mapdh8b - Mathbox for Norm Megill

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Theorem mapdh8b 37069

Description: Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-May-2015.)

**Hypotheses**
Ref	Expression
mapdh8a.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdh8a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdh8a.v	⊢ 𝑉 = (Base‘𝑈)
mapdh8a.s	⊢ − = (-_g‘𝑈)
mapdh8a.o	⊢ 0 = (0_g‘𝑈)
mapdh8a.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdh8a.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdh8a.d	⊢ 𝐷 = (Base‘𝐶)
mapdh8a.r	⊢ 𝑅 = (-_g‘𝐶)
mapdh8a.q	⊢ 𝑄 = (0_g‘𝐶)
mapdh8a.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdh8a.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdh8a.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)})))))
mapdh8a.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdh8b.f	⊢ (𝜑 → 𝐺 ∈ 𝐷)
mapdh8b.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
mapdh8b.a	⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑤⟩) = 𝐸)
mapdh8b.x	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh8b.y	⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
mapdh8b.yz	⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))
mapdh8b.xt	⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
mapdh8b.vw	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
mapdh8b.e	⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
mapdh8b.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))

**Assertion**
Ref	Expression
mapdh8b	⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑌, 𝐺, 𝑇⟩))

Distinct variable groups: 𝑥,ℎ, − 0 ,ℎ,𝑥 𝐶,ℎ 𝐷,ℎ,𝑥 ℎ,𝐼 ℎ,𝐺,𝑥 ℎ,𝐽,𝑥 ℎ,𝑀,𝑥 ℎ,𝑁,𝑥 𝜑,ℎ 𝑅,ℎ,𝑥 𝑥,𝑄 𝑇,ℎ,𝑥 𝑈,ℎ ℎ,𝑋,𝑥 ℎ,𝑌,𝑥 ℎ,𝐸,𝑥 𝑤,ℎ,𝑥 Allowed substitution hints: 𝜑(𝑥,𝑤) 𝐶(𝑥,𝑤) 𝐷(𝑤) 𝑄(𝑤,ℎ) 𝑅(𝑤) 𝑇(𝑤) 𝑈(𝑥,𝑤) 𝐸(𝑤) 𝐺(𝑤) 𝐻(𝑥,𝑤,ℎ) 𝐼(𝑥,𝑤) 𝐽(𝑤) 𝐾(𝑥,𝑤,ℎ) 𝑀(𝑤) − (𝑤) 𝑁(𝑤) 𝑉(𝑥,𝑤,ℎ) 𝑊(𝑥,𝑤,ℎ) 𝑋(𝑤) 𝑌(𝑤) 0 (𝑤)

**Proof of Theorem mapdh8b**
Step	Hyp	Ref	Expression
1		mapdh8a.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdh8a.u	. 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdh8a.v	. 2 ⊢ 𝑉 = (Base‘𝑈)
4		mapdh8a.s	. 2 ⊢ − = (-_g‘𝑈)
5		mapdh8a.o	. 2 ⊢ 0 = (0_g‘𝑈)
6		mapdh8a.n	. 2 ⊢ 𝑁 = (LSpan‘𝑈)
7		mapdh8a.c	. 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		mapdh8a.d	. 2 ⊢ 𝐷 = (Base‘𝐶)
9		mapdh8a.r	. 2 ⊢ 𝑅 = (-_g‘𝐶)
10		mapdh8a.q	. 2 ⊢ 𝑄 = (0_g‘𝐶)
11		mapdh8a.j	. 2 ⊢ 𝐽 = (LSpan‘𝐶)
12		mapdh8a.m	. 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		mapdh8a.i	. 2 ⊢ 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)})))))
14		mapdh8a.k	. 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		mapdh8b.f	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
16		mapdh8b.mn	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
17		mapdh8b.a	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑤⟩) = 𝐸)
18		mapdh8b.x	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
19		mapdh8b.y	. 2 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
20		mapdh8b.yz	. 2 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))
21		mapdh8b.xt	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
22	1, 2, 14	dvhlvec 36398	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec)
23	18	eldifad 3586	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
24	19	eldifad 3586	. . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
25	21	eldifad 3586	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
26		mapdh8b.e	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
27		mapdh8b.xn	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))
28	3, 6, 22, 23, 24, 25, 26, 27	lspindp5 37059	. . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ (𝑁‘{𝑌, 𝑤}))
29		prcom 4267	. . . . . 6 ⊢ {𝑤, 𝑇} = {𝑇, 𝑤}
30	29	fveq2i 6194	. . . . 5 ⊢ (𝑁‘{𝑤, 𝑇}) = (𝑁‘{𝑇, 𝑤})
31	30	eleq2i 2693	. . . 4 ⊢ (𝑌 ∈ (𝑁‘{𝑤, 𝑇}) ↔ 𝑌 ∈ (𝑁‘{𝑇, 𝑤}))
32	22	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑈 ∈ LVec)
33	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
34	25	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑇 ∈ 𝑉)
35	24	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑤 ∈ 𝑉)
36		mapdh8b.vw	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
37	36	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
38		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑌 ∈ (𝑁‘{𝑇, 𝑤}))
39	3, 5, 6, 32, 33, 34, 35, 37, 38	lspexch 19129	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤}))
40	39	ex 450	. . . 4 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘{𝑇, 𝑤}) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤})))
41	31, 40	syl5bi 232	. . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘{𝑤, 𝑇}) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤})))
42	28, 41	mtod 189	. 2 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑤, 𝑇}))
43	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 42	mapdh8a 37064	1 ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑌, 𝐺, 𝑇⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∖ cdif 3571 ifcif 4086 {csn 4177 {cpr 4179 ⟨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 LVecclvec 19102 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: mapdh8c 37070 mapdh8d0N 37071 mapdh8d 37072

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