mapdh6bN - Mathbox for Norm Megill

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Theorem mapdh6bN 42007

Description: Lemmma for mapdh6N 42017. (Contributed by NM, 24-Apr-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)
mapdh6b.y	⊢ (𝜑 → 𝑌 = 0 )
mapdh6b.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
mapdh6b.ne	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

**Assertion**
Ref	Expression
mapdh6bN	⊢ (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 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) + (𝑥,ℎ) ✚ (𝑥,ℎ) 𝑄(ℎ) 𝑈(𝑥) 𝐻(𝑥,ℎ) 𝐼(𝑥,ℎ) 𝐾(𝑥,ℎ) 𝑉(𝑥,ℎ) 𝑊(𝑥,ℎ)

**Proof of Theorem mapdh6bN**
Step	Hyp	Ref	Expression
1		mapdh.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdh.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
3		mapdh.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	lcdlmod 41862	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		lmodgrp 20818	. . . 4 ⊢ (𝐶 ∈ LMod → 𝐶 ∈ Grp)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Grp)
7		mapdh.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
8		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)})))))
9		mapdh.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
10		mapdh.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
11		mapdh.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
12		mapdh.s	. . . 4 ⊢ − = (-_g‘𝑈)
13		mapdhc.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
14		mapdh.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
15		mapdh.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
16		mapdh.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
17		mapdh.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
18		mapdhc.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
19		mapdh.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
20		mapdhcl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21		mapdh6b.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
22	1, 10, 3	dvhlvec 41379	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
23	20	eldifad 3913	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
24		mapdh6b.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 = 0 )
25	1, 10, 3	dvhlmod 41380	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
26	11, 13	lmod0vcl 20842	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉)
27	25, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑉)
28	24, 27	eqeltrd 2836	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
29		mapdh6b.ne	. . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
30	11, 14, 22, 23, 28, 21, 29	lspindpi 21087	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
31	30	simprd 495	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
32	7, 8, 1, 9, 10, 11, 12, 13, 14, 2, 15, 16, 17, 3, 18, 19, 20, 21, 31	mapdhcl 41997	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
33		mapdh.a	. . . 4 ⊢ ✚ = (+_g‘𝐶)
34	15, 33, 7	grplid 18897	. . 3 ⊢ ((𝐶 ∈ Grp ∧ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷) → (𝑄 ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))
35	6, 32, 34	syl2anc 584	. 2 ⊢ (𝜑 → (𝑄 ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))
36	24	oteq3d 4843	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝐹, 𝑌⟩ = ⟨𝑋, 𝐹, 0 ⟩)
37	36	fveq2d 6838	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐼‘⟨𝑋, 𝐹, 0 ⟩))
38	7, 8, 13, 20, 18	mapdhval0 41995	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
39	37, 38	eqtrd 2771	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑄)
40	39	oveq1d 7373	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)) = (𝑄 ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
41	24	oveq1d 7373	. . . . 5 ⊢ (𝜑 → (𝑌 + 𝑍) = ( 0 + 𝑍))
42		lmodgrp 20818	. . . . . . 7 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp)
43	25, 42	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Grp)
44		mapdh.p	. . . . . . 7 ⊢ + = (+_g‘𝑈)
45	11, 44, 13	grplid 18897	. . . . . 6 ⊢ ((𝑈 ∈ Grp ∧ 𝑍 ∈ 𝑉) → ( 0 + 𝑍) = 𝑍)
46	43, 21, 45	syl2anc 584	. . . . 5 ⊢ (𝜑 → ( 0 + 𝑍) = 𝑍)
47	41, 46	eqtrd 2771	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = 𝑍)
48	47	oteq3d 4843	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩ = ⟨𝑋, 𝐹, 𝑍⟩)
49	48	fveq2d 6838	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))
50	35, 40, 49	3eqtr4rd 2782	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 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 Vcvv 3440 ∖ cdif 3898 ifcif 4479 {csn 4580 {cpr 4582 ⟨cotp 4588 ↦ cmpt 5179 ‘cfv 6492 ℩crio 7314 (class class class)co 7358 1^st c1st 7931 2^nd c2nd 7932 Basecbs 17136 +_gcplusg 17177 0_gc0g 17359 Grpcgrp 18863 -_gcsg 18865 LModclmod 20811 LSpanclspn 20922 HLchlt 39620 LHypclh 40254 DVecHcdvh 41348 LCDualclcd 41856 mapdcmpd 41894

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-riotaBAD 39223

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-ot 4589 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-0g 17361 df-mre 17505 df-mrc 17506 df-acs 17508 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-cntz 19246 df-oppg 19275 df-lsm 19565 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-nzr 20446 df-rlreg 20627 df-domn 20628 df-drng 20664 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lvec 21055 df-lsatoms 39246 df-lshyp 39247 df-lcv 39289 df-lfl 39328 df-lkr 39356 df-ldual 39394 df-oposet 39446 df-ol 39448 df-oml 39449 df-covers 39536 df-ats 39537 df-atl 39568 df-cvlat 39592 df-hlat 39621 df-llines 39768 df-lplanes 39769 df-lvols 39770 df-lines 39771 df-psubsp 39773 df-pmap 39774 df-padd 40066 df-lhyp 40258 df-laut 40259 df-ldil 40374 df-ltrn 40375 df-trl 40429 df-tgrp 41013 df-tendo 41025 df-edring 41027 df-dveca 41273 df-disoa 41299 df-dvech 41349 df-dib 41409 df-dic 41443 df-dih 41499 df-doch 41618 df-djh 41665 df-lcdual 41857 df-mapd 41895

This theorem is referenced by: mapdh6kN 42016

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