hdmap14lem13 - Mathbox for Norm Megill

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Theorem hdmap14lem13 41859

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 41858	. 2 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
18		velsn 4593	. . . . . 6 ⊢ (𝑦 ∈ { 0 } ↔ 𝑦 = 0 )
19	1, 7, 10	lcdlmod 41571	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ LMod)
20		eqid 2729	. . . . . . . . . 10 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
21	12, 8, 13, 20	lmodvs0 20799	. . . . . . . . 9 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐴) → (𝐺 ∙ (0_g‘𝐶)) = (0_g‘𝐶))
22	19, 16, 21	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∙ (0_g‘𝐶)) = (0_g‘𝐶))
23	1, 2, 14, 7, 20, 9, 10	hdmapval0 41812	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘ 0 ) = (0_g‘𝐶))
24	23	oveq2d 7365	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∙ (𝑆‘ 0 )) = (𝐺 ∙ (0_g‘𝐶)))
25	1, 2, 10	dvhlmod 41089	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
26	5, 4, 6, 14	lmodvs0 20799	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵) → (𝐹 · 0 ) = 0 )
27	25, 11, 26	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 · 0 ) = 0 )
28	27	fveq2d 6826	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (𝑆‘ 0 ))
29	28, 23	eqtrd 2764	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (0_g‘𝐶))
30	22, 24, 29	3eqtr4rd 2775	. . . . . . 7 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (𝐺 ∙ (𝑆‘ 0 )))
31		oveq2 7357	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝐹 · 𝑦) = (𝐹 · 0 ))
32	31	fveq2d 6826	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝑆‘(𝐹 · 𝑦)) = (𝑆‘(𝐹 · 0 )))
33		fveq2 6822	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑆‘𝑦) = (𝑆‘ 0 ))
34	33	oveq2d 7365	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝐺 ∙ (𝑆‘𝑦)) = (𝐺 ∙ (𝑆‘ 0 )))
35	32, 34	eqeq12d 2745	. . . . . . 7 ⊢ (𝑦 = 0 → ((𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (𝑆‘(𝐹 · 0 )) = (𝐺 ∙ (𝑆‘ 0 ))))
36	30, 35	syl5ibrcom 247	. . . . . 6 ⊢ (𝜑 → (𝑦 = 0 → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
37	18, 36	biimtrid 242	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ { 0 } → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
38	37	ralrimiv 3120	. . . 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 4148	. . 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 289	. 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 20794	. . . 4 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉)
43		difsnid 4761	. . . 4 ⊢ ( 0 ∈ 𝑉 → ((𝑉 ∖ { 0 }) ∪ { 0 }) = 𝑉)
44	25, 42, 43	3syl 18	. . 3 ⊢ (𝜑 → ((𝑉 ∖ { 0 }) ∪ { 0 }) = 𝑉)
45	44	raleqdv 3289	. 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 305	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∖ cdif 3900 ∪ cun 3901 {csn 4577 ‘cfv 6482 (class class class)co 7349 Basecbs 17120 Scalarcsca 17164 ·_𝑠 cvsca 17165 0_gc0g 17343 LModclmod 20763 HLchlt 39329 LHypclh 39963 DVecHcdvh 41057 LCDualclcd 41565 HDMapchdma 41771

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-riotaBAD 38932

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-tpos 8159 df-undef 8206 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-n0 12385 df-z 12472 df-uz 12736 df-fz 13411 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-grp 18815 df-minusg 18816 df-sbg 18817 df-subg 19002 df-cntz 19196 df-oppg 19225 df-lsm 19515 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-nzr 20398 df-rlreg 20579 df-domn 20580 df-drng 20616 df-lmod 20765 df-lss 20835 df-lsp 20875 df-lvec 21007 df-lsatoms 38955 df-lshyp 38956 df-lcv 38998 df-lfl 39037 df-lkr 39065 df-ldual 39103 df-oposet 39155 df-ol 39157 df-oml 39158 df-covers 39245 df-ats 39246 df-atl 39277 df-cvlat 39301 df-hlat 39330 df-llines 39477 df-lplanes 39478 df-lvols 39479 df-lines 39480 df-psubsp 39482 df-pmap 39483 df-padd 39775 df-lhyp 39967 df-laut 39968 df-ldil 40083 df-ltrn 40084 df-trl 40138 df-tgrp 40722 df-tendo 40734 df-edring 40736 df-dveca 40982 df-disoa 41008 df-dvech 41058 df-dib 41118 df-dic 41152 df-dih 41208 df-doch 41327 df-djh 41374 df-lcdual 41566 df-mapd 41604 df-hvmap 41736 df-hdmap1 41772 df-hdmap 41773

This theorem is referenced by: hdmap14lem14 41860

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