mapdh8 - Mathbox for Norm Megill

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Theorem mapdh8 41784

Description: Part (8) in [Baer] p. 48. Given a reference vector 𝑋, the value of function 𝐼 at a vector 𝑇 is independent of the choice of auxiliary vectors 𝑌 and 𝑍. Unlike Baer's, our version does not require 𝑋, 𝑌, and 𝑍 to be independent, and also is defined for all 𝑌 and 𝑍 that are not colinear with 𝑋 or 𝑇. We do this to make the definition of Baer's sigma function more straightforward. (This part eliminates 𝑇 ≠ 0.) (Contributed by NM, 13-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))
mapdh8h.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh8h.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdh8i.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdh8i.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh8i.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
mapdh8i.xy	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdh8i.xz	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
mapdh8i.yt	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
mapdh8i.zt	⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))
mapdh8.t	⊢ (𝜑 → 𝑇 ∈ 𝑉)

**Assertion**
Ref	Expression
mapdh8	⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))

Distinct variable groups: 𝑥,ℎ, − 0 ,ℎ,𝑥 𝐶,ℎ 𝐷,ℎ,𝑥 ℎ,𝐹,𝑥 ℎ,𝐼 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,𝑍,𝑥 𝑥,𝐼 ℎ,𝑉 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) 𝑄(ℎ) 𝑈(𝑥) 𝐻(𝑥,ℎ) 𝐾(𝑥,ℎ) 𝑉(𝑥) 𝑊(𝑥,ℎ)

**Proof of Theorem mapdh8**
Step	Hyp	Ref	Expression
1		mapdh8a.q	. . . . . 6 ⊢ 𝑄 = (0_g‘𝐶)
2		mapdh8a.i	. . . . . 6 ⊢ 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)})))))
3		mapdh8a.o	. . . . . 6 ⊢ 0 = (0_g‘𝑈)
4		mapdh8i.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
5		fvexd 6831	. . . . . 6 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ V)
6	1, 2, 3, 4, 5	mapdhval0 41721	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩) = 𝑄)
7		mapdh8i.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
8		fvexd 6831	. . . . . 6 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ V)
9	1, 2, 3, 7, 8	mapdhval0 41721	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩) = 𝑄)
10	6, 9	eqtr4d 2767	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
12		oteq3 4833	. . . . 5 ⊢ (𝑇 = 0 → ⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩ = ⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩)
13	12	fveq2d 6820	. . . 4 ⊢ (𝑇 = 0 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩))
14	13	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩))
15		oteq3 4833	. . . . 5 ⊢ (𝑇 = 0 → ⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩ = ⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩)
16	15	fveq2d 6820	. . . 4 ⊢ (𝑇 = 0 → (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
17	16	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
18	11, 14, 17	3eqtr4d 2774	. 2 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))
19		mapdh8a.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
20		mapdh8a.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
21		mapdh8a.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
22		mapdh8a.s	. . 3 ⊢ − = (-_g‘𝑈)
23		mapdh8a.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
24		mapdh8a.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
25		mapdh8a.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
26		mapdh8a.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
27		mapdh8a.j	. . 3 ⊢ 𝐽 = (LSpan‘𝐶)
28		mapdh8a.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
29		mapdh8a.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
30	29	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31		mapdh8h.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝐹 ∈ 𝐷)
33		mapdh8h.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
34	33	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
35		mapdh8i.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
36	35	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
37	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 }))
38	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑍 ∈ (𝑉 ∖ { 0 }))
39		mapdh8i.xy	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
40	39	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
41		mapdh8i.xz	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
42	41	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
43		mapdh8i.yt	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
44	43	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
45		mapdh8i.zt	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))
46	45	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))
47		mapdh8.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
48	47	anim1i 615	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 ))
49		eldifsn 4735	. . . 4 ⊢ (𝑇 ∈ (𝑉 ∖ { 0 }) ↔ (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 ))
50	48, 49	sylibr 234	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑇 ∈ (𝑉 ∖ { 0 }))
51	19, 20, 21, 22, 3, 23, 24, 25, 26, 1, 27, 28, 2, 30, 32, 34, 36, 37, 38, 40, 42, 44, 46, 50	mapdh8j 41783	. 2 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))
52	18, 51	pm2.61dane 3012	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3433 ∖ cdif 3896 ifcif 4472 {csn 4573 ⟨cotp 4581 ↦ cmpt 5169 ‘cfv 6476 ℩crio 7296 (class class class)co 7340 1^st c1st 7913 2^nd c2nd 7914 Basecbs 17107 0_gc0g 17330 -_gcsg 18801 LSpanclspn 20858 HLchlt 39346 LHypclh 39980 DVecHcdvh 41074 LCDualclcd 41582 mapdcmpd 41620

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5214 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662 ax-cnex 11053 ax-resscn 11054 ax-1cn 11055 ax-icn 11056 ax-addcl 11057 ax-addrcl 11058 ax-mulcl 11059 ax-mulrcl 11060 ax-mulcom 11061 ax-addass 11062 ax-mulass 11063 ax-distr 11064 ax-i2m1 11065 ax-1ne0 11066 ax-1rid 11067 ax-rnegex 11068 ax-rrecex 11069 ax-cnre 11070 ax-pre-lttri 11071 ax-pre-lttrn 11072 ax-pre-ltadd 11073 ax-pre-mulgt0 11074 ax-riotaBAD 38949

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-ot 4582 df-uni 4857 df-int 4895 df-iun 4940 df-iin 4941 df-br 5089 df-opab 5151 df-mpt 5170 df-tr 5196 df-id 5508 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5566 df-we 5568 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7297 df-ov 7343 df-oprab 7344 df-mpo 7345 df-of 7604 df-om 7791 df-1st 7915 df-2nd 7916 df-tpos 8150 df-undef 8197 df-frecs 8205 df-wrecs 8236 df-recs 8285 df-rdg 8323 df-1o 8379 df-2o 8380 df-er 8616 df-map 8746 df-en 8864 df-dom 8865 df-sdom 8866 df-fin 8867 df-pnf 11139 df-mnf 11140 df-xr 11141 df-ltxr 11142 df-le 11143 df-sub 11337 df-neg 11338 df-nn 12117 df-2 12179 df-3 12180 df-4 12181 df-5 12182 df-6 12183 df-n0 12373 df-z 12460 df-uz 12724 df-fz 13399 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17108 df-ress 17129 df-plusg 17161 df-mulr 17162 df-sca 17164 df-vsca 17165 df-0g 17332 df-mre 17475 df-mrc 17476 df-acs 17478 df-proset 18187 df-poset 18206 df-plt 18221 df-lub 18237 df-glb 18238 df-join 18239 df-meet 18240 df-p0 18316 df-p1 18317 df-lat 18325 df-clat 18392 df-mgm 18501 df-sgrp 18580 df-mnd 18596 df-submnd 18645 df-grp 18802 df-minusg 18803 df-sbg 18804 df-subg 18989 df-cntz 19183 df-oppg 19212 df-lsm 19502 df-cmn 19648 df-abl 19649 df-mgp 20013 df-rng 20025 df-ur 20054 df-ring 20107 df-oppr 20209 df-dvdsr 20229 df-unit 20230 df-invr 20260 df-dvr 20273 df-nzr 20382 df-rlreg 20563 df-domn 20564 df-drng 20600 df-lmod 20749 df-lss 20819 df-lsp 20859 df-lvec 20991 df-lsatoms 38972 df-lshyp 38973 df-lcv 39015 df-lfl 39054 df-lkr 39082 df-ldual 39120 df-oposet 39172 df-ol 39174 df-oml 39175 df-covers 39262 df-ats 39263 df-atl 39294 df-cvlat 39318 df-hlat 39347 df-llines 39494 df-lplanes 39495 df-lvols 39496 df-lines 39497 df-psubsp 39499 df-pmap 39500 df-padd 39792 df-lhyp 39984 df-laut 39985 df-ldil 40100 df-ltrn 40101 df-trl 40155 df-tgrp 40739 df-tendo 40751 df-edring 40753 df-dveca 40999 df-disoa 41025 df-dvech 41075 df-dib 41135 df-dic 41169 df-dih 41225 df-doch 41344 df-djh 41391 df-lcdual 41583 df-mapd 41621

This theorem is referenced by: mapdh9a 41785 mapdh9aOLDN 41786

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