mapdh8c - Mathbox for Norm Megill

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Theorem mapdh8c 41953

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))
mapdh8c.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh8c.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdh8c.a	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) = 𝐸)
mapdh8c.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdh8c.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh8c.xt	⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
mapdh8c.yz	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
mapdh8c.w	⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
mapdh8c.wt	⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))
mapdh8c.ut	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))
mapdh8c.vw	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
mapdh8c.e	⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
mapdh8c.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))

**Assertion**
Ref	Expression
mapdh8c	⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 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,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) 𝐹(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 mapdh8c**
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		mapdh8c.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
16		mapdh8c.mn	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
17		mapdh8c.a	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) = 𝐸)
18		mapdh8c.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19		mapdh8c.w	. 2 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
20		mapdh8c.wt	. 2 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))
21		mapdh8c.xt	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
22	1, 2, 14	dvhlvec 41281	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec)
23	18	eldifad 3910	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
24		mapdh8c.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
25	24	eldifad 3910	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
26	19	eldifad 3910	. . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
27		mapdh8c.xn	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))
28	3, 6, 22, 23, 25, 26, 27	lspindpi 21078	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤})))
29	28	simprd 495	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤}))
30	21	eldifad 3910	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
31		mapdh8c.ut	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))
32		mapdh8c.e	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
33	3, 5, 6, 22, 18, 25, 30, 31, 32	lspexch 21075	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑇}))
34	22	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑈 ∈ LVec)
35	24	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
36	23	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑋 ∈ 𝑉)
37	26	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑤 ∈ 𝑉)
38		mapdh8c.vw	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
39	38	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
40		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑌 ∈ (𝑁‘{𝑋, 𝑤}))
41	3, 5, 6, 34, 35, 36, 37, 39, 40	lspexch 21075	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))
42	27, 41	mtand 815	. 2 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑤}))
43	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 29, 33, 42	mapdh8b 41952	1 ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 Vcvv 3437 ∖ cdif 3895 ifcif 4476 {csn 4577 {cpr 4579 ⟨cotp 4585 ↦ cmpt 5176 ‘cfv 6489 ℩crio 7311 (class class class)co 7355 1^st c1st 7928 2^nd c2nd 7929 Basecbs 17127 0_gc0g 17350 -_gcsg 18856 LSpanclspn 20913 LVecclvec 21045 HLchlt 39522 LHypclh 40156 DVecHcdvh 41250 LCDualclcd 41758 mapdcmpd 41796

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-riotaBAD 39125

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-tpos 8165 df-undef 8212 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-n0 12393 df-z 12480 df-uz 12743 df-fz 13415 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-sca 17184 df-vsca 17185 df-0g 17352 df-mre 17496 df-mrc 17497 df-acs 17499 df-proset 18208 df-poset 18227 df-plt 18242 df-lub 18258 df-glb 18259 df-join 18260 df-meet 18261 df-p0 18337 df-p1 18338 df-lat 18346 df-clat 18413 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-grp 18857 df-minusg 18858 df-sbg 18859 df-subg 19044 df-cntz 19237 df-oppg 19266 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20067 df-rng 20079 df-ur 20108 df-ring 20161 df-oppr 20264 df-dvdsr 20284 df-unit 20285 df-invr 20315 df-dvr 20328 df-nzr 20437 df-rlreg 20618 df-domn 20619 df-drng 20655 df-lmod 20804 df-lss 20874 df-lsp 20914 df-lvec 21046 df-lsatoms 39148 df-lshyp 39149 df-lcv 39191 df-lfl 39230 df-lkr 39258 df-ldual 39296 df-oposet 39348 df-ol 39350 df-oml 39351 df-covers 39438 df-ats 39439 df-atl 39470 df-cvlat 39494 df-hlat 39523 df-llines 39670 df-lplanes 39671 df-lvols 39672 df-lines 39673 df-psubsp 39675 df-pmap 39676 df-padd 39968 df-lhyp 40160 df-laut 40161 df-ldil 40276 df-ltrn 40277 df-trl 40331 df-tgrp 40915 df-tendo 40927 df-edring 40929 df-dveca 41175 df-disoa 41201 df-dvech 41251 df-dib 41311 df-dic 41345 df-dih 41401 df-doch 41520 df-djh 41567 df-lcdual 41759 df-mapd 41797

This theorem is referenced by: mapdh8d0N 41954 mapdh8d 41955

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