Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap14lem6 Structured version   Visualization version   GIF version

Theorem hdmap14lem6 37482
Description: Case where 𝐹 is zero. (Contributed by NM, 1-Jun-2015.)
Hypotheses
Ref Expression
hdmap14lem1.h 𝐻 = (LHyp‘𝐾)
hdmap14lem1.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap14lem1.v 𝑉 = (Base‘𝑈)
hdmap14lem1.t · = ( ·𝑠𝑈)
hdmap14lem3.o 0 = (0g𝑈)
hdmap14lem1.r 𝑅 = (Scalar‘𝑈)
hdmap14lem1.b 𝐵 = (Base‘𝑅)
hdmap14lem1.z 𝑍 = (0g𝑅)
hdmap14lem1.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap14lem2.e = ( ·𝑠𝐶)
hdmap14lem1.l 𝐿 = (LSpan‘𝐶)
hdmap14lem2.p 𝑃 = (Scalar‘𝐶)
hdmap14lem2.a 𝐴 = (Base‘𝑃)
hdmap14lem2.q 𝑄 = (0g𝑃)
hdmap14lem1.s 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap14lem1.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
hdmap14lem3.x (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap14lem6.f (𝜑𝐹 = 𝑍)
Assertion
Ref Expression
hdmap14lem6 (𝜑 → ∃!𝑔𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)))
Distinct variable groups:   𝐴,𝑔   𝐶,𝑔   ,𝑔   𝑄,𝑔   𝑆,𝑔   𝑔,𝑋   𝜑,𝑔
Allowed substitution hints:   𝐵(𝑔)   𝑃(𝑔)   𝑅(𝑔)   · (𝑔)   𝑈(𝑔)   𝐹(𝑔)   𝐻(𝑔)   𝐾(𝑔)   𝐿(𝑔)   𝑉(𝑔)   𝑊(𝑔)   0 (𝑔)   𝑍(𝑔)

Proof of Theorem hdmap14lem6
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 hdmap14lem1.h . . . . . 6 𝐻 = (LHyp‘𝐾)
2 hdmap14lem1.c . . . . . 6 𝐶 = ((LCDual‘𝐾)‘𝑊)
3 hdmap14lem1.k . . . . . 6 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
41, 2, 3lcdlmod 37198 . . . . 5 (𝜑𝐶 ∈ LMod)
5 hdmap14lem2.p . . . . . 6 𝑃 = (Scalar‘𝐶)
6 hdmap14lem2.a . . . . . 6 𝐴 = (Base‘𝑃)
7 hdmap14lem2.q . . . . . 6 𝑄 = (0g𝑃)
85, 6, 7lmod0cl 18937 . . . . 5 (𝐶 ∈ LMod → 𝑄𝐴)
94, 8syl 17 . . . 4 (𝜑𝑄𝐴)
10 hdmap14lem1.u . . . . . . 7 𝑈 = ((DVecH‘𝐾)‘𝑊)
11 hdmap14lem1.v . . . . . . 7 𝑉 = (Base‘𝑈)
12 eqid 2651 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
13 hdmap14lem1.s . . . . . . 7 𝑆 = ((HDMap‘𝐾)‘𝑊)
14 hdmap14lem3.x . . . . . . . 8 (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
1514eldifad 3619 . . . . . . 7 (𝜑𝑋𝑉)
161, 10, 11, 2, 12, 13, 3, 15hdmapcl 37439 . . . . . 6 (𝜑 → (𝑆𝑋) ∈ (Base‘𝐶))
17 hdmap14lem2.e . . . . . . 7 = ( ·𝑠𝐶)
18 eqid 2651 . . . . . . 7 (0g𝐶) = (0g𝐶)
1912, 5, 17, 7, 18lmod0vs 18944 . . . . . 6 ((𝐶 ∈ LMod ∧ (𝑆𝑋) ∈ (Base‘𝐶)) → (𝑄 (𝑆𝑋)) = (0g𝐶))
204, 16, 19syl2anc 694 . . . . 5 (𝜑 → (𝑄 (𝑆𝑋)) = (0g𝐶))
2120eqcomd 2657 . . . 4 (𝜑 → (0g𝐶) = (𝑄 (𝑆𝑋)))
22 oveq1 6697 . . . . . 6 (𝑔 = 𝑄 → (𝑔 (𝑆𝑋)) = (𝑄 (𝑆𝑋)))
2322eqeq2d 2661 . . . . 5 (𝑔 = 𝑄 → ((0g𝐶) = (𝑔 (𝑆𝑋)) ↔ (0g𝐶) = (𝑄 (𝑆𝑋))))
2423rspcev 3340 . . . 4 ((𝑄𝐴 ∧ (0g𝐶) = (𝑄 (𝑆𝑋))) → ∃𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋)))
259, 21, 24syl2anc 694 . . 3 (𝜑 → ∃𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋)))
26 hdmap14lem3.o . . . . . . . . . . 11 0 = (0g𝑈)
271, 10, 11, 26, 2, 18, 12, 13, 3, 14hdmapnzcl 37454 . . . . . . . . . 10 (𝜑 → (𝑆𝑋) ∈ ((Base‘𝐶) ∖ {(0g𝐶)}))
28 eldifsni 4353 . . . . . . . . . 10 ((𝑆𝑋) ∈ ((Base‘𝐶) ∖ {(0g𝐶)}) → (𝑆𝑋) ≠ (0g𝐶))
2927, 28syl 17 . . . . . . . . 9 (𝜑 → (𝑆𝑋) ≠ (0g𝐶))
3029neneqd 2828 . . . . . . . 8 (𝜑 → ¬ (𝑆𝑋) = (0g𝐶))
31303ad2ant1 1102 . . . . . . 7 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ¬ (𝑆𝑋) = (0g𝐶))
32 simp3l 1109 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (0g𝐶) = (𝑔 (𝑆𝑋)))
3332eqcomd 2657 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (𝑔 (𝑆𝑋)) = (0g𝐶))
341, 2, 3lcdlvec 37197 . . . . . . . . . . . 12 (𝜑𝐶 ∈ LVec)
35343ad2ant1 1102 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → 𝐶 ∈ LVec)
36 simp2l 1107 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → 𝑔𝐴)
37163ad2ant1 1102 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (𝑆𝑋) ∈ (Base‘𝐶))
3812, 17, 5, 6, 7, 18, 35, 36, 37lvecvs0or 19156 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ((𝑔 (𝑆𝑋)) = (0g𝐶) ↔ (𝑔 = 𝑄 ∨ (𝑆𝑋) = (0g𝐶))))
3933, 38mpbid 222 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (𝑔 = 𝑄 ∨ (𝑆𝑋) = (0g𝐶)))
4039orcomd 402 . . . . . . . 8 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ((𝑆𝑋) = (0g𝐶) ∨ 𝑔 = 𝑄))
4140ord 391 . . . . . . 7 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (¬ (𝑆𝑋) = (0g𝐶) → 𝑔 = 𝑄))
4231, 41mpd 15 . . . . . 6 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → 𝑔 = 𝑄)
43 simp3r 1110 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (0g𝐶) = ( (𝑆𝑋)))
4443eqcomd 2657 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ( (𝑆𝑋)) = (0g𝐶))
45 simp2r 1108 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → 𝐴)
4612, 17, 5, 6, 7, 18, 35, 45, 37lvecvs0or 19156 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (( (𝑆𝑋)) = (0g𝐶) ↔ ( = 𝑄 ∨ (𝑆𝑋) = (0g𝐶))))
4744, 46mpbid 222 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ( = 𝑄 ∨ (𝑆𝑋) = (0g𝐶)))
4847orcomd 402 . . . . . . . 8 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ((𝑆𝑋) = (0g𝐶) ∨ = 𝑄))
4948ord 391 . . . . . . 7 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (¬ (𝑆𝑋) = (0g𝐶) → = 𝑄))
5031, 49mpd 15 . . . . . 6 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → = 𝑄)
5142, 50eqtr4d 2688 . . . . 5 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → 𝑔 = )
52513exp 1283 . . . 4 (𝜑 → ((𝑔𝐴𝐴) → (((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋))) → 𝑔 = )))
5352ralrimivv 2999 . . 3 (𝜑 → ∀𝑔𝐴𝐴 (((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋))) → 𝑔 = ))
54 oveq1 6697 . . . . 5 (𝑔 = → (𝑔 (𝑆𝑋)) = ( (𝑆𝑋)))
5554eqeq2d 2661 . . . 4 (𝑔 = → ((0g𝐶) = (𝑔 (𝑆𝑋)) ↔ (0g𝐶) = ( (𝑆𝑋))))
5655reu4 3433 . . 3 (∃!𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋)) ↔ (∃𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋)) ∧ ∀𝑔𝐴𝐴 (((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋))) → 𝑔 = )))
5725, 53, 56sylanbrc 699 . 2 (𝜑 → ∃!𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋)))
58 hdmap14lem6.f . . . . . . . 8 (𝜑𝐹 = 𝑍)
5958oveq1d 6705 . . . . . . 7 (𝜑 → (𝐹 · 𝑋) = (𝑍 · 𝑋))
601, 10, 3dvhlmod 36716 . . . . . . . 8 (𝜑𝑈 ∈ LMod)
61 hdmap14lem1.r . . . . . . . . 9 𝑅 = (Scalar‘𝑈)
62 hdmap14lem1.t . . . . . . . . 9 · = ( ·𝑠𝑈)
63 hdmap14lem1.z . . . . . . . . 9 𝑍 = (0g𝑅)
6411, 61, 62, 63, 26lmod0vs 18944 . . . . . . . 8 ((𝑈 ∈ LMod ∧ 𝑋𝑉) → (𝑍 · 𝑋) = 0 )
6560, 15, 64syl2anc 694 . . . . . . 7 (𝜑 → (𝑍 · 𝑋) = 0 )
6659, 65eqtrd 2685 . . . . . 6 (𝜑 → (𝐹 · 𝑋) = 0 )
6766fveq2d 6233 . . . . 5 (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝑆0 ))
681, 10, 26, 2, 18, 13, 3hdmapval0 37442 . . . . 5 (𝜑 → (𝑆0 ) = (0g𝐶))
6967, 68eqtrd 2685 . . . 4 (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (0g𝐶))
7069eqeq1d 2653 . . 3 (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)) ↔ (0g𝐶) = (𝑔 (𝑆𝑋))))
7170reubidv 3156 . 2 (𝜑 → (∃!𝑔𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)) ↔ ∃!𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋))))
7257, 71mpbird 247 1 (𝜑 → ∃!𝑔𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  ∃!wreu 2943  cdif 3604  {csn 4210  cfv 5926  (class class class)co 6690  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  LModclmod 18911  LSpanclspn 19019  LVecclvec 19150  HLchlt 34955  LHypclh 35588  DVecHcdvh 36684  LCDualclcd 37192  HDMapchdma 37399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-riotaBAD 34557
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-undef 7444  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-0g 16149  df-mre 16293  df-mrc 16294  df-acs 16296  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-subg 17638  df-cntz 17796  df-oppg 17822  df-lsm 18097  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-lmod 18913  df-lss 18981  df-lsp 19020  df-lvec 19151  df-lsatoms 34581  df-lshyp 34582  df-lcv 34624  df-lfl 34663  df-lkr 34691  df-ldual 34729  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764  df-tgrp 36348  df-tendo 36360  df-edring 36362  df-dveca 36608  df-disoa 36635  df-dvech 36685  df-dib 36745  df-dic 36779  df-dih 36835  df-doch 36954  df-djh 37001  df-lcdual 37193  df-mapd 37231  df-hvmap 37363  df-hdmap1 37400  df-hdmap 37401
This theorem is referenced by:  hdmap14lem7  37483
  Copyright terms: Public domain W3C validator