mapdh8g - Mathbox for Norm Megill

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Theorem mapdh8g 41782

Description: Part of Part (8) in [Baer] p. 48. Eliminate 𝑋 ∈ (𝑁‘{𝑌, 𝑇}). TODO: break out 𝑇 ≠ 0 in mapdh8e 41781 so we can share hypotheses. Also, look at hypothesis sharing for earlier mapdh8* and mapdh75* stuff. (Contributed by NM, 10-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))
mapdh8e.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh8e.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdh8e.eg	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
mapdh8e.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdh8e.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh8e.t	⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
mapdh8e.xy	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdh8e.xt	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))
mapdh8e.yt	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))

**Assertion**
Ref	Expression
mapdh8g	⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Distinct variable groups: 𝑥,ℎ, − 0 ,ℎ,𝑥 𝐶,ℎ 𝐷,ℎ,𝑥 ℎ,𝐹,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 𝑥,𝐼 ℎ,𝑉 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) 𝑄(ℎ) 𝑈(𝑥) 𝐻(𝑥,ℎ) 𝐾(𝑥,ℎ) 𝑉(𝑥) 𝑊(𝑥,ℎ)

**Proof of Theorem mapdh8g**
Step	Hyp	Ref	Expression
1		mapdh8a.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdh8a.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdh8a.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		mapdh8a.s	. . 3 ⊢ − = (-_g‘𝑈)
5		mapdh8a.o	. . 3 ⊢ 0 = (0_g‘𝑈)
6		mapdh8a.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
7		mapdh8a.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		mapdh8a.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
9		mapdh8a.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
10		mapdh8a.q	. . 3 ⊢ 𝑄 = (0_g‘𝐶)
11		mapdh8a.j	. . 3 ⊢ 𝐽 = (LSpan‘𝐶)
12		mapdh8a.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		mapdh8a.i	. . 3 ⊢ 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	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		mapdh8e.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝐹 ∈ 𝐷)
18		mapdh8e.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
19	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
20		mapdh8e.eg	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
21	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
22		mapdh8e.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
23	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
24		mapdh8e.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
25	24	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
26		mapdh8e.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
27	26	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑇 ∈ (𝑉 ∖ { 0 }))
28		mapdh8e.xy	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
29	28	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30		mapdh8e.xt	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))
31	30	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))
32		mapdh8e.yt	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
33	32	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
34		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 34	mapdh8e 41781	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))
36	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
37	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝐹 ∈ 𝐷)
38	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
39	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
40	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
41	24	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
42	32	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
43	26	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑇 ∈ (𝑉 ∖ { 0 }))
44		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
45	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 36, 37, 38, 39, 40, 41, 42, 43, 44	mapdh8a 41772	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))
46	35, 45	pm2.61dan 813	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2940 Vcvv 3481 ∖ cdif 3963 ifcif 4534 {csn 4634 {cpr 4636 ⟨cotp 4642 ↦ cmpt 5234 ‘cfv 6569 ℩crio 7394 (class class class)co 7438 1^st c1st 8020 2^nd c2nd 8021 Basecbs 17254 0_gc0g 17495 -_gcsg 18975 LSpanclspn 20996 HLchlt 39346 LHypclh 39981 DVecHcdvh 41075 LCDualclcd 41583 mapdcmpd 41621

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239 ax-riotaBAD 38949

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-ot 4643 df-uni 4916 df-int 4955 df-iun 5001 df-iin 5002 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-of 7704 df-om 7895 df-1st 8022 df-2nd 8023 df-tpos 8259 df-undef 8306 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-2o 8515 df-er 8753 df-map 8876 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-nn 12274 df-2 12336 df-3 12337 df-4 12338 df-5 12339 df-6 12340 df-n0 12534 df-z 12621 df-uz 12886 df-fz 13554 df-struct 17190 df-sets 17207 df-slot 17225 df-ndx 17237 df-base 17255 df-ress 17284 df-plusg 17320 df-mulr 17321 df-sca 17323 df-vsca 17324 df-0g 17497 df-mre 17640 df-mrc 17641 df-acs 17643 df-proset 18361 df-poset 18380 df-plt 18397 df-lub 18413 df-glb 18414 df-join 18415 df-meet 18416 df-p0 18492 df-p1 18493 df-lat 18499 df-clat 18566 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-cntz 19357 df-oppg 19386 df-lsm 19678 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-nzr 20539 df-rlreg 20720 df-domn 20721 df-drng 20757 df-lmod 20886 df-lss 20957 df-lsp 20997 df-lvec 21129 df-lsatoms 38972 df-lshyp 38973 df-lcv 39015 df-lfl 39054 df-lkr 39082 df-ldual 39120 df-oposet 39172 df-ol 39174 df-oml 39175 df-covers 39262 df-ats 39263 df-atl 39294 df-cvlat 39318 df-hlat 39347 df-llines 39495 df-lplanes 39496 df-lvols 39497 df-lines 39498 df-psubsp 39500 df-pmap 39501 df-padd 39793 df-lhyp 39985 df-laut 39986 df-ldil 40101 df-ltrn 40102 df-trl 40156 df-tgrp 40740 df-tendo 40752 df-edring 40754 df-dveca 41000 df-disoa 41026 df-dvech 41076 df-dib 41136 df-dic 41170 df-dih 41226 df-doch 41345 df-djh 41392 df-lcdual 41584 df-mapd 41622

This theorem is referenced by: mapdh8i 41783

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