hdmap14lem4a - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem4a		Structured version Visualization version GIF version

Theorem hdmap14lem4a 42131

Description: Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 42130 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 2736	. . . . . . . . 9 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
7		eqid 2736	. . . . . . . . 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 41370	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
11		hdmap14lem1.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍}))
12	11	eldifad 3913	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
13		hdmap14lem3.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3913	. . . . . . . . . . 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 20829	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉)
19	10, 12, 14, 18	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉)
20		eldifsni 4746	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐵 ∖ {𝑍}) → 𝐹 ≠ 𝑍)
21	11, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ≠ 𝑍)
22		eldifsni 4746	. . . . . . . . . . . 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 41369	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
26	3, 16, 15, 17, 24, 4, 25, 12, 14	lvecvsn0 21064	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 · 𝑋) ≠ 0 ↔ (𝐹 ≠ 𝑍 ∧ 𝑋 ≠ 0 )))
27	21, 23, 26	mpbir2and 713	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 · 𝑋) ≠ 0 )
28		eldifsn 4742	. . . . . . . . . 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 42105	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) ∈ ((Base‘𝐶) ∖ {(0_g‘𝐶)}))
31		eldifsni 4746	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → (𝑆‘(𝐹 · 𝑋)) ≠ (0_g‘𝐶))
34		elsni 4597	. . . . . . . 8 ⊢ (𝑔 ∈ {𝑄} → 𝑔 = 𝑄)
35	34	oveq1d 7373	. . . . . . 7 ⊢ (𝑔 ∈ {𝑄} → (𝑔 ∙ (𝑆‘𝑋)) = (𝑄 ∙ (𝑆‘𝑋)))
36	1, 5, 9	lcdlmod 41852	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ LMod)
37	1, 2, 3, 5, 7, 8, 9, 14	hdmapcl 42090	. . . . . . . 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 20846	. . . . . . . 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 2793	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → (𝑔 ∙ (𝑆‘𝑋)) = (0_g‘𝐶))
44	33, 43	neeqtrrd 3006	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → (𝑆‘(𝐹 · 𝑋)) ≠ (𝑔 ∙ (𝑆‘𝑋)))
45	44	neneqd 2937	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → ¬ (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
46	45	nrexdv 3131	. . 3 ⊢ (𝜑 → ¬ ∃𝑔 ∈ {𝑄} (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
47		reuun2 4277	. . 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 20839	. . 3 ⊢ (𝐶 ∈ LMod → 𝑄 ∈ 𝐴)
51		difsnid 4766	. . 3 ⊢ (𝑄 ∈ 𝐴 → ((𝐴 ∖ {𝑄}) ∪ {𝑄}) = 𝐴)
52		reueq1 3382	. . 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 281	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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∃wrex 3060 ∃!wreu 3348 ∖ cdif 3898 ∪ cun 3899 {csn 4580 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 Scalarcsca 17180 ·_𝑠 cvsca 17181 0_gc0g 17359 LModclmod 20811 LSpanclspn 20922 HLchlt 39610 LHypclh 40244 DVecHcdvh 41338 LCDualclcd 41846 HDMapchdma 42052

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 39213

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 39236 df-lshyp 39237 df-lcv 39279 df-lfl 39318 df-lkr 39346 df-ldual 39384 df-oposet 39436 df-ol 39438 df-oml 39439 df-covers 39526 df-ats 39527 df-atl 39558 df-cvlat 39582 df-hlat 39611 df-llines 39758 df-lplanes 39759 df-lvols 39760 df-lines 39761 df-psubsp 39763 df-pmap 39764 df-padd 40056 df-lhyp 40248 df-laut 40249 df-ldil 40364 df-ltrn 40365 df-trl 40419 df-tgrp 41003 df-tendo 41015 df-edring 41017 df-dveca 41263 df-disoa 41289 df-dvech 41339 df-dib 41399 df-dic 41433 df-dih 41489 df-doch 41608 df-djh 41655 df-lcdual 41847 df-mapd 41885 df-hvmap 42017 df-hdmap1 42053 df-hdmap 42054

This theorem is referenced by: hdmap14lem4 42132

W3C validator