hdmap14lem4a - Mathbox for Norm Megill

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Theorem hdmap14lem4a 39885

Description: Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 39884 to provide a slightly simpler definition later. (Contributed by NM, 31-May-2015.)

**Hypotheses**
Ref	Expression
hdmap14lem1.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap14lem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap14lem1.v	⊢ 𝑉 = (Base‘𝑈)
hdmap14lem1.t	⊢ · = ( ·_𝑠 ‘𝑈)
hdmap14lem3.o	⊢ 0 = (0_g‘𝑈)
hdmap14lem1.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmap14lem1.b	⊢ 𝐵 = (Base‘𝑅)
hdmap14lem1.z	⊢ 𝑍 = (0_g‘𝑅)
hdmap14lem1.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap14lem2.e	⊢ ∙ = ( ·_𝑠 ‘𝐶)
hdmap14lem1.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap14lem2.p	⊢ 𝑃 = (Scalar‘𝐶)
hdmap14lem2.a	⊢ 𝐴 = (Base‘𝑃)
hdmap14lem2.q	⊢ 𝑄 = (0_g‘𝑃)
hdmap14lem1.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap14lem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap14lem3.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap14lem1.f	⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍}))

**Assertion**
Ref	Expression
hdmap14lem4a	⊢ (𝜑 → (∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))

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

**Proof of Theorem hdmap14lem4a**
Step	Hyp	Ref	Expression
1		hdmap14lem1.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap14lem1.u	. . . . . . . . 9 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap14lem1.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap14lem3.o	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑈)
5		hdmap14lem1.c	. . . . . . . . 9 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
6		eqid 2738	. . . . . . . . 9 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
7		eqid 2738	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		hdmap14lem1.s	. . . . . . . . 9 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
9		hdmap14lem1.k	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	1, 2, 9	dvhlmod 39124	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
11		hdmap14lem1.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍}))
12	11	eldifad 3899	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
13		hdmap14lem3.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3899	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
15		hdmap14lem1.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Scalar‘𝑈)
16		hdmap14lem1.t	. . . . . . . . . . . 12 ⊢ · = ( ·_𝑠 ‘𝑈)
17		hdmap14lem1.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑅)
18	3, 15, 16, 17	lmodvscl 20140	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉)
19	10, 12, 14, 18	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉)
20		eldifsni 4723	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐵 ∖ {𝑍}) → 𝐹 ≠ 𝑍)
21	11, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ≠ 𝑍)
22		eldifsni 4723	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
23	13, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ 0 )
24		hdmap14lem1.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (0_g‘𝑅)
25	1, 2, 9	dvhlvec 39123	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
26	3, 16, 15, 17, 24, 4, 25, 12, 14	lvecvsn0 20371	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 · 𝑋) ≠ 0 ↔ (𝐹 ≠ 𝑍 ∧ 𝑋 ≠ 0 )))
27	21, 23, 26	mpbir2and 710	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 · 𝑋) ≠ 0 )
28		eldifsn 4720	. . . . . . . . . 10 ⊢ ((𝐹 · 𝑋) ∈ (𝑉 ∖ { 0 }) ↔ ((𝐹 · 𝑋) ∈ 𝑉 ∧ (𝐹 · 𝑋) ≠ 0 ))
29	19, 27, 28	sylanbrc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ (𝑉 ∖ { 0 }))
30	1, 2, 3, 4, 5, 6, 7, 8, 9, 29	hdmapnzcl 39859	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) ∈ ((Base‘𝐶) ∖ {(0_g‘𝐶)}))
31		eldifsni 4723	. . . . . . . 8 ⊢ ((𝑆‘(𝐹 · 𝑋)) ∈ ((Base‘𝐶) ∖ {(0_g‘𝐶)}) → (𝑆‘(𝐹 · 𝑋)) ≠ (0_g‘𝐶))
32	30, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) ≠ (0_g‘𝐶))
33	32	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → (𝑆‘(𝐹 · 𝑋)) ≠ (0_g‘𝐶))
34		elsni 4578	. . . . . . . 8 ⊢ (𝑔 ∈ {𝑄} → 𝑔 = 𝑄)
35	34	oveq1d 7290	. . . . . . 7 ⊢ (𝑔 ∈ {𝑄} → (𝑔 ∙ (𝑆‘𝑋)) = (𝑄 ∙ (𝑆‘𝑋)))
36	1, 5, 9	lcdlmod 39606	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ LMod)
37	1, 2, 3, 5, 7, 8, 9, 14	hdmapcl 39844	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶))
38		hdmap14lem2.p	. . . . . . . . 9 ⊢ 𝑃 = (Scalar‘𝐶)
39		hdmap14lem2.e	. . . . . . . . 9 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
40		hdmap14lem2.q	. . . . . . . . 9 ⊢ 𝑄 = (0_g‘𝑃)
41	7, 38, 39, 40, 6	lmod0vs 20156	. . . . . . . 8 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑋) ∈ (Base‘𝐶)) → (𝑄 ∙ (𝑆‘𝑋)) = (0_g‘𝐶))
42	36, 37, 41	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∙ (𝑆‘𝑋)) = (0_g‘𝐶))
43	35, 42	sylan9eqr 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → (𝑔 ∙ (𝑆‘𝑋)) = (0_g‘𝐶))
44	33, 43	neeqtrrd 3018	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → (𝑆‘(𝐹 · 𝑋)) ≠ (𝑔 ∙ (𝑆‘𝑋)))
45	44	neneqd 2948	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → ¬ (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
46	45	nrexdv 3198	. . 3 ⊢ (𝜑 → ¬ ∃𝑔 ∈ {𝑄} (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
47		reuun2 4248	. . 3 ⊢ (¬ ∃𝑔 ∈ {𝑄} (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) → (∃!𝑔 ∈ ((𝐴 ∖ {𝑄}) ∪ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
48	46, 47	syl 17	. 2 ⊢ (𝜑 → (∃!𝑔 ∈ ((𝐴 ∖ {𝑄}) ∪ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
49		hdmap14lem2.a	. . . 4 ⊢ 𝐴 = (Base‘𝑃)
50	38, 49, 40	lmod0cl 20149	. . 3 ⊢ (𝐶 ∈ LMod → 𝑄 ∈ 𝐴)
51		difsnid 4743	. . 3 ⊢ (𝑄 ∈ 𝐴 → ((𝐴 ∖ {𝑄}) ∪ {𝑄}) = 𝐴)
52		reueq1 3344	. . 3 ⊢ (((𝐴 ∖ {𝑄}) ∪ {𝑄}) = 𝐴 → (∃!𝑔 ∈ ((𝐴 ∖ {𝑄}) ∪ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
53	36, 50, 51, 52	4syl 19	. 2 ⊢ (𝜑 → (∃!𝑔 ∈ ((𝐴 ∖ {𝑄}) ∪ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
54	48, 53	bitr3d 280	1 ⊢ (𝜑 → (∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 ∃!wreu 3066 ∖ cdif 3884 ∪ cun 3885 {csn 4561 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 Scalarcsca 16965 ·_𝑠 cvsca 16966 0_gc0g 17150 LModclmod 20123 LSpanclspn 20233 HLchlt 37364 LHypclh 37998 DVecHcdvh 39092 LCDualclcd 39600 HDMapchdma 39806

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-mre 17295 df-mrc 17296 df-acs 17298 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-oppg 18950 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-lshyp 36991 df-lcv 37033 df-lfl 37072 df-lkr 37100 df-ldual 37138 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tgrp 38757 df-tendo 38769 df-edring 38771 df-dveca 39017 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243 df-doch 39362 df-djh 39409 df-lcdual 39601 df-mapd 39639 df-hvmap 39771 df-hdmap1 39807 df-hdmap 39808

This theorem is referenced by: hdmap14lem4 39886

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