mapdh6iN - Mathbox for Norm Megill

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Theorem mapdh6iN 42486

Description: Lemmma for mapdh6N 42489. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)

**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 }))
mapdh.p	⊢ + = (+_g‘𝑈)
mapdh.a	⊢ ✚ = (+_g‘𝐶)
mapdh6i.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
mapdh6i.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh6i.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
mapdh6i.yz	⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))

**Assertion**
Ref	Expression
mapdh6iN	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

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,𝑍,𝑥 ✚ ,ℎ ℎ,𝐼,𝑥 + ,ℎ,𝑥 ℎ,𝑉 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) ✚ (𝑥) 𝑄(ℎ) 𝑈(𝑥) 𝐻(𝑥,ℎ) 𝐾(𝑥,ℎ) 𝑉(𝑥) 𝑊(𝑥,ℎ)

Proof of Theorem mapdh6iN Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapdh.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdh.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdh.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		mapdh.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
5		mapdh.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		mapdhcl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
7	6	eldifad 3916	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
8		mapdh6i.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3916	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10	1, 2, 3, 4, 5, 7, 9	dvh3dim 42188	. 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
11		mapdh.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
12		mapdh.i	. . . 4 ⊢ 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)})))))
13		mapdh.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
14		mapdh.s	. . . 4 ⊢ − = (-_g‘𝑈)
15		mapdhc.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
16		mapdh.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
17		mapdh.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
18		mapdh.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
19		mapdh.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
20	5	3ad2ant1 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21		mapdhc.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
22	21	3ad2ant1 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝐹 ∈ 𝐷)
23		mapdh.mn	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
24	23	3ad2ant1 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
25	6	3ad2ant1 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
26		mapdh.p	. . . 4 ⊢ + = (+_g‘𝑈)
27		mapdh.a	. . . 4 ⊢ ✚ = (+_g‘𝐶)
28		mapdh6i.xn	. . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
29	28	3ad2ant1 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
30		mapdh6i.yz	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
31	30	3ad2ant1 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
32	8	3ad2ant1 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
33		mapdh6i.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
34	33	3ad2ant1 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑍 ∈ (𝑉 ∖ { 0 }))
35		eqid 2761	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
36	1, 2, 5	dvhlmod 41852	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
37	36	3ad2ant1 1149	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LMod)
38	3, 35, 4, 36, 7, 9	lspprcl 21078	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
39	38	3ad2ant1 1149	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
40		simp2 1153	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ 𝑉)
41		simp3 1154	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
42	15, 35, 37, 39, 40, 41	lssneln0 21053	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ (𝑉 ∖ { 0 }))
43	11, 12, 1, 13, 2, 3, 14, 15, 4, 16, 17, 18, 19, 20, 22, 24, 25, 26, 27, 29, 31, 32, 34, 42, 41	mapdh6hN 42485	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
44	43	rexlimdv3a 3168	. 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))))
45	10, 44	mpd 16	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ∃wrex 3087 Vcvv 3453 ∖ cdif 3901 ifcif 4486 {csn 4588 {cpr 4590 ⟨cotp 4596 ↦ cmpt 5191 ‘cfv 6536 ℩crio 7366 (class class class)co 7410 1^st c1st 7983 2^nd c2nd 7984 Basecbs 17268 +_gcplusg 17309 0_gc0g 17491 -_gcsg 19001 LModclmod 20960 LSubSpclss 21031 LSpanclspn 21071 HLchlt 40092 LHypclh 40726 DVecHcdvh 41820 LCDualclcd 42328 mapdcmpd 42366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-riotaBAD 39695

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 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 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-ot 4597 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7862 df-1st 7985 df-2nd 7986 df-tpos 8221 df-undef 8268 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-n0 12504 df-z 12591 df-uz 12862 df-fz 13535 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-sca 17325 df-vsca 17326 df-0g 17493 df-mre 17637 df-mrc 17638 df-acs 17640 df-proset 18349 df-poset 18368 df-plt 18383 df-lub 18399 df-glb 18400 df-join 18401 df-meet 18402 df-p0 18478 df-p1 18479 df-lat 18487 df-clat 18554 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-grp 19002 df-minusg 19003 df-sbg 19004 df-subg 19188 df-cntz 19386 df-oppg 19415 df-lsm 19705 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-invr 20469 df-dvr 20482 df-nzr 20595 df-rlreg 20778 df-domn 20779 df-drng 20814 df-lmod 20962 df-lss 21032 df-lsp 21072 df-lvec 21203 df-lsatoms 39718 df-lshyp 39719 df-lcv 39761 df-lfl 39800 df-lkr 39828 df-ldual 39866 df-oposet 39918 df-ol 39920 df-oml 39921 df-covers 40008 df-ats 40009 df-atl 40040 df-cvlat 40064 df-hlat 40093 df-llines 40240 df-lplanes 40241 df-lvols 40242 df-lines 40243 df-psubsp 40245 df-pmap 40246 df-padd 40538 df-lhyp 40730 df-laut 40731 df-ldil 40846 df-ltrn 40847 df-trl 40901 df-tgrp 41485 df-tendo 41497 df-edring 41499 df-dveca 41745 df-disoa 41771 df-dvech 41821 df-dib 41881 df-dic 41915 df-dih 41971 df-doch 42090 df-djh 42137 df-lcdual 42329 df-mapd 42367

This theorem is referenced by: mapdh6jN 42487

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