mapdh8ab - Mathbox for Norm Megill

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Theorem mapdh8ab 35882

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 35214	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
23	19	eldifad 3546	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
24	20	eldifad 3546	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
25	21	eldifad 3546	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
26		mapdh8ab.xn	. . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
27	3, 6, 22, 23, 24, 25, 26	lspindpi 18894	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
28	27	simprd 477	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
29	28	necomd 2831	. . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}))
30		mapdh8ab.yn	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇}))
31	29, 30	neeqtrd 2845	. 2 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))
32		mapdh8ab.t	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
33	30	sseq1d 3589	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑍})))
34		eqid 2604	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
35	1, 2, 14	dvhlmod 35215	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
36	3, 34, 6, 35, 24, 25	lspprcl 18740	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈))
37	3, 34, 6, 35, 36, 23	lspsnel5 18757	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
38	32	eldifad 3546	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
39	3, 34, 6, 35, 36, 38	lspsnel5 18757	. . . . 5 ⊢ (𝜑 → (𝑇 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑍})))
40	33, 37, 39	3bitr4d 298	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ 𝑇 ∈ (𝑁‘{𝑌, 𝑍})))
41	26, 40	mtbid 312	. . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ (𝑁‘{𝑌, 𝑍}))
42	22	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑈 ∈ LVec)
43	20	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
44	38	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ 𝑉)
45	25	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑍 ∈ 𝑉)
46		mapdh8ab.yz	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
47	46	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
48		simpr 475	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑍, 𝑇}))
49		prcom 4205	. . . . . 6 ⊢ {𝑍, 𝑇} = {𝑇, 𝑍}
50	49	fveq2i 6086	. . . . 5 ⊢ (𝑁‘{𝑍, 𝑇}) = (𝑁‘{𝑇, 𝑍})
51	48, 50	syl6eleq 2692	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑇, 𝑍}))
52	3, 5, 6, 42, 43, 44, 45, 47, 51	lspexch 18891	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ (𝑁‘{𝑌, 𝑍}))
53	41, 52	mtand 688	. 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 35881	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 = wceq 1474 ∈ wcel 1975 ≠ wne 2774 Vcvv 3167 ∖ cdif 3531 ⊆ wss 3534 ifcif 4030 {csn 4119 {cpr 4121 ⟨cotp 4127 ↦ cmpt 4632 ‘cfv 5785 ℩crio 6483 (class class class)co 6522 1^st c1st 7029 2^nd c2nd 7030 Basecbs 15636 0_gc0g 15864 -_gcsg 17188 LSubSpclss 18694 LSpanclspn 18733 LVecclvec 18864 HLchlt 33453 LHypclh 34086 DVecHcdvh 35183 LCDualclcd 35691 mapdcmpd 35729

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1711 ax-4 1726 ax-5 1825 ax-6 1873 ax-7 1920 ax-8 1977 ax-9 1984 ax-10 2004 ax-11 2019 ax-12 2031 ax-13 2227 ax-ext 2584 ax-rep 4688 ax-sep 4698 ax-nul 4707 ax-pow 4759 ax-pr 4823 ax-un 6819 ax-cnex 9843 ax-resscn 9844 ax-1cn 9845 ax-icn 9846 ax-addcl 9847 ax-addrcl 9848 ax-mulcl 9849 ax-mulrcl 9850 ax-mulcom 9851 ax-addass 9852 ax-mulass 9853 ax-distr 9854 ax-i2m1 9855 ax-1ne0 9856 ax-1rid 9857 ax-rnegex 9858 ax-rrecex 9859 ax-cnre 9860 ax-pre-lttri 9861 ax-pre-lttrn 9862 ax-pre-ltadd 9863 ax-pre-mulgt0 9864 ax-riotaBAD 33055

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-fal 1480 df-ex 1695 df-nf 1700 df-sb 1866 df-eu 2456 df-mo 2457 df-clab 2591 df-cleq 2597 df-clel 2600 df-nfc 2734 df-ne 2776 df-nel 2777 df-ral 2895 df-rex 2896 df-reu 2897 df-rmo 2898 df-rab 2899 df-v 3169 df-sbc 3397 df-csb 3494 df-dif 3537 df-un 3539 df-in 3541 df-ss 3548 df-pss 3550 df-nul 3869 df-if 4031 df-pw 4104 df-sn 4120 df-pr 4122 df-tp 4124 df-op 4126 df-ot 4128 df-uni 4362 df-int 4400 df-iun 4446 df-iin 4447 df-br 4573 df-opab 4633 df-mpt 4634 df-tr 4670 df-eprel 4934 df-id 4938 df-po 4944 df-so 4945 df-fr 4982 df-we 4984 df-xp 5029 df-rel 5030 df-cnv 5031 df-co 5032 df-dm 5033 df-rn 5034 df-res 5035 df-ima 5036 df-pred 5578 df-ord 5624 df-on 5625 df-lim 5626 df-suc 5627 df-iota 5749 df-fun 5787 df-fn 5788 df-f 5789 df-f1 5790 df-fo 5791 df-f1o 5792 df-fv 5793 df-riota 6484 df-ov 6525 df-oprab 6526 df-mpt2 6527 df-of 6767 df-om 6930 df-1st 7031 df-2nd 7032 df-tpos 7211 df-undef 7258 df-wrecs 7266 df-recs 7327 df-rdg 7365 df-1o 7419 df-oadd 7423 df-er 7601 df-map 7718 df-en 7814 df-dom 7815 df-sdom 7816 df-fin 7817 df-pnf 9927 df-mnf 9928 df-xr 9929 df-ltxr 9930 df-le 9931 df-sub 10114 df-neg 10115 df-nn 10863 df-2 10921 df-3 10922 df-4 10923 df-5 10924 df-6 10925 df-n0 11135 df-z 11206 df-uz 11515 df-fz 12148 df-struct 15638 df-ndx 15639 df-slot 15640 df-base 15641 df-sets 15642 df-ress 15643 df-plusg 15722 df-mulr 15723 df-sca 15725 df-vsca 15726 df-0g 15866 df-mre 16010 df-mrc 16011 df-acs 16013 df-preset 16692 df-poset 16710 df-plt 16722 df-lub 16738 df-glb 16739 df-join 16740 df-meet 16741 df-p0 16803 df-p1 16804 df-lat 16810 df-clat 16872 df-mgm 17006 df-sgrp 17048 df-mnd 17059 df-submnd 17100 df-grp 17189 df-minusg 17190 df-sbg 17191 df-subg 17355 df-cntz 17514 df-oppg 17540 df-lsm 17815 df-cmn 17959 df-abl 17960 df-mgp 18254 df-ur 18266 df-ring 18313 df-oppr 18387 df-dvdsr 18405 df-unit 18406 df-invr 18436 df-dvr 18447 df-drng 18513 df-lmod 18629 df-lss 18695 df-lsp 18734 df-lvec 18865 df-lsatoms 33079 df-lshyp 33080 df-lcv 33122 df-lfl 33161 df-lkr 33189 df-ldual 33227 df-oposet 33279 df-ol 33281 df-oml 33282 df-covers 33369 df-ats 33370 df-atl 33401 df-cvlat 33425 df-hlat 33454 df-llines 33600 df-lplanes 33601 df-lvols 33602 df-lines 33603 df-psubsp 33605 df-pmap 33606 df-padd 33898 df-lhyp 34090 df-laut 34091 df-ldil 34206 df-ltrn 34207 df-trl 34262 df-tgrp 34847 df-tendo 34859 df-edring 34861 df-dveca 35107 df-disoa 35134 df-dvech 35184 df-dib 35244 df-dic 35278 df-dih 35334 df-doch 35453 df-djh 35500 df-lcdual 35692 df-mapd 35730

This theorem is referenced by: mapdh8ac 35883

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