hdmap14lem13 - Mathbox for Norm Megill

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Theorem hdmap14lem13 39821

Description: Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)

**Hypotheses**
Ref	Expression
hdmap14lem12.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap14lem12.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap14lem12.v	⊢ 𝑉 = (Base‘𝑈)
hdmap14lem12.t	⊢ · = ( ·_𝑠 ‘𝑈)
hdmap14lem12.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmap14lem12.b	⊢ 𝐵 = (Base‘𝑅)
hdmap14lem12.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap14lem12.e	⊢ ∙ = ( ·_𝑠 ‘𝐶)
hdmap14lem12.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap14lem12.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap14lem12.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
hdmap14lem12.p	⊢ 𝑃 = (Scalar‘𝐶)
hdmap14lem12.a	⊢ 𝐴 = (Base‘𝑃)
hdmap14lem12.o	⊢ 0 = (0_g‘𝑈)
hdmap14lem12.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap14lem12.g	⊢ (𝜑 → 𝐺 ∈ 𝐴)

**Assertion**
Ref	Expression
hdmap14lem13	⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ 𝑉 (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))

Distinct variable groups: 𝑦,𝐴 𝑦, ∙ 𝑦,𝐹 𝑦,𝐺 𝑦, 0 𝑦,𝑆 𝑦, · 𝑦,𝑈 𝑦,𝑉 𝑦,𝑋 𝜑,𝑦 Allowed substitution hints: 𝐵(𝑦) 𝐶(𝑦) 𝑃(𝑦) 𝑅(𝑦) 𝐻(𝑦) 𝐾(𝑦) 𝑊(𝑦)

**Proof of Theorem hdmap14lem13**
Step	Hyp	Ref	Expression
1		hdmap14lem12.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap14lem12.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap14lem12.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap14lem12.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
5		hdmap14lem12.r	. . 3 ⊢ 𝑅 = (Scalar‘𝑈)
6		hdmap14lem12.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
7		hdmap14lem12.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		hdmap14lem12.e	. . 3 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
9		hdmap14lem12.s	. . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
10		hdmap14lem12.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		hdmap14lem12.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
12		hdmap14lem12.p	. . 3 ⊢ 𝑃 = (Scalar‘𝐶)
13		hdmap14lem12.a	. . 3 ⊢ 𝐴 = (Base‘𝑃)
14		hdmap14lem12.o	. . 3 ⊢ 0 = (0_g‘𝑈)
15		hdmap14lem12.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
16		hdmap14lem12.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	hdmap14lem12 39820	. 2 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
18		velsn 4574	. . . . . 6 ⊢ (𝑦 ∈ { 0 } ↔ 𝑦 = 0 )
19	1, 7, 10	lcdlmod 39533	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ LMod)
20		eqid 2738	. . . . . . . . . 10 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
21	12, 8, 13, 20	lmodvs0 20072	. . . . . . . . 9 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐴) → (𝐺 ∙ (0_g‘𝐶)) = (0_g‘𝐶))
22	19, 16, 21	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∙ (0_g‘𝐶)) = (0_g‘𝐶))
23	1, 2, 14, 7, 20, 9, 10	hdmapval0 39774	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘ 0 ) = (0_g‘𝐶))
24	23	oveq2d 7271	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∙ (𝑆‘ 0 )) = (𝐺 ∙ (0_g‘𝐶)))
25	1, 2, 10	dvhlmod 39051	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
26	5, 4, 6, 14	lmodvs0 20072	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵) → (𝐹 · 0 ) = 0 )
27	25, 11, 26	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 · 0 ) = 0 )
28	27	fveq2d 6760	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (𝑆‘ 0 ))
29	28, 23	eqtrd 2778	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (0_g‘𝐶))
30	22, 24, 29	3eqtr4rd 2789	. . . . . . 7 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (𝐺 ∙ (𝑆‘ 0 )))
31		oveq2 7263	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝐹 · 𝑦) = (𝐹 · 0 ))
32	31	fveq2d 6760	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝑆‘(𝐹 · 𝑦)) = (𝑆‘(𝐹 · 0 )))
33		fveq2 6756	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑆‘𝑦) = (𝑆‘ 0 ))
34	33	oveq2d 7271	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝐺 ∙ (𝑆‘𝑦)) = (𝐺 ∙ (𝑆‘ 0 )))
35	32, 34	eqeq12d 2754	. . . . . . 7 ⊢ (𝑦 = 0 → ((𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (𝑆‘(𝐹 · 0 )) = (𝐺 ∙ (𝑆‘ 0 ))))
36	30, 35	syl5ibrcom 246	. . . . . 6 ⊢ (𝜑 → (𝑦 = 0 → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
37	18, 36	syl5bi 241	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ { 0 } → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
38	37	ralrimiv 3106	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ { 0 } (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))
39	38	biantrud 531	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ∧ ∀𝑦 ∈ { 0 } (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))))
40		ralunb 4121	. . 3 ⊢ (∀𝑦 ∈ ((𝑉 ∖ { 0 }) ∪ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ∧ ∀𝑦 ∈ { 0 } (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
41	39, 40	bitr4di 288	. 2 ⊢ (𝜑 → (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ ∀𝑦 ∈ ((𝑉 ∖ { 0 }) ∪ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
42	3, 14	lmod0vcl 20067	. . . 4 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉)
43		difsnid 4740	. . . 4 ⊢ ( 0 ∈ 𝑉 → ((𝑉 ∖ { 0 }) ∪ { 0 }) = 𝑉)
44	25, 42, 43	3syl 18	. . 3 ⊢ (𝜑 → ((𝑉 ∖ { 0 }) ∪ { 0 }) = 𝑉)
45	44	raleqdv 3339	. 2 ⊢ (𝜑 → (∀𝑦 ∈ ((𝑉 ∖ { 0 }) ∪ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ ∀𝑦 ∈ 𝑉 (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
46	17, 41, 45	3bitrd 304	1 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ 𝑉 (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∖ cdif 3880 ∪ cun 3881 {csn 4558 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 Scalarcsca 16891 ·_𝑠 cvsca 16892 0_gc0g 17067 LModclmod 20038 HLchlt 37291 LHypclh 37925 DVecHcdvh 39019 LCDualclcd 39527 HDMapchdma 39733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-mre 17212 df-mrc 17213 df-acs 17215 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-oppg 18865 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lsatoms 36917 df-lshyp 36918 df-lcv 36960 df-lfl 36999 df-lkr 37027 df-ldual 37065 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tgrp 38684 df-tendo 38696 df-edring 38698 df-dveca 38944 df-disoa 38970 df-dvech 39020 df-dib 39080 df-dic 39114 df-dih 39170 df-doch 39289 df-djh 39336 df-lcdual 39528 df-mapd 39566 df-hvmap 39698 df-hdmap1 39734 df-hdmap 39735

This theorem is referenced by: hdmap14lem14 39822

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