mapdh8c - Mathbox for Norm Megill

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Theorem mapdh8c 37856

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 37184	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec)
23	18	eldifad 3810	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
24		mapdh8c.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
25	24	eldifad 3810	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
26	19	eldifad 3810	. . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
27		mapdh8c.xn	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))
28	3, 6, 22, 23, 25, 26, 27	lspindpi 19492	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤})))
29	28	simprd 491	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤}))
30	21	eldifad 3810	. . 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 19489	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑇}))
34	22	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑈 ∈ LVec)
35	24	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
36	23	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑋 ∈ 𝑉)
37	26	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑤 ∈ 𝑉)
38		mapdh8c.vw	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
39	38	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
40		simpr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑌 ∈ (𝑁‘{𝑋, 𝑤}))
41	3, 5, 6, 34, 35, 36, 37, 39, 40	lspexch 19489	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))
42	27, 41	mtand 852	. 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 37855	1 ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 Vcvv 3414 ∖ cdif 3795 ifcif 4306 {csn 4397 {cpr 4399 ⟨cotp 4405 ↦ cmpt 4952 ‘cfv 6123 ℩crio 6865 (class class class)co 6905 1^st c1st 7426 2^nd c2nd 7427 Basecbs 16222 0_gc0g 16453 -_gcsg 17778 LSpanclspn 19330 LVecclvec 19461 HLchlt 35425 LHypclh 36059 DVecHcdvh 37153 LCDualclcd 37661 mapdcmpd 37699

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-riotaBAD 35028

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-ot 4406 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-undef 7664 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-0g 16455 df-mre 16599 df-mrc 16600 df-acs 16602 df-proset 17281 df-poset 17299 df-plt 17311 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-p1 17393 df-lat 17399 df-clat 17461 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 df-oppg 18126 df-lsm 18402 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-oppr 18977 df-dvdsr 18995 df-unit 18996 df-invr 19026 df-dvr 19037 df-drng 19105 df-lmod 19221 df-lss 19289 df-lsp 19331 df-lvec 19462 df-lsatoms 35051 df-lshyp 35052 df-lcv 35094 df-lfl 35133 df-lkr 35161 df-ldual 35199 df-oposet 35251 df-ol 35253 df-oml 35254 df-covers 35341 df-ats 35342 df-atl 35373 df-cvlat 35397 df-hlat 35426 df-llines 35573 df-lplanes 35574 df-lvols 35575 df-lines 35576 df-psubsp 35578 df-pmap 35579 df-padd 35871 df-lhyp 36063 df-laut 36064 df-ldil 36179 df-ltrn 36180 df-trl 36234 df-tgrp 36818 df-tendo 36830 df-edring 36832 df-dveca 37078 df-disoa 37104 df-dvech 37154 df-dib 37214 df-dic 37248 df-dih 37304 df-doch 37423 df-djh 37470 df-lcdual 37662 df-mapd 37700

This theorem is referenced by: mapdh8d0N 37857 mapdh8d 37858

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