mapdheq4 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdheq4		Structured version Visualization version GIF version

Theorem mapdheq4 36039

Description: Lemma for ~? mapdh . Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.)

**Hypotheses**
Ref	Expression
mapdh.q	⊢ 𝑄 = (0_g‘𝐶)
mapdh.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)})))))
mapdh.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdh.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdh.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdh.v	⊢ 𝑉 = (Base‘𝑈)
mapdh.s	⊢ − = (-_g‘𝑈)
mapdhc.o	⊢ 0 = (0_g‘𝑈)
mapdh.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdh.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdh.d	⊢ 𝐷 = (Base‘𝐶)
mapdh.r	⊢ 𝑅 = (-_g‘𝐶)
mapdh.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdh.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdhc.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdhcl.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdhe4.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdhe.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
mapdh.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
mapdh.yz	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
mapdh.eg	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
mapdh.ee	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)

**Assertion**
Ref	Expression
mapdheq4	⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸)

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

**Proof of Theorem mapdheq4**
Step	Hyp	Ref	Expression
1		mapdh.ee	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)
2		mapdh.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝐶)
3		mapdh.i	. . . . 5 ⊢ 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)})))))
4		mapdh.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		mapdh.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		mapdh.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		mapdh.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
8		mapdh.s	. . . . 5 ⊢ − = (-_g‘𝑈)
9		mapdhc.o	. . . . 5 ⊢ 0 = (0_g‘𝑈)
10		mapdh.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
11		mapdh.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
12		mapdh.d	. . . . 5 ⊢ 𝐷 = (Base‘𝐶)
13		mapdh.r	. . . . 5 ⊢ 𝑅 = (-_g‘𝐶)
14		mapdh.j	. . . . 5 ⊢ 𝐽 = (LSpan‘𝐶)
15		mapdh.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		mapdhc.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
17		mapdh.mn	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
18		mapdhcl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19		mapdhe.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
20	19	eldifad 3552	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
21	4, 6, 15	dvhlvec 35416	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LVec)
22		mapdhe4.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
23	18	eldifad 3552	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
24		mapdh.yz	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
25		mapdh.xn	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
26	7, 9, 10, 21, 22, 20, 23, 24, 25	lspindp1 18954	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
27	26	simpld 474	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
28	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 27	mapdhcl 36034	. . . . . 6 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
29	1, 28	eqeltrrd 2689	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐷)
30	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 29, 27	mapdheq 36035	. . . 4 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)}))))
31	1, 30	mpbid 221	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)})))
32	31	simpld 474	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}))
33		mapdh.eg	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
34	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 19, 25, 24, 33, 1	mapdheq4lem 36038	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 − 𝑍)})) = (𝐽‘{(𝐺𝑅𝐸)}))
35	22	eldifad 3552	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
36	7, 9, 10, 21, 35, 19, 23, 24, 25	lspindp2 18956	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})))
37	36	simpld 474	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
38	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 35, 37	mapdhcl 36034	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
39	33, 38	eqeltrrd 2689	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
40	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 39, 37	mapdheq 36035	. . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))))
41	33, 40	mpbid 221	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))
42	41	simpld 474	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
43	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 42, 22, 19, 29, 24	mapdheq 36035	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑌 − 𝑍)})) = (𝐽‘{(𝐺𝑅𝐸)}))))
44	32, 34, 43	mpbir2and 959	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∖ cdif 3537 ifcif 4036 {csn 4125 {cpr 4127 ⟨cotp 4133 ↦ cmpt 4643 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 Basecbs 15695 0_gc0g 15923 -_gcsg 17247 LSpanclspn 18792 HLchlt 33655 LHypclh 34288 DVecHcdvh 35385 LCDualclcd 35893 mapdcmpd 35931

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-mre 16069 df-mrc 16070 df-acs 16072 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-oppg 17599 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-lshyp 33282 df-lcv 33324 df-lfl 33363 df-lkr 33391 df-ldual 33429 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tgrp 35049 df-tendo 35061 df-edring 35063 df-dveca 35309 df-disoa 35336 df-dvech 35386 df-dib 35446 df-dic 35480 df-dih 35536 df-doch 35655 df-djh 35702 df-lcdual 35894 df-mapd 35932

This theorem is referenced by: mapdh7dN 36057 mapdh75d 36061 mapdh8a 36082 hdmap1eq4N 36114

W3C validator