hdmap14lem2a - Mathbox for Norm Megill

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Theorem hdmap14lem2a 39808

Description: Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 0 so it can be used in hdmap14lem10 39818. (Contributed by NM, 31-May-2015.)

**Hypotheses**
Ref	Expression
hdmap14lem1a.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap14lem1a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap14lem1a.v	⊢ 𝑉 = (Base‘𝑈)
hdmap14lem1a.t	⊢ · = ( ·_𝑠 ‘𝑈)
hdmap14lem1a.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmap14lem1a.b	⊢ 𝐵 = (Base‘𝑅)
hdmap14lem1a.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap14lem2a.e	⊢ ∙ = ( ·_𝑠 ‘𝐶)
hdmap14lem1a.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap14lem2a.p	⊢ 𝑃 = (Scalar‘𝐶)
hdmap14lem2a.a	⊢ 𝐴 = (Base‘𝑃)
hdmap14lem1a.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap14lem1a.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap14lem3a.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hdmap14lem1a.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)

**Assertion**
Ref	Expression
hdmap14lem2a	⊢ (𝜑 → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))

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

**Proof of Theorem hdmap14lem2a**
Step	Hyp	Ref	Expression
1		fvoveq1 7278	. . . 4 ⊢ (𝐹 = (0_g‘𝑅) → (𝑆‘(𝐹 · 𝑋)) = (𝑆‘((0_g‘𝑅) · 𝑋)))
2	1	eqeq1d 2740	. . 3 ⊢ (𝐹 = (0_g‘𝑅) → ((𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ (𝑆‘((0_g‘𝑅) · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
3	2	rexbidv 3225	. 2 ⊢ (𝐹 = (0_g‘𝑅) → (∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃𝑔 ∈ 𝐴 (𝑆‘((0_g‘𝑅) · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
4		difss 4062	. . 3 ⊢ (𝐴 ∖ {(0_g‘𝑃)}) ⊆ 𝐴
5		hdmap14lem1a.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
6		hdmap14lem1a.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		hdmap14lem1a.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
8		hdmap14lem1a.t	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑈)
9		hdmap14lem1a.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈)
10		hdmap14lem1a.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
11		hdmap14lem1a.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
12		hdmap14lem2a.e	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
13		hdmap14lem1a.l	. . . . . 6 ⊢ 𝐿 = (LSpan‘𝐶)
14		hdmap14lem2a.p	. . . . . 6 ⊢ 𝑃 = (Scalar‘𝐶)
15		hdmap14lem2a.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝑃)
16		hdmap14lem1a.s	. . . . . 6 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
17		hdmap14lem1a.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
18	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ≠ (0_g‘𝑅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
19		hdmap14lem3a.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
20	19	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ≠ (0_g‘𝑅)) → 𝑋 ∈ 𝑉)
21		hdmap14lem1a.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ≠ (0_g‘𝑅)) → 𝐹 ∈ 𝐵)
23		eqid 2738	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
24		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ≠ (0_g‘𝑅)) → 𝐹 ≠ (0_g‘𝑅))
25	5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 23, 24	hdmap14lem1a 39807	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ≠ (0_g‘𝑅)) → (𝐿‘{(𝑆‘𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))}))
26	25	eqcomd 2744	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ≠ (0_g‘𝑅)) → (𝐿‘{(𝑆‘(𝐹 · 𝑋))}) = (𝐿‘{(𝑆‘𝑋)}))
27		eqid 2738	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
28		eqid 2738	. . . . . 6 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
29	5, 11, 17	lcdlvec 39532	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LVec)
30	5, 6, 17	dvhlmod 39051	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
31	7, 9, 8, 10	lmodvscl 20055	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉)
32	30, 21, 19, 31	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉)
33	5, 6, 7, 11, 27, 16, 17, 32	hdmapcl 39771	. . . . . 6 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) ∈ (Base‘𝐶))
34	5, 6, 7, 11, 27, 16, 17, 19	hdmapcl 39771	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶))
35	27, 14, 15, 28, 12, 13, 29, 33, 34	lspsneq 20299	. . . . 5 ⊢ (𝜑 → ((𝐿‘{(𝑆‘(𝐹 · 𝑋))}) = (𝐿‘{(𝑆‘𝑋)}) ↔ ∃𝑔 ∈ (𝐴 ∖ {(0_g‘𝑃)})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
36	35	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ≠ (0_g‘𝑅)) → ((𝐿‘{(𝑆‘(𝐹 · 𝑋))}) = (𝐿‘{(𝑆‘𝑋)}) ↔ ∃𝑔 ∈ (𝐴 ∖ {(0_g‘𝑃)})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
37	26, 36	mpbid 231	. . 3 ⊢ ((𝜑 ∧ 𝐹 ≠ (0_g‘𝑅)) → ∃𝑔 ∈ (𝐴 ∖ {(0_g‘𝑃)})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
38		ssrexv 3984	. . 3 ⊢ ((𝐴 ∖ {(0_g‘𝑃)}) ⊆ 𝐴 → (∃𝑔 ∈ (𝐴 ∖ {(0_g‘𝑃)})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
39	4, 37, 38	mpsyl 68	. 2 ⊢ ((𝜑 ∧ 𝐹 ≠ (0_g‘𝑅)) → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
40	5, 11, 17	lcdlmod 39533	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
41	14, 15, 28	lmod0cl 20064	. . . 4 ⊢ (𝐶 ∈ LMod → (0_g‘𝑃) ∈ 𝐴)
42	40, 41	syl 17	. . 3 ⊢ (𝜑 → (0_g‘𝑃) ∈ 𝐴)
43		eqid 2738	. . . . 5 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
44		eqid 2738	. . . . 5 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
45	5, 6, 43, 11, 44, 16, 17	hdmapval0 39774	. . . 4 ⊢ (𝜑 → (𝑆‘(0_g‘𝑈)) = (0_g‘𝐶))
46	7, 9, 8, 23, 43	lmod0vs 20071	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((0_g‘𝑅) · 𝑋) = (0_g‘𝑈))
47	30, 19, 46	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((0_g‘𝑅) · 𝑋) = (0_g‘𝑈))
48	47	fveq2d 6760	. . . 4 ⊢ (𝜑 → (𝑆‘((0_g‘𝑅) · 𝑋)) = (𝑆‘(0_g‘𝑈)))
49	27, 14, 12, 28, 44	lmod0vs 20071	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑋) ∈ (Base‘𝐶)) → ((0_g‘𝑃) ∙ (𝑆‘𝑋)) = (0_g‘𝐶))
50	40, 34, 49	syl2anc 583	. . . 4 ⊢ (𝜑 → ((0_g‘𝑃) ∙ (𝑆‘𝑋)) = (0_g‘𝐶))
51	45, 48, 50	3eqtr4d 2788	. . 3 ⊢ (𝜑 → (𝑆‘((0_g‘𝑅) · 𝑋)) = ((0_g‘𝑃) ∙ (𝑆‘𝑋)))
52		oveq1 7262	. . . 4 ⊢ (𝑔 = (0_g‘𝑃) → (𝑔 ∙ (𝑆‘𝑋)) = ((0_g‘𝑃) ∙ (𝑆‘𝑋)))
53	52	rspceeqv 3567	. . 3 ⊢ (((0_g‘𝑃) ∈ 𝐴 ∧ (𝑆‘((0_g‘𝑅) · 𝑋)) = ((0_g‘𝑃) ∙ (𝑆‘𝑋))) → ∃𝑔 ∈ 𝐴 (𝑆‘((0_g‘𝑅) · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
54	42, 51, 53	syl2anc 583	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 (𝑆‘((0_g‘𝑅) · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
55	3, 39, 54	pm2.61ne 3029	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 ∖ cdif 3880 ⊆ wss 3883 {csn 4558 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 Scalarcsca 16891 ·_𝑠 cvsca 16892 0_gc0g 17067 LModclmod 20038 LSpanclspn 20148 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: hdmap14lem10 39818 hdmap14lem11 39819 hdmap14lem12 39820

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