mapdh8 - Mathbox for Norm Megill

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Theorem mapdh8 42289

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 6843	. . . . . 6 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ V)
6	1, 2, 3, 4, 5	mapdhval0 42226	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩) = 𝑄)
7		mapdh8i.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
8		fvexd 6843	. . . . . 6 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ V)
9	1, 2, 3, 7, 8	mapdhval0 42226	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩) = 𝑄)
10	6, 9	eqtr4d 2777	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
11	10	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
12		oteq3 4816	. . . . 5 ⊢ (𝑇 = 0 → ⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩ = ⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩)
13	12	fveq2d 6832	. . . 4 ⊢ (𝑇 = 0 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩))
14	13	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 0 ⟩))
15		oteq3 4816	. . . . 5 ⊢ (𝑇 = 0 → ⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩ = ⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩)
16	15	fveq2d 6832	. . . 4 ⊢ (𝑇 = 0 → (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
17	16	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 0 ⟩))
18	11, 14, 17	3eqtr4d 2784	. 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 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31		mapdh8h.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
32	31	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝐹 ∈ 𝐷)
33		mapdh8h.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
34	33	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
35		mapdh8i.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
36	35	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
37	4	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 }))
38	7	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑍 ∈ (𝑉 ∖ { 0 }))
39		mapdh8i.xy	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
40	39	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
41		mapdh8i.xz	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
42	41	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
43		mapdh8i.yt	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
44	43	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
45		mapdh8i.zt	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))
46	45	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))
47		mapdh8.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
48	47	anim1i 621	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 ))
49		eldifsn 4720	. . . 4 ⊢ (𝑇 ∈ (𝑉 ∖ { 0 }) ↔ (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 ))
50	48, 49	sylibr 235	. . 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 42288	. 2 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))
52	18, 51	pm2.61dane 3021	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 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 Vcvv 3431 ∖ cdif 3880 ifcif 4455 {csn 4556 ⟨cotp 4564 ↦ cmpt 5154 ‘cfv 6486 ℩crio 7313 (class class class)co 7357 1^st c1st 7930 2^nd c2nd 7931 Basecbs 17171 0_gc0g 17394 -_gcsg 18903 LSpanclspn 20962 HLchlt 39851 LHypclh 40485 DVecHcdvh 41579 LCDualclcd 42087 mapdcmpd 42125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-riotaBAD 39454

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-ot 4565 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-om 7808 df-1st 7932 df-2nd 7933 df-tpos 8167 df-undef 8214 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-n0 12430 df-z 12517 df-uz 12781 df-fz 13454 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-sca 17228 df-vsca 17229 df-0g 17396 df-mre 17540 df-mrc 17541 df-acs 17543 df-proset 18252 df-poset 18271 df-plt 18286 df-lub 18302 df-glb 18303 df-join 18304 df-meet 18305 df-p0 18381 df-p1 18382 df-lat 18390 df-clat 18457 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-grp 18904 df-minusg 18905 df-sbg 18906 df-subg 19091 df-cntz 19284 df-oppg 19313 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20114 df-rng 20126 df-ur 20155 df-ring 20208 df-oppr 20309 df-dvdsr 20329 df-unit 20330 df-invr 20360 df-dvr 20373 df-nzr 20486 df-rlreg 20667 df-domn 20668 df-drng 20704 df-lmod 20853 df-lss 20923 df-lsp 20963 df-lvec 21094 df-lsatoms 39477 df-lshyp 39478 df-lcv 39520 df-lfl 39559 df-lkr 39587 df-ldual 39625 df-oposet 39677 df-ol 39679 df-oml 39680 df-covers 39767 df-ats 39768 df-atl 39799 df-cvlat 39823 df-hlat 39852 df-llines 39999 df-lplanes 40000 df-lvols 40001 df-lines 40002 df-psubsp 40004 df-pmap 40005 df-padd 40297 df-lhyp 40489 df-laut 40490 df-ldil 40605 df-ltrn 40606 df-trl 40660 df-tgrp 41244 df-tendo 41256 df-edring 41258 df-dveca 41504 df-disoa 41530 df-dvech 41580 df-dib 41640 df-dic 41674 df-dih 41730 df-doch 41849 df-djh 41896 df-lcdual 42088 df-mapd 42126

This theorem is referenced by: mapdh9a 42290 mapdh9aOLDN 42291

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