mapdh8 - Mathbox for Norm Megill

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Theorem mapdh8 42675

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 6895	. . . . . 6 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ V)
6	1, 2, 3, 4, 5	mapdhval0 42612	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩) = 𝑄)
7		mapdh8i.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
8		fvexd 6895	. . . . . 6 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ V)
9	1, 2, 3, 7, 8	mapdhval0 42612	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩) = 𝑄)
10	6, 9	eqtr4d 2798	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
11	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
12		oteq3 4844	. . . . 5 ⊢ (𝑇 = 0 → ⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩ = ⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩)
13	12	fveq2d 6884	. . . 4 ⊢ (𝑇 = 0 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩))
14	13	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩))
15		oteq3 4844	. . . . 5 ⊢ (𝑇 = 0 → ⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩ = ⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩)
16	15	fveq2d 6884	. . . 4 ⊢ (𝑇 = 0 → (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
17	16	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
18	11, 14, 17	3eqtr4d 2805	. 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 486	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31		mapdh8h.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
32	31	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝐹 ∈ 𝐷)
33		mapdh8h.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
34	33	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
35		mapdh8i.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
36	35	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
37	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 }))
38	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑍 ∈ (𝑉 ∖ { 0 }))
39		mapdh8i.xy	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
40	39	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
41		mapdh8i.xz	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
42	41	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
43		mapdh8i.yt	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
44	43	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
45		mapdh8i.zt	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))
46	45	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))
47		mapdh8.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
48	47	anim1i 627	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 ))
49		eldifsn 4748	. . . 4 ⊢ (𝑇 ∈ (𝑉 ∖ { 0 }) ↔ (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 ))
50	48, 49	sylibr 237	. . 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 42674	. 2 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))
52	18, 51	pm2.61dane 3042	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 Vcvv 3450 ∖ cdif 3896 ifcif 4482 {csn 4584 ⟨cotp 4592 ↦ cmpt 5186 ‘cfv 6534 ℩crio 7371 (class class class)co 7415 1^st c1st 7986 2^nd c2nd 7987 Basecbs 17323 0_gc0g 17546 -_gcsg 19082 LSpanclspn 21182 HLchlt 40237 LHypclh 40871 DVecHcdvh 41965 LCDualclcd 42473 mapdcmpd 42511

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7738 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-riotaBAD 39840

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7372 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-tpos 8226 df-undef 8273 df-frecs 8282 df-wrecs 8313 df-recs 8362 df-rdg 8401 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11489 df-neg 11490 df-nn 12280 df-2 12349 df-3 12350 df-4 12351 df-5 12352 df-6 12353 df-n0 12551 df-z 12638 df-uz 12910 df-fz 13584 df-struct 17261 df-sets 17278 df-slot 17296 df-ndx 17308 df-base 17324 df-ress 17345 df-plusg 17377 df-mulr 17378 df-sca 17380 df-vsca 17381 df-0g 17548 df-mre 17692 df-mrc 17693 df-acs 17695 df-proset 18404 df-poset 18423 df-plt 18438 df-lub 18454 df-glb 18455 df-join 18456 df-meet 18457 df-p0 18533 df-p1 18534 df-lat 18542 df-clat 18609 df-mgm 18752 df-sgrp 18844 df-mnd 18860 df-submnd 18915 df-grp 19083 df-minusg 19084 df-sbg 19085 df-subg 19269 df-cntz 19467 df-oppg 19496 df-lsm 19786 df-cmn 19932 df-abl 19933 df-mgp 20297 df-rng 20311 df-ur 20344 df-ring 20397 df-oppr 20503 df-dvdsr 20523 df-unit 20524 df-invr 20554 df-dvr 20567 df-nzr 20699 df-rlreg 20882 df-domn 20883 df-drng 20918 df-lmod 21073 df-lss 21143 df-lsp 21183 df-lvec 21314 df-lsatoms 39863 df-lshyp 39864 df-lcv 39906 df-lfl 39945 df-lkr 39973 df-ldual 40011 df-oposet 40063 df-ol 40065 df-oml 40066 df-covers 40153 df-ats 40154 df-atl 40185 df-cvlat 40209 df-hlat 40238 df-llines 40385 df-lplanes 40386 df-lvols 40387 df-lines 40388 df-psubsp 40390 df-pmap 40391 df-padd 40683 df-lhyp 40875 df-laut 40876 df-ldil 40991 df-ltrn 40992 df-trl 41046 df-tgrp 41630 df-tendo 41642 df-edring 41644 df-dveca 41890 df-disoa 41916 df-dvech 41966 df-dib 42026 df-dic 42060 df-dih 42116 df-doch 42235 df-djh 42282 df-lcdual 42474 df-mapd 42512

This theorem is used by: mapdh9a 42676 mapdh9aOLDN 42677

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