mapdh8c - Mathbox for Norm Megill

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Theorem mapdh8c 42840

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 42168	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec)
23	18	eldifad 3909	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
24		mapdh8c.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
25	24	eldifad 3909	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
26	19	eldifad 3909	. . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
27		mapdh8c.xn	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))
28	3, 6, 22, 23, 25, 26, 27	lspindpi 21411	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤})))
29	28	simprd 501	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤}))
30	21	eldifad 3909	. . 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 21408	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑇}))
34	22	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑈 ∈ LVec)
35	24	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
36	23	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑋 ∈ 𝑉)
37	26	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑤 ∈ 𝑉)
38		mapdh8c.vw	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
39	38	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
40		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑌 ∈ (𝑁‘{𝑋, 𝑤}))
41	3, 5, 6, 34, 35, 36, 37, 39, 40	lspexch 21408	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑤})) → 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))
42	27, 41	mtand 828	. 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 42839	1 ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 Vcvv 3451 ∖ cdif 3896 ifcif 4482 {csn 4584 {cpr 4586 ⟨cotp 4592 ↦ cmpt 5186 ‘cfv 6539 ℩crio 7377 (class class class)co 7421 1^st c1st 8000 2^nd c2nd 8001 Basecbs 17388 0_gc0g 17611 -_gcsg 19147 LSpanclspn 21247 LVecclvec 21378 HLchlt 40409 LHypclh 41043 DVecHcdvh 42137 LCDualclcd 42645 mapdcmpd 42683

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7752 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-riotaBAD 40012

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3900 df-un 3902 df-in 3904 df-ss 3915 df-pss 4035 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-tpos 8244 df-undef 8291 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-2o 8478 df-er 8718 df-map 8850 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-n0 12608 df-z 12695 df-uz 12967 df-fz 13641 df-struct 17326 df-sets 17343 df-slot 17361 df-ndx 17373 df-base 17389 df-ress 17410 df-plusg 17442 df-mulr 17443 df-sca 17445 df-vsca 17446 df-0g 17613 df-mre 17757 df-mrc 17758 df-acs 17760 df-proset 18469 df-poset 18488 df-plt 18503 df-lub 18519 df-glb 18520 df-join 18521 df-meet 18522 df-p0 18598 df-p1 18599 df-lat 18607 df-clat 18674 df-mgm 18817 df-sgrp 18909 df-mnd 18925 df-submnd 18980 df-grp 19148 df-minusg 19149 df-sbg 19150 df-subg 19334 df-cntz 19532 df-oppg 19561 df-lsm 19851 df-cmn 19997 df-abl 19998 df-mgp 20362 df-rng 20376 df-ur 20409 df-ring 20462 df-oppr 20568 df-dvdsr 20588 df-unit 20589 df-invr 20619 df-dvr 20632 df-nzr 20764 df-rlreg 20947 df-domn 20948 df-drng 20983 df-lmod 21138 df-lss 21208 df-lsp 21248 df-lvec 21379 df-lsatoms 40035 df-lshyp 40036 df-lcv 40078 df-lfl 40117 df-lkr 40145 df-ldual 40183 df-oposet 40235 df-ol 40237 df-oml 40238 df-covers 40325 df-ats 40326 df-atl 40357 df-cvlat 40381 df-hlat 40410 df-llines 40557 df-lplanes 40558 df-lvols 40559 df-lines 40560 df-psubsp 40562 df-pmap 40563 df-padd 40855 df-lhyp 41047 df-laut 41048 df-ldil 41163 df-ltrn 41164 df-trl 41218 df-tgrp 41802 df-tendo 41814 df-edring 41816 df-dveca 42062 df-disoa 42088 df-dvech 42138 df-dib 42198 df-dic 42232 df-dih 42288 df-doch 42407 df-djh 42454 df-lcdual 42646 df-mapd 42684

This theorem is used by: mapdh8d0N 42841 mapdh8d 42842

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