hdmapip0 - Mathbox for Norm Megill

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Theorem hdmapip0 42285

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 2737	. . . . . . . 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 4744	. . . . . . . . 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 41763	. . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
16		eqid 2737	. . . . . . . . . . . 12 ⊢ (LKer‘𝑈) = (LKer‘𝑈)
17	1, 3, 6	dvhlmod 41480	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LMod)
18		eqid 2737	. . . . . . . . . . . . 13 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
19		eqid 2737	. . . . . . . . . . . . 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 42200	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
22	1, 18, 19, 3, 15, 6, 21	lcdvbaselfl 41965	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (LFnl‘𝑈))
23	4, 13, 14, 15, 16, 17, 22, 8	ellkr2 39461	. . . . . . . . . . 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 42283	. . . . . . . . . . 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 2839	. . . . . . . . 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 2940	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ((𝑆‘𝑋)‘𝑋) ≠ 𝑍)
32	31	ex 412	. . . 4 ⊢ (𝜑 → (𝑋 ≠ 0 → ((𝑆‘𝑋)‘𝑋) ≠ 𝑍))
33	32	necon4d 2957	. . 3 ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 → 𝑋 = 0 ))
34	33	imp 406	. 2 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → 𝑋 = 0 )
35		fveq2 6842	. . 3 ⊢ (𝑋 = 0 → ((𝑆‘𝑋)‘𝑋) = ((𝑆‘𝑋)‘ 0 ))
36	13, 14, 5, 15	lfl0 39435	. . . 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 2794	. 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 2933 ∖ cdif 3900 {csn 4582 ‘cfv 6500 Basecbs 17148 Scalarcsca 17192 0_gc0g 17371 LModclmod 20823 LFnlclfn 39427 LKerclk 39455 HLchlt 39720 LHypclh 40354 DVecHcdvh 41448 ocHcoch 41717 LCDualclcd 41956 HDMapchdma 42162

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 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 39323

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-ot 4591 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-undef 8225 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-0g 17373 df-mre 17517 df-mrc 17518 df-acs 17520 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cntz 19258 df-oppg 19287 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-nzr 20458 df-rlreg 20639 df-domn 20640 df-drng 20676 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lvec 21067 df-lsatoms 39346 df-lshyp 39347 df-lcv 39389 df-lfl 39428 df-lkr 39456 df-ldual 39494 df-oposet 39546 df-ol 39548 df-oml 39549 df-covers 39636 df-ats 39637 df-atl 39668 df-cvlat 39692 df-hlat 39721 df-llines 39868 df-lplanes 39869 df-lvols 39870 df-lines 39871 df-psubsp 39873 df-pmap 39874 df-padd 40166 df-lhyp 40358 df-laut 40359 df-ldil 40474 df-ltrn 40475 df-trl 40529 df-tgrp 41113 df-tendo 41125 df-edring 41127 df-dveca 41373 df-disoa 41399 df-dvech 41449 df-dib 41509 df-dic 41543 df-dih 41599 df-doch 41718 df-djh 41765 df-lcdual 41957 df-mapd 41995 df-hvmap 42127 df-hdmap1 42163 df-hdmap 42164

This theorem is referenced by: hdmapip1 42286 hgmapvvlem3 42295 hlhilphllem 42329

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