mapdh8b - Mathbox for Norm Megill

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Theorem mapdh8b 39794

Description: Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-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))
mapdh8b.f	⊢ (𝜑 → 𝐺 ∈ 𝐷)
mapdh8b.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
mapdh8b.a	⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑤⟩) = 𝐸)
mapdh8b.x	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh8b.y	⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
mapdh8b.yz	⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))
mapdh8b.xt	⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
mapdh8b.vw	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
mapdh8b.e	⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
mapdh8b.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))

**Assertion**
Ref	Expression
mapdh8b	⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑌, 𝐺, 𝑇⟩))

Distinct variable groups: 𝑥,ℎ, − 0 ,ℎ,𝑥 𝐶,ℎ 𝐷,ℎ,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 𝑤,ℎ,𝑥 Allowed substitution hints: 𝜑(𝑥,𝑤) 𝐶(𝑥,𝑤) 𝐷(𝑤) 𝑄(𝑤,ℎ) 𝑅(𝑤) 𝑇(𝑤) 𝑈(𝑥,𝑤) 𝐸(𝑤) 𝐺(𝑤) 𝐻(𝑥,𝑤,ℎ) 𝐼(𝑥,𝑤) 𝐽(𝑤) 𝐾(𝑥,𝑤,ℎ) 𝑀(𝑤) − (𝑤) 𝑁(𝑤) 𝑉(𝑥,𝑤,ℎ) 𝑊(𝑥,𝑤,ℎ) 𝑋(𝑤) 𝑌(𝑤) 0 (𝑤)

**Proof of Theorem mapdh8b**
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		mapdh8b.f	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
16		mapdh8b.mn	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
17		mapdh8b.a	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑤⟩) = 𝐸)
18		mapdh8b.x	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
19		mapdh8b.y	. 2 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
20		mapdh8b.yz	. 2 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))
21		mapdh8b.xt	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
22	1, 2, 14	dvhlvec 39123	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec)
23	18	eldifad 3899	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
24	19	eldifad 3899	. . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
25	21	eldifad 3899	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
26		mapdh8b.e	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
27		mapdh8b.xn	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))
28	3, 6, 22, 23, 24, 25, 26, 27	lspindp5 39784	. . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ (𝑁‘{𝑌, 𝑤}))
29		prcom 4668	. . . . . 6 ⊢ {𝑤, 𝑇} = {𝑇, 𝑤}
30	29	fveq2i 6777	. . . . 5 ⊢ (𝑁‘{𝑤, 𝑇}) = (𝑁‘{𝑇, 𝑤})
31	30	eleq2i 2830	. . . 4 ⊢ (𝑌 ∈ (𝑁‘{𝑤, 𝑇}) ↔ 𝑌 ∈ (𝑁‘{𝑇, 𝑤}))
32	22	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑈 ∈ LVec)
33	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
34	25	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑇 ∈ 𝑉)
35	24	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑤 ∈ 𝑉)
36		mapdh8b.vw	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
37	36	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
38		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑌 ∈ (𝑁‘{𝑇, 𝑤}))
39	3, 5, 6, 32, 33, 34, 35, 37, 38	lspexch 20391	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤}))
40	39	ex 413	. . . 4 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘{𝑇, 𝑤}) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤})))
41	31, 40	syl5bi 241	. . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘{𝑤, 𝑇}) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤})))
42	28, 41	mtod 197	. 2 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑤, 𝑇}))
43	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 42	mapdh8a 39789	1 ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑌, 𝐺, 𝑇⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 ∖ cdif 3884 ifcif 4459 {csn 4561 {cpr 4563 ⟨cotp 4569 ↦ cmpt 5157 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 Basecbs 16912 0_gc0g 17150 -_gcsg 18579 LSpanclspn 20233 LVecclvec 20364 HLchlt 37364 LHypclh 37998 DVecHcdvh 39092 LCDualclcd 39600 mapdcmpd 39638

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-mre 17295 df-mrc 17296 df-acs 17298 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-oppg 18950 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-lshyp 36991 df-lcv 37033 df-lfl 37072 df-lkr 37100 df-ldual 37138 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tgrp 38757 df-tendo 38769 df-edring 38771 df-dveca 39017 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243 df-doch 39362 df-djh 39409 df-lcdual 39601 df-mapd 39639

This theorem is referenced by: mapdh8c 39795 mapdh8d0N 39796 mapdh8d 39797

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