 37080 hdmap1cl 37094 hdmap1eulem 37113 hdmap1eulemOLDN 37114

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