hdmapip0 - Mathbox for Norm Megill

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Theorem hdmapip0 41614

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 2726	. . . . . . . 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 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		hdmapip0.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
9	8	anim1i 613	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
10		eldifsn 4795	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
11	9, 10	sylibr 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
12	1, 2, 3, 4, 5, 7, 11	dochnel 41092	. . . . . . 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 2726	. . . . . . . . . . . 12 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
16		eqid 2726	. . . . . . . . . . . 12 ⊢ (LKer‘𝑈) = (LKer‘𝑈)
17	1, 3, 6	dvhlmod 40809	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LMod)
18		eqid 2726	. . . . . . . . . . . . 13 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
19		eqid 2726	. . . . . . . . . . . . 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 41529	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
22	1, 18, 19, 3, 15, 6, 21	lcdvbaselfl 41294	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (LFnl‘𝑈))
23	4, 13, 14, 15, 16, 17, 22, 8	ellkr2 38789	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((LKer‘𝑈)‘(𝑆‘𝑋)) ↔ ((𝑆‘𝑋)‘𝑋) = 𝑍))
24	23	biimpar 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → 𝑋 ∈ ((LKer‘𝑈)‘(𝑆‘𝑋)))
25	1, 2, 3, 4, 15, 16, 20, 6, 8	hdmaplkr 41612	. . . . . . . . . . 11 ⊢ (𝜑 → ((LKer‘𝑈)‘(𝑆‘𝑋)) = (((ocH‘𝐾)‘𝑊)‘{𝑋}))
26	25	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → ((LKer‘𝑈)‘(𝑆‘𝑋)) = (((ocH‘𝐾)‘𝑊)‘{𝑋}))
27	24, 26	eleqtrd 2828	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → 𝑋 ∈ (((ocH‘𝐾)‘𝑊)‘{𝑋}))
28	27	ex 411	. . . . . . . 8 ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 → 𝑋 ∈ (((ocH‘𝐾)‘𝑊)‘{𝑋})))
29	28	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (((𝑆‘𝑋)‘𝑋) = 𝑍 → 𝑋 ∈ (((ocH‘𝐾)‘𝑊)‘{𝑋})))
30	12, 29	mtod 197	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ¬ ((𝑆‘𝑋)‘𝑋) = 𝑍)
31	30	neqned 2937	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ((𝑆‘𝑋)‘𝑋) ≠ 𝑍)
32	31	ex 411	. . . 4 ⊢ (𝜑 → (𝑋 ≠ 0 → ((𝑆‘𝑋)‘𝑋) ≠ 𝑍))
33	32	necon4d 2954	. . 3 ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 → 𝑋 = 0 ))
34	33	imp 405	. 2 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → 𝑋 = 0 )
35		fveq2 6901	. . 3 ⊢ (𝑋 = 0 → ((𝑆‘𝑋)‘𝑋) = ((𝑆‘𝑋)‘ 0 ))
36	13, 14, 5, 15	lfl0 38763	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑆‘𝑋) ∈ (LFnl‘𝑈)) → ((𝑆‘𝑋)‘ 0 ) = 𝑍)
37	17, 22, 36	syl2anc 582	. . 3 ⊢ (𝜑 → ((𝑆‘𝑋)‘ 0 ) = 𝑍)
38	35, 37	sylan9eqr 2788	. 2 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ((𝑆‘𝑋)‘𝑋) = 𝑍)
39	34, 38	impbida 799	1 ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 ↔ 𝑋 = 0 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3944 {csn 4633 ‘cfv 6554 Basecbs 17213 Scalarcsca 17269 0_gc0g 17454 LModclmod 20836 LFnlclfn 38755 LKerclk 38783 HLchlt 39048 LHypclh 39683 DVecHcdvh 40777 ocHcoch 41046 LCDualclcd 41285 HDMapchdma 41491

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-riotaBAD 38651

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-ot 4642 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-undef 8288 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-0g 17456 df-mre 17599 df-mrc 17600 df-acs 17602 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-p1 18451 df-lat 18457 df-clat 18524 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-subg 19117 df-cntz 19311 df-oppg 19340 df-lsm 19634 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-nzr 20495 df-rlreg 20672 df-domn 20673 df-drng 20709 df-lmod 20838 df-lss 20909 df-lsp 20949 df-lvec 21081 df-lsatoms 38674 df-lshyp 38675 df-lcv 38717 df-lfl 38756 df-lkr 38784 df-ldual 38822 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-llines 39197 df-lplanes 39198 df-lvols 39199 df-lines 39200 df-psubsp 39202 df-pmap 39203 df-padd 39495 df-lhyp 39687 df-laut 39688 df-ldil 39803 df-ltrn 39804 df-trl 39858 df-tgrp 40442 df-tendo 40454 df-edring 40456 df-dveca 40702 df-disoa 40728 df-dvech 40778 df-dib 40838 df-dic 40872 df-dih 40928 df-doch 41047 df-djh 41094 df-lcdual 41286 df-mapd 41324 df-hvmap 41456 df-hdmap1 41492 df-hdmap 41493

This theorem is referenced by: hdmapip1 41615 hgmapvvlem3 41624 hlhilphllem 41662

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