mapdh8ab - Mathbox for Norm Megill

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Theorem mapdh8ab 42928

Description: Part of Part (8) in [Baer] p. 48. (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))
mapdh8ab.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh8ab.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdh8ab.eg	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
mapdh8ab.ee	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)
mapdh8ab.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdh8ab.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh8ab.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
mapdh8ab.t	⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
mapdh8ab.yz	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
mapdh8ab.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
mapdh8ab.yn	⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇}))

**Assertion**
Ref	Expression
mapdh8ab	⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩))

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

**Proof of Theorem mapdh8ab**
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		mapdh8ab.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
16		mapdh8ab.mn	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
17		mapdh8ab.eg	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
18		mapdh8ab.ee	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)
19		mapdh8ab.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
20		mapdh8ab.y	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
21		mapdh8ab.z	. 2 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
22	1, 2, 14	dvhlvec 42260	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
23	19	eldifad 4051	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
24	20	eldifad 4051	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
25	21	eldifad 4051	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
26		mapdh8ab.xn	. . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
27	3, 6, 22, 23, 24, 25, 26	lspindpi 21416	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
28	27	simprd 501	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
29	28	necomd 3010	. . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}))
30		mapdh8ab.yn	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇}))
31	29, 30	neeqtrd 3024	. 2 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))
32		mapdh8ab.t	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
33	30	sseq1d 3939	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑍})))
34		eqid 2760	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
35	1, 2, 14	dvhlmod 42261	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
36	3, 34, 6, 35, 24, 25	lspprcl 21259	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈))
37	3, 34, 6, 35, 36, 23	ellspsn5b 21276	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
38	32	eldifad 4051	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
39	3, 34, 6, 35, 36, 38	ellspsn5b 21276	. . . . 5 ⊢ (𝜑 → (𝑇 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑍})))
40	33, 37, 39	3bitr4d 314	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ 𝑇 ∈ (𝑁‘{𝑌, 𝑍})))
41	26, 40	mtbid 327	. . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ (𝑁‘{𝑌, 𝑍}))
42	22	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑈 ∈ LVec)
43	20	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
44	38	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ 𝑉)
45	25	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑍 ∈ 𝑉)
46		mapdh8ab.yz	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
47	46	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
48		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑍, 𝑇}))
49		prcom 4689	. . . . . 6 ⊢ {𝑍, 𝑇} = {𝑇, 𝑍}
50	49	fveq2i 6889	. . . . 5 ⊢ (𝑁‘{𝑍, 𝑇}) = (𝑁‘{𝑇, 𝑍})
51	48, 50	eleqtrdi 2870	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑇, 𝑍}))
52	3, 5, 6, 42, 43, 44, 45, 47, 51	lspexch 21413	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ (𝑁‘{𝑌, 𝑍}))
53	41, 52	mtand 828	. 2 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇}))
54	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 31, 32, 53, 26	mapdh8aa 42927	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ifcif 4481 {csn 4584 {cpr 4586 ⟨cotp 4857 ↦ cmpt 5185 ‘cfv 6539 ℩crio 7377 (class class class)co 7421 1^st c1st 8000 2^nd c2nd 8001 Basecbs 17393 0_gc0g 17616 -_gcsg 19152 LSubSpclss 21212 LSpanclspn 21252 LVecclvec 21383 HLchlt 40501 LHypclh 41135 DVecHcdvh 42229 LCDualclcd 42737 mapdcmpd 42775

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-riotaBAD 40104

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-ot 4858 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-tpos 8247 df-undef 8294 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-n0 12611 df-z 12698 df-uz 12970 df-fz 13644 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-sca 17450 df-vsca 17451 df-0g 17618 df-mre 17762 df-mrc 17763 df-acs 17765 df-proset 18474 df-poset 18493 df-plt 18508 df-lub 18524 df-glb 18525 df-join 18526 df-meet 18527 df-p0 18603 df-p1 18604 df-lat 18612 df-clat 18679 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-grp 19153 df-minusg 19154 df-sbg 19155 df-subg 19339 df-cntz 19537 df-oppg 19566 df-lsm 19856 df-cmn 20002 df-abl 20003 df-mgp 20367 df-rng 20381 df-ur 20414 df-ring 20467 df-oppr 20573 df-dvdsr 20593 df-unit 20594 df-invr 20624 df-dvr 20637 df-nzr 20769 df-rlreg 20952 df-domn 20953 df-drng 20988 df-lmod 21143 df-lss 21213 df-lsp 21253 df-lvec 21384 df-lsatoms 40127 df-lshyp 40128 df-lcv 40170 df-lfl 40209 df-lkr 40237 df-ldual 40275 df-oposet 40327 df-ol 40329 df-oml 40330 df-covers 40417 df-ats 40418 df-atl 40449 df-cvlat 40473 df-hlat 40502 df-llines 40649 df-lplanes 40650 df-lvols 40651 df-lines 40652 df-psubsp 40654 df-pmap 40655 df-padd 40947 df-lhyp 41139 df-laut 41140 df-ldil 41255 df-ltrn 41256 df-trl 41310 df-tgrp 41894 df-tendo 41906 df-edring 41908 df-dveca 42154 df-disoa 42180 df-dvech 42230 df-dib 42290 df-dic 42324 df-dih 42380 df-doch 42499 df-djh 42546 df-lcdual 42738 df-mapd 42776

This theorem is used by: mapdh8ac 42929

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