hdmapip0 - Mathbox for Norm Megill

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Theorem hdmapip0 42361

Description: Zero property that will be used for inner product. (Contributed by NM, 9-Jun-2015.)

**Hypotheses**
Ref	Expression
hdmapip0.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapip0.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapip0.v	⊢ 𝑉 = (Base‘𝑈)
hdmapip0.o	⊢ 0 = (0_g‘𝑈)
hdmapip0.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmapip0.z	⊢ 𝑍 = (0_g‘𝑅)
hdmapip0.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapip0.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmapip0.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)

**Assertion**
Ref	Expression
hdmapip0	⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 ↔ 𝑋 = 0 ))

**Proof of Theorem hdmapip0**
Step	Hyp	Ref	Expression
1		hdmapip0.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2736	. . . . . . . 8 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
3		hdmapip0.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		hdmapip0.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈)
5		hdmapip0.o	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
6		hdmapip0.k	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		hdmapip0.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
9	8	anim1i 616	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
10		eldifsn 4731	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
11	9, 10	sylibr 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
12	1, 2, 3, 4, 5, 7, 11	dochnel 41839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ¬ 𝑋 ∈ (((ocH‘𝐾)‘𝑊)‘{𝑋}))
13		hdmapip0.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Scalar‘𝑈)
14		hdmapip0.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (0_g‘𝑅)
15		eqid 2736	. . . . . . . . . . . 12 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
16		eqid 2736	. . . . . . . . . . . 12 ⊢ (LKer‘𝑈) = (LKer‘𝑈)
17	1, 3, 6	dvhlmod 41556	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LMod)
18		eqid 2736	. . . . . . . . . . . . 13 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
19		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
20		hdmapip0.s	. . . . . . . . . . . . . 14 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
21	1, 3, 4, 18, 19, 20, 6, 8	hdmapcl 42276	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
22	1, 18, 19, 3, 15, 6, 21	lcdvbaselfl 42041	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (LFnl‘𝑈))
23	4, 13, 14, 15, 16, 17, 22, 8	ellkr2 39537	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((LKer‘𝑈)‘(𝑆‘𝑋)) ↔ ((𝑆‘𝑋)‘𝑋) = 𝑍))
24	23	biimpar 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → 𝑋 ∈ ((LKer‘𝑈)‘(𝑆‘𝑋)))
25	1, 2, 3, 4, 15, 16, 20, 6, 8	hdmaplkr 42359	. . . . . . . . . . 11 ⊢ (𝜑 → ((LKer‘𝑈)‘(𝑆‘𝑋)) = (((ocH‘𝐾)‘𝑊)‘{𝑋}))
26	25	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → ((LKer‘𝑈)‘(𝑆‘𝑋)) = (((ocH‘𝐾)‘𝑊)‘{𝑋}))
27	24, 26	eleqtrd 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → 𝑋 ∈ (((ocH‘𝐾)‘𝑊)‘{𝑋}))
28	27	ex 412	. . . . . . . 8 ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 → 𝑋 ∈ (((ocH‘𝐾)‘𝑊)‘{𝑋})))
29	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (((𝑆‘𝑋)‘𝑋) = 𝑍 → 𝑋 ∈ (((ocH‘𝐾)‘𝑊)‘{𝑋})))
30	12, 29	mtod 198	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ¬ ((𝑆‘𝑋)‘𝑋) = 𝑍)
31	30	neqned 2939	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ((𝑆‘𝑋)‘𝑋) ≠ 𝑍)
32	31	ex 412	. . . 4 ⊢ (𝜑 → (𝑋 ≠ 0 → ((𝑆‘𝑋)‘𝑋) ≠ 𝑍))
33	32	necon4d 2956	. . 3 ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 → 𝑋 = 0 ))
34	33	imp 406	. 2 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → 𝑋 = 0 )
35		fveq2 6840	. . 3 ⊢ (𝑋 = 0 → ((𝑆‘𝑋)‘𝑋) = ((𝑆‘𝑋)‘ 0 ))
36	13, 14, 5, 15	lfl0 39511	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑆‘𝑋) ∈ (LFnl‘𝑈)) → ((𝑆‘𝑋)‘ 0 ) = 𝑍)
37	17, 22, 36	syl2anc 585	. . 3 ⊢ (𝜑 → ((𝑆‘𝑋)‘ 0 ) = 𝑍)
38	35, 37	sylan9eqr 2793	. 2 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ((𝑆‘𝑋)‘𝑋) = 𝑍)
39	34, 38	impbida 801	1 ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 ↔ 𝑋 = 0 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∖ cdif 3886 {csn 4567 ‘cfv 6498 Basecbs 17179 Scalarcsca 17223 0_gc0g 17402 LModclmod 20855 LFnlclfn 39503 LKerclk 39531 HLchlt 39796 LHypclh 40430 DVecHcdvh 41524 ocHcoch 41793 LCDualclcd 42032 HDMapchdma 42238

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39399

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 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 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-undef 8223 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-mre 17548 df-mrc 17549 df-acs 17551 df-proset 18260 df-poset 18279 df-plt 18294 df-lub 18310 df-glb 18311 df-join 18312 df-meet 18313 df-p0 18389 df-p1 18390 df-lat 18398 df-clat 18465 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-cntz 19292 df-oppg 19321 df-lsm 19611 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-nzr 20490 df-rlreg 20671 df-domn 20672 df-drng 20708 df-lmod 20857 df-lss 20927 df-lsp 20967 df-lvec 21098 df-lsatoms 39422 df-lshyp 39423 df-lcv 39465 df-lfl 39504 df-lkr 39532 df-ldual 39570 df-oposet 39622 df-ol 39624 df-oml 39625 df-covers 39712 df-ats 39713 df-atl 39744 df-cvlat 39768 df-hlat 39797 df-llines 39944 df-lplanes 39945 df-lvols 39946 df-lines 39947 df-psubsp 39949 df-pmap 39950 df-padd 40242 df-lhyp 40434 df-laut 40435 df-ldil 40550 df-ltrn 40551 df-trl 40605 df-tgrp 41189 df-tendo 41201 df-edring 41203 df-dveca 41449 df-disoa 41475 df-dvech 41525 df-dib 41585 df-dic 41619 df-dih 41675 df-doch 41794 df-djh 41841 df-lcdual 42033 df-mapd 42071 df-hvmap 42203 df-hdmap1 42239 df-hdmap 42240

This theorem is referenced by: hdmapip1 42362 hgmapvvlem3 42371 hlhilphllem 42405

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